2011 Lightning Calculation Calendar

A 2011 Calendar

Lightning

Calculation

Fast mental techniques for arithmetic operations and elementary functions

Introduction

My philosophy on mental calculation:1. Numbers are not just collections of digits;

they have properties that can be exploited2. It helps greatly to work from left-to-right.3. Try to use convenient numbers near the

actual ones and correct the result at the end.4. Have confidence! Don’t limit yourself by saying it’s “too hard.”

Don’t worry about speed at first—just completing a calculationis often a real achievement, and speed comes naturally with practice.

The art of lightning calculation reveals the remarkable potential of the mind.

Arthur F. Griffith (1880-1911)

Individuals with preternatural abilities to calculate arithmetic results without pen, paper or other instruments, and to do so at astonishing speed, are the stuff of mathematical and psychological lore. These “lightning calculators” were sometimes of limited mental ability, sometimes illiterate but of average intelligence, and often exceptionally bright, this despite the popular notion of the idiot savant. The techniques used by these people are not generally well known.

The types of calculations performed by lightning calculators were historically quite limited, notable mainly for the size of the numbers and the speed at which they were manipulated. But remember that the questioner had to verify every calculation by hand, making higher powers and roots (particularly inexact roots) much less feasible. The dawn of calculators and computers propelled some of these tasks into hitherto uncharted territories such as 13th or 23rd roots, deep roots of inexact powers, and so forth, much of it supported by more sophisticated

mathematics. In this calendar we will explore the methods of calculation used in the past, many of them not commonly known, as well as other techniques that are relatively new.

The history of lightning calculators is interesting from a human standpoint, but it’s perhaps more intriguing because the methods they learned or developed are uniquely suited for fast mental calculation. These methods are different from the ones taught in school for pencil-and-paper solution, and therefore most people are quite surprised when they find out that other algorithms such as these exist.

Each month in this calendar is devoted to a different type of mental calculation. Exercises are provided within the dates displayed for each month, as described in the Legend for that month. The answers are contained within the boxes connected with the dates involved, but small enough that they are not visible from a distance. Every day starting in February also poses a day-date calculation, a very popularpastime of current and historical mental calculators that is discussed below.

The answers here are coded in a simple manner to avoid spoiling your fun! The calendar format naturally encourages daily exercise, but I encourage you to skip ahead to other days and months. After all, the topic of a particular month may not appeal to you, and in any event if you work ahead you will be able to breeze through the exercises when you get there.

It is important to realize that lightning calculators were highly individual in how they approached these tasks, and most calculators have such a vast knowledge of number facts that answers are often obtained immediately from memory or following only slight adjustment. As one example, Wim Klein learned through experience the multiplication table through 100x100 and used it to great advantage doing cross-multiplication in 2-digit by 2-digit groups. He also knew squares of integers up to 1000, cubes up to 100, and roughly all primes below 10,000. He also knew logarithms to 5 digits for integers up to 150. Sometimes calculators used a mnemonic scheme, often of their own design, to help remember these number facts. Use whatever knowledge you can muster!

Have a happy 2011!

Ron Doerfler ([email protected])

www.myreckonings.com/wordpress

Using the Calendar

18

12

18^249^279^2

Jan 181925

(09)

12^243^273^2

Nov 121799

(08)

Legend

The boxes contain products of the connected dates.

Blue End: Double the date Red End: Triple the date

Example for the box at the bottom:

1. Left date: Blue end 2 x 18 = 36

2. Right date: Blue end 2 x 12 = 24

3. 36 x 24 = 864 (answer in box)

Single-Date Rules

Add 31: 18 + 31 = 4949^2 = 2401

Add 61: 18 + 61 = 7979^2 = 6241

18^2 = 324

Day of the Week Code: Add to the date, sum the digits, check day-code table:

18 + 09 = 27, 2 + 7 = 9Jan 18, 1925, was a Sunday

12 + 08 = 202 + 0 = 2

Nov 12, 1799, was a

Tuesday

1 Mon

2 Tue

3 Wed

4 Thu

5 Fri

6 Sat

9 Sun

Two-Date Rules

There are 3958 mental calculation exercises in this calendar.

Thought you would want to know that.

This calendar is packed with challenging mental calculations! We will explore a different field of mental calculation each month .The upper half of each calendar page describes methods that can be used to solve problems of that month, and the lower half offers opportunities to test yourself and practice these techniques daily using the actual dates shown in the calendar. Of course, you should feel free to try all the dates at once, or bounce around between months, or do whatever you want. After all, this is all for fun!

In nearly all cases the answers are provided right in the calendar itself. That’s where the legend comes in. Every month includes a legend that describes how the dates are used in the calculations. The calculations in some months use 3- or 4-digit numbers , and in that case the legend describes how to form those numbers from the 2-digit dates.

12

18

32424016241

May 182022

(12)

14418495329

Feb 121994

(12)

So take a minute to look over the legend for a typical month shown above along with the corresponding date blocks on the left. The colored end of each box is a key to the operation to perform first on the date at that end of the box. Here we are told to double the date if a tip is blue and triple it if it is red, then multiply it by the result of the operation on the date at the other end of the box. Sometimes the colored ends add values to the date rather than double or triple them, so please look at the legend for before starting a month. Here we also square the date, (date + 31), and (date + 61), and those answers are shown right above the date. It’s all much easier than you might think at first.

The answers for calculations that involve pairs of dates appear in boxes that connect the two dates, or connect a date on the edge of the calendar to a date printed next to it. These answers are printed in very small font to avoid revealing them too readily. Since it would be too easy to accidentally detect the answer for the day of the week in the calendar calculations (on months after January ), a simple code based on the date is used. This is consistent throughout the year but is described in the legend for every month.

Doomsday Method: Find a nearby “doomsday” and adjust it.

Days are numbered 0-6 for Sunday-Saturday.

Each century has a “Doomsday” for its first year:

1700,2100,2500, … Sunday (0)1800,2200,2600, … Friday (5)1900,2300,2700, … Wednesday (3)2000,2400,2800, … Tuesday (2)

For y = last two digits of the year you are calculating,

where [ ] means to round down to the integer and R is the remainder of [y/12]. Subtract multiples of 7 to get a number less than 7 to find the Doomsday for your year.

Then remember the table of dates that fall on the Doomsday andcalculate the difference for the date you are calculating, or use this:

For M = the number of the month, Doomsday is

- The last day of February and January (where Jan 32 = Feb 1)- M for an even month- M+4 for an odd month with 31 days- M-4 for an odd month with 30 days- “Work from 9 to 5 at the 7-11” for Sep 5, May 9, July 11 and Nov 7

Doomsday is Monday for 2011.

May 17, 1889: Anchor = 5[89/12] = 7 with remainder R= 5 [5/4] = 1 and forget the remainder7+5+1+5 = 18, subtract 14 to get 4, so the Doomsday for 1889 is Thursday

So May 9 is a Thursday, and May 17 is 8 days later, or a Friday.

Tip: Every 12 years the Doomsday repeats, so we can simplify: y 89 – 84 = 5

Day-Date Calendar CalculationsFinding the Day of the Week is a very common task.

Pericles Diamandi (1868-?) visualized a table with a rotating grille for day-date calculations.

Zeller’s Congruence: A straightforward calculation.

d = day of week numbered 0-6 for Saturday-Fridayn = day of the month

m = month (3-14, where Jan/Feb are 13/14 of the previous year)y = last two digits of the year (or previous year if m = 13 or 14)

C = first two digits of the year

where [ ] means to round down to the integer and mod 7 indicates the remainder when divided by 7. To simplify the addition we can find this remainder for each term as we go.

May 17, 1889:

d = 3 (since 17 mod 7 = 3)+ 1 (since 6x26/10 mod 7 = 1)+ 5 (since 89 mod 7 = 5)+ 1 (since 89/4 mod 7 = 1)+ 4 (since 18/4 mod 7 = 2)– 1 (since 2x18 mod 7 = 1)

= 13 mod 7 = 6, or a Friday

You probably know the day of the week for many personal, family and historical events. Calculate from one of these if it is convenient!

Some Historical Date Exercises in this CalendarJul 4, 1776 United States IndependenceFeb 4, 1789 George Washington electedJul 14, 1789 Storming of the BastilleDec 2, 1804 Napoleon declares himself emperorSep 16, 1810 Mexico Cry of IndependenceNov 19, 1863 Gettysburg AddressApr 14, 1865 Abraham Lincoln assassinatedJul 1, 1867 Canada IndependenceJan 1, 1901 Australia IndependenceDec 17, 1903 Wright brother's first powered flightApr 15, 1912 Titanic sinksNov 7, 1917 Bolshevik RevolutionNov 4, 1922 Howard Carter discovers King Tut's tombMay 21, 1927 Charles Lindbergh lands in ParisMar 12, 1930 Gandhi begins Dandi Salt MarchMay 6, 1937 Hindenburg disasterDec 7, 1941 Pearl HarborJun 6, 1944 D-DayMay 8, 1945 Victory in EuropeAug 6, 1945 HiroshimaAug 15, 1947 India IndependenceMay 14, 1948 Israel IndependenceMay 29, 1953 Edmund Hillary / Tenzing Norgay reach Mt. Everest summitFeb 6, 1952 Elizabeth II becomes queenOct 4, 1957 Sputnik launchedAug 28, 1963 March on Washington / ML King "I Have a Dream" speechNov 22, 1963 John F. Kennedy assassinatedApr 4, 1968 Martin Luther King assassinatedJul 21, 1969 First man on the moonApr 26, 1986 Chernobyl disasterJun 4, 1989 Tiananmen Square massacreNov 9, 1989 Fall of Berlin WallOct 3, 1990 Reunification of GermanyFeb 1, 2003 Challenger space shuttle disasterDec 21, 2012 Mayan calendar endsApr 13, 2029 Asteroid Apophos near-miss of EarthJan 19, 2038 Unix time ends

Every day after January contains a date between the adoption of the Gregorian calendar in Britain

and its colonies (1752) and the year 2099. What day of the week does that date fall on?

The answer is found from a simple code described in the Legend for each month.

Shown here are two common methods used by lightning calculators for day-date calculations.

Jan 31/32 Jul 11

Feb 28/29 Aug 8

Mar 7 Sep 5

Apr 4 Oct 10

May 9 Nov 7

Jun 6 Dec 12

The Doomsday method was

invented by John Conway in 1982

Mechanical

2-Digit MultiplicationMultiplication is the most common and useful type of lightning calculation. There are a variety of methods you can choose among to instantly find the product of any pair of two-digit numbers.

The two most common techniques used by lightning calculators

Legend

The boxes contain products of the

connected dates.

Blue End: Double the date

Red End: Triple the date

Example for the rightmost box here:

1. Bottom date: Blue end 2 x 19 = 38

2. Top date: Blue end 2 x 12 = 24

3. 38 x 24 = 912 (answer in box)

12

19

38 x

12

57 x

36

57 x

12

19 x

12

38 x

36

38 x

24

Partial Products are the combinations of the individual digit multiplications. They are added from left to right to find the product:

46 x 58 = 40x50 + 40x8 + 6x50 + 6x8= 2000 + 320 + 300 + 48= 2668

The terms are added as they are calculated, so when 40x8 is calculated, it is added to 2000 to get 2320, then 6x50 is added to get 2620, and finally 6x8 is added to yield 2668. Fast, with only one running total to remember!

Cross Multiplication adds single-digit products that contribute to each digit of the result, including carries:

46 x 58:1: 6x8 = 48 , or 8 with a carry of 42: 4x8 + 6x5 + 4 = 66 , or 6 with a carry of 63: 4x5 + 6 = 26 Answer: 2668

Easily scaled to larger numbers, but the answer is foundfrom right to left so it must be reversed to recite it.

5 8

4 6

26 6 8

12

3

Mechanical Methods

12 x 13 = 10x15 + 2x3 = 156

18 x 16 = 20x14 + 2x4 = 288

18 x 24 = 20x22 − 2x4 = 432

Good Neighbor Method: Is one of the numbers neara very round number? Multiply by the round number insteadand adjust for the difference at the end:

29x34 = 30x34 − 34 = 98642x56 = 40x56 + 2x56 = 2352

To find 30x34 here, we would multiply from left to right: 30x30 + 30x4 = 1020. Then subtract 40 and add 6 to subtract 34. In the second example, 40x56 = 40x50 + 40x6 = 2240, and add 2x56 = 112.

Is a number a multiple of 9 or 11? Use nearby multiple of 10 and subtract or add 1/10:

44x52: Find 40x52 = 2080, then 2080 + 208 = 228836x52: Find 40x52 = 2080, then 2080 − 208 = 1872 (subtract 200, then 8 more!)

TIPS

… more multiplication methods in February’s Calendar---take a look!

Becoming Neighbors: Bringing two multipliers nearer can sometimes allow use of other methods. For example:

1. Subtract one number from a very round number (or add it to a veryround number) to bring it closer to the other number:

23x67 = 23(100−33) = 2300 − 23x33 = 2300 − (20x36 + 3x13)

2. Divide or multiply one number by a low integer and add a correction:23x 67 = 23x33x2 + 23 = 2(20x36 + 3x13) + 23

3. Break one number into two convenient parts: 23x67 = 23(50+17) = 2300/2 + 23x17 = 1150 + 202 − 32

Anchor Method: Anchor one multiplier at a nearby round number a, and then string out the differences (plus or minus) of the original numbers from this anchor to find the other multiplier. Then add the product of the differences. Algebraically this is represented as

This is much easier to use than it might appear! Visualize “anchoring” one multiplier at the round number, and then literally stringing out the differences (plus or minus) of the original numbers from this anchor to find the other multiplier. If the differences have the same sign, the correction at the end will be added, and if one is positive and the other is negative the correction will be subtracted.

52x64 = 50x66 + 2x14 = 332897x92 = 100x89 + 3x8 = 892437x45 = 40x42 - 3x5 = 1665

In one performance, Maurice Dagbert (1913-1995) extracted a fifth root (answer: 243) in 14 seconds; a seventh root (answer: 125) in 15 seconds; a cube root (answer: 78,517) in 2 minutes 15 seconds; a fifth root (answer: 2189) in 2 minutes 3 seconds; and raised 827 to its cube in 55 seconds.

JA

NU

AR

YSunday Monday Tuesday Wednesday Thursday Friday Saturday

February 2011

S M T W T F S1 2 3 4 5

6 7 8 9 10 11 1213 14 15 16 17 18 1920 21 22 23 24 24 2627 28

December 2010

S M T W T F S1 2 3 4

5 6 7 8 9 10 1112 13 14 15 16 17 1819 20 21 22 23 24 2526 27 28 29 30 31

05 06

12 13

19 20

26 27

07 08

14 15

21 22

28 29

02 03

09 10

16 17

23 24

04

11

18

25

01

61218543624

1224361087248

90180270810540360

110220330990660440

272544816244816321088

306612918275418361224

20406018012080

1322643961188792528

3426841026307820521368

56112168504336224

4284126378252168

21042063018901260840

18236454616381092728

4629241386415827721848

4208401260378025201680

306090270180120

1563124681404936624

3807601140342022801520

55211041656496833122208

60012001800540036002400

65013001950585039002600

81216242436730848723248

70214042106631842122808

1632481449664

1352704051215810540

3527041056316821121408

66713342001600340022668

8 16 24 72 48 32

30 60 90 270

180

120

44 88 132

396

264

176

78 156

234

702

468

312

60 120

180

540

360

240

18 36 54 162

108

72 98 196

294

882

588

392

120

240

360

1080

720

480

170

340

510

1530

1020

680

198

396

594

1782

1188

792

260

520

780

2340

1560

1040

228

456

684

2052

1368

912

144

288

432

1296

864

576

294

588

882

2646

1764

1176

330

660

990

2970

1980

1320

408

816

1224

3672

2448

1632

450

900

1350

4050

2700

1800

540

1080

1620

4860

3240

2160

494

988

1482

4446

2964

1976

368

736

1104

3312

2208

1472

588

1176

1764

5292

3528

2352

638

1276

1914

5742

3828

2552

800

1600

2400

7200

4800

3200

918

1836

2754

8262

5508

3672

858

1716

2574

7722

5148

3432

980

1960

2940

8820

5880

3920

1044

2088

3132

9396

6264

4176

30 3193018602790837055803720

1178

2356

3534

1060

270

6847

12

1110

2220

3330

9990

6660

4440

02

09

16

23

3635343332

744

1488

2232

6696

4464

2976

690

1380

2070

6210

4140

2760

37 38

7561512

680445363024

2268

An anchor of 100 is very common, say, 842 = 100x68 + 162. With an anchor of 100, we can find the value 68 simply by doubling 84 and using the last two digits of the resulting 168 rather than by finding the difference between 100 and 84 and subtracting this again from 84.

Reverse Midpoint Method: To calculate a square, we can split it into the product of two numbers equidistant from the original number, and add the square of that distance (one scenario of the Anchor Method). For example, we can find

652 = 60x70 + 52 = 4225132 = 10x16 + 32 = 169

It is helpful to remember that the average squared will always be larger than the spread numbers multiplied, so when spreading a square to the product of two numbers you addthe correction, and when collapsing two multipliers to a square you subtract the correction.

Cool Facts: For squares of two-digit numbers ending in 7, 8 or 9:

(10a+7)2 = 100a(a+1) + 40(a+1) + 9(10a+8)2 = 100a(a+1) + 60(a+1) + 4(10a+9)2 = 100a(a+1) + 80(a+1) + 1

where the red digits make up the squares of the units digits.

792 = 5600 + 640 + 1 = 6241872 = 7200 + 360 + 9 = 7569

Squares and 2-Digit Multiplication

18

12

18^249^279^2

Jan 181925

(09)

12^243^273^2

Nov 121799

(08)

Legend

The boxes contain products of the connected dates.





3. 36 x 24 = 864 (answer in box)

Single-Date Rules

Add 31: 18 + 31 = 4949^2 = 2401

Add 61: 18 + 61 = 7979^2 = 6241

18^2 = 324


18 + 09 = 27, 2 + 7 = 9Jan 18, 1925, was a Sunday

12 + 08 = 202 + 0 = 2

Nov 12, 1799, was a Tuesday

1 Mon

2 Tue

3 Wed

4 Thu

5 Fri

6 Sat

9 Sun

Two-Date Rules

Squares and general multiplication share a strange and productive relationship. Squares can be used to greatly ease general multiplication, while general multiplication can be used to greatly ease squaring.

Recognize the trick for squaring numbers ending in 5? Multiply the number left of the units digit by that number plus one, and then append 25, as in 652 = 6x7 | 25 = 4225.

Jedidiah Buxton (1707–1772) was an illiterate lightning calculator who once calculated the product of a farthing doubled 139 times., which expressed in pounds has thirty-nine digits.

Special Neighbors: To find the square of a number near 50, add the difference from 50 to 25, multiply by 100, and add the difference squared. If the number is within 10 of 50, we can add the difference to 25 and simply append the distance squared rather than adding it. In this notation,

(25+b)2 = (6 + b/2) | (25 + b2)(50+b)2 = (25 + b) | b2

(75+b)2 = (56 + b + b/2) | (25 + b2)

272 = (6 + 1) | (25 + 22) = 729522 = (25 + 2) | 22 = 2704382 = (25−12) | 122 = 13 | 144 = 1444782 = (56 + 3 + 1.5) | (25 + 32) = 60.5 | 34 = 6084

The vertical bar “|” separates two-digit groups. If we end up with a 3-digit result in a grouping, its most significant digit would be added to the group to its left. A 0.5 in a group adds 50 to the group to its right.

Neighbors of Squares: Since (a+1)2 = a2 + a + (a+1), we can find 312 = 302 + 30 + 31 = 961. Similarly, 292 = 302 − 30 − 29 = 841. For other neighboring numbers we can find the square of the convenient number, then add or subtract the original number, the final number, and twice each number in between, so 322 = 302 + 30 + 2x31 + 32 = 1024. A shortcut for squaring a number ending in 1 is a12 = a2 || (2xa) || 1 where “||” means to limit the middle value to one digit by merging any upper digit to the left, as in 312 = 32 || 6 || 1 = 961 or 612 = 62 || 12 || 1 = (36+1) || 2 || 1 = 3721.

No Midpoint Method: We might have the case where there is no midpoint of the two multipliers—here we can adjust one of the multipliers by one, do the calculation, and then provide a correction to account for the original adjustment, as for

28x33 = 28x32 + 28 = 302 − 22 + 28

but in this particular case it may be easier to use the Anchor Method (see January) to get

28x33 = 30x31 − 2x3

Binomial Expansion for Squares: We can express the number to be squared as the sum of two other numbers that are more easily squared:

342 = (30+4)2 = 302 + 2x30x4 + 42 = 1156692 = (70−1)2 = 702 − 2x70x1 + 12 = 4761

Very useful for larger numbers

Midpoint Method: One of the most powerful tools in mental calculation is converting the multiplication of two different numbers into the square of the average minus the square of the distance to the average. This is the Midpoint Method, an algebraic identity:

where a is the average of the two numbers, (a+c) is one of the numbers, and (a−c) is the other number. This is equivalent to the Anchor Method when the anchor is midway between the two multipliers. For example,

28x32 = 302 − 22 52x78 = 652 − 132 46 x 58 = 522 − 62

Very useful!

These follow from the binomial expansions,such as(50+b)2 = 2500 + 100b + b2

FE

BR

UA

RY

Sunday Monday Tuesday Wednesday Thursday Friday Saturday

09 10

16 17

23 24

11 12

18 19

25 26

06 07

13 14

20 21

27 28

08

15

22

02 03 04 0501

March 2011

S M T W T F S1 2 3 4 5

6 7 8 9 10 11 1213 14 15 16 17 18 1920 21 22 23 24 24 2627 28 29 30 31

January 2011

S M T W T F S1

2 3 4 5 6 7 89 10 11 12 13 14 15

16 17 18 19 20 21 2223 24 25 26 27 28 2930 31

30 30

3635

31 32 33

13

06

0128

29

20

6415214761

Mar 81825

(12)

110243844

June 11923

(22)

410893969

Dec 21804

(16)

911564096

Nov 32012

(21)

8116004900

Sept 92026

(21)

10016815041

Apr 101986

(21)

22521165776

Oct 151800

(06)

25622095929

Aug 161768

(04)

28923046084

Jan 171967

(03)

4914444624

July 71903

(13)

19620255625

May 141948

(09)

44127046724

Jul 212041

(15)

3613694489

Jan 61834

(04)

16919365476

Feb 132061

(14)

40026016561

Dec 201897

(80)

48428096889

Mar 221845

(11)

72933647744

Apr 272003

(09)

78434817921

June 281778

(08)

57630257225

Feb 241814

(16)

62531367396

Jan 251974

(07)

67632497569

Nov 261854

(10)

32424016241

May 182022

(12)

36125006400

Aug 191882

(05)

1612254225

Feb 41789

(08)

2512964356

Aug 51931

(25)

12117645184

Dec 111789

(30)

14418495329

Feb 121994

(12)

52929167056

Nov 231866

(09)

Reciprocals and Division

17

11

1/171/481/78

Jan 171918

(05)

1/111/421/72

Nov 111798

(16)

Legend

The boxes contain quotients of the connected dates.

Blue End: Add 31 to date Red End: Add 61 to date


1. Left date: Blue end 17 + 31 = 48

2. Right date: Blue end 11 + 31 = 42

3. 48 / 42 = 1.14285714 (answer in box)

Single-Date Rules

Add 31: 17 + 31 = 481/ 48 = 0.02083333

Add 61: 17 + 61 = 781/78 = 0.012820513

1/17 = 0.05882353


17 + 05 = 22, 2 + 2 = 4Jan 17, 1918, was a Thursday

11 + 16 = 272 + 7 = 9

Nov 11, 1798, was a Sunday

1 Mon

2 Tue

3 Wed

4 Thu

5 Fri

6 Sat

9 Sun

Two-Date Rules

We can usually manipulate divisions to 1-digit or 2-digit divisors depending on the accuracy needed. Division is an important skill.

Salo Finkelstein (1896/7-?) was notable for his mental addition and number memorization abilities, and was one of the many lightning calculators who used cross multiplication to calculate products of large numbers.

Simplify the Denominator: Try to reduce the denominator to an integer of one or two digits and then do short division.

1. Shift decimal point:4.657 / 0.07 = 465.7 / 7 = 66.53

2. Divide the numerator and denominator by low common factors:0.2420 / 7.2 = 2.420 / 72 = 0.605 / 18 = 0.0336

3. Divide by low factors of the denominator:0.605/18 = (0.605/2) / 9 = 0.3025 / 9 = 0.033635 / 36 = (35/6) / 6 = 5.833 / 6 = 0.9722

Friendly Neighbor—Exact:

Divide by nearby round number and adjust the remainder in each step ofShort division. If rounding up, add (adjustment x quotient digit), otherwise subtract.

124172 / 78:Adjusting 78 up to 80 will add 2 x quotient digit to each remainder :

124 / 80 = 1 Remainder = 44 + 2x1 = 46461 / 80 = 5 Remainder = 61 + 2x5 = 71717 / 80 = 8 Remainder = 77 + 2x8 = 93

But 93>80, so we change the quotient 8 to 9:717 / 80 = 9 Remainder = -3 + 2x8 = 15152 / 80 = 1 Remainder = 72 + 2x1 = 74740 / 80 = 9 Remainder = 20 + 2x9 = 38, etc.

Placing the decimal point, we have 1591.9…

274 / 603:Adjusting 603 down to 600 will subtract 3 x quotient digit from each remainder :

2740 / 600 = 4 Remainder = 340 – 3x4 = 3283280 / 600 = 5 Remainder = 280 – 3x5 = 2652650 / 600 = 4 Remainder = 250 – 3x4 = 2382380 / 600 = 3 Remainder = 580 – 3x3 = 5715710 / 600 = 9 Remainder = 310 – 3x9 = 283, etc.


4267 / 3492:Multiply top and bottom by 2 8534 / 6984 will provide single-digit divisor.Adjusting 6984 up to 7000 will add 16 x quotient digit to each remainder :

8534 / 7000 = 1 Remainder = 1534 + 16x1 = 155015500 / 7000 = 2 Remainder = 1500 + 16x2 = 153215320 / 7000 = 2 Remainder = 1320 + 16x2 = 135213520 / 7000 = 1 Remainder = 6520 + 16x1 = 653665360 / 7000 = 9 Remainder = 2360 + 16x9 = 2505, etc.


Adjust the Denominator: Adjust the denominator to a round number and then adjust the numerator by the same percentage, or roughly so for an approximation:

247 / 119:Adjusting 119 up to 120 is a change of about 1/120, so we adjust 247 by 2:

249 / 120 = 24.9 / 12 = 2.0750 Actual Value: 2.0756For more accuracy, notice that 247 is twice 119 plus ~10%, so adjust 247 by 2.1:

24.91 / 12 = 2.0758

9152 / 3533:Adjusting 3533 down to 3500 is a change of just less than 1%, so we adjust9152 down by 91 for a first estimate:

9061 / 3500 = 9.062 / 3.5 = 18.124 / 7 = 2.5889 Since 1% of 3500 = 35 rather than 33, an error of 2/3500 ≈ 6/10000, we shouldadjust 9061 down by 6 more to get a more accurate answer:

9065 / 3500 = 9.065 / 3.5 = 18.130 / 7 = 2.5900 Actual Value: 2.5904

Friendly Neighbor—Approximation: For small b, 1/(1+b) ≈ 1 – b , so

and

274 / 603 ≈ (274 / 600) (1 – 3/600) = 0.4567 (1 – 1/200)= 0.4567 – 0.0023 = 0.4544 Actual Value: 0.45395

6243 / 5412 ≈ (6243 / 5400) (1 – 12/5400) ≈ 1.1561 (1 – 0.002)= 1.1561 – 0.0023 = 1.1538 Actual Value: 1.1535

MA

RC

H

April 2011

S M T W T F S1 2

3 4 5 6 7 8 910 11 12 13 14 15 1617 18 19 20 21 22 2324 25 26 27 28 29 30

February 2011

S M T W T F S1 2 3 4 5

6 7 8 9 10 11 1213 14 15 16 17 18 1920 21 22 23 24 25 2627 28

0.1250.0256410260.014492754

Jan 81937

(06)

1.00.03125

0.016129032

Oct 11812

(21)

0.50.03030303

0.015873016

May 22053

(30)

0.3333333330.029411765

0.015625

Dec 31752

(15)

0.111111110.025

0.014285714

June 92033

(13)

0.10.02439024

0.014084507

Oct 101784

(17)

0.0666666670.02173913

0.013157895

July 151893

(09)

0.06250.02127660

0.012987013

June 161781

(17)

0.058823530.02083333

0.012820513

Mar 171989

(05)

0.1428571430.0263157890.014705882

Dec 71871

(15)

0.0714285710.0222222220.013333333

Feb 141945

(16)

0.0476190480.0192307690.012195122

Sep 212029

(11)

0.1666666670.0270270270.014925373

Sep 62007

(07)

0.0769230770.0227272730.013513514

Nov 131782

(08)

0.050.0196078430.012345679

July 201926

(00)

0.0454545450.0188679250.012048193

May 221772

(28)

0.0370370370.0172413790.011363636

Feb 271774

(09)

0.0357142860.0169491530.011235955

May 281995

(08)

0.041666670.01818182

0.011764706

Apr 242014

(07)

0.040.017857140.011627907

Aug 251984

(08)

0.038461540.017543860.011494253

Dec 261877

(04)

0.055555560.020408160.012658228

Aug 181808

(18)

0.052631580.02

0.0125

Jan 192038

(01)

0.250.0285714290.015384615

Mar 41907

(11)

0.20.0277777780.015151515

Mar 51998

(08)

0.090909090.023809520.013888889

Mar 111890

(09)

0.083333330.023255810.01369863

Mar 121930

(09)

0.043478260.01851852

0.011904762

Feb 231888

(08)

0.0344827590.0166666670.011111111

Oct 291780

(16)

0.033333330.01639344

0.010989011

Aug 301829

(06)

0.032258060.01612903

0.010869565

July 311977

(14)

09 10

16 17

23 24

30 31

11 12

18 19

25 26

06 07

13 14

20 21

27 28

08

15

22

29

02 03 04 0501

34 35 36 02 37

32 33

20

13

06

1628

38


Square Roots

15

09

151/2

461/2

761/2

Jan 151925

(16)

91/2

401/2

701/2

Nov 91799

(06)

Legend

The boxes contain square roots of the connected dates separated by a decimal point.





3. (30.18)1/2 = 5.49363268 (answer in box)

Single-Date Rules

Add 31: 15 + 31 = 46461/2 = 6.78232998

Add 61: 15 + 61 = 76761/2 = 8.717797887

151/2 = 3.87298335



09 + 06 = 151 + 5 = 6

Nov 9, 1799, was a Saturday

1 Mon

2 Tue

3 Wed

4 Thu

5 Fri

6 Sat

9 Sun

Two-Date Rules

One-Digit Endings

PowerSquare Root

0 01 1 9 2

3

4 2 8 5 5

6 4 6

7

8

9 3 7

Square roots are often encountered in technical work, and the ability to quickly calculate them is easily acquired.

John Wallis (1616-1703) wrote that one night in 1669 he mentally extracted the square root of 3x10^45 to 21 digits and two months later did the same for a 53-digit number to 27 digits.

Is N a Perfect Square? 1. If the last two digits of N are one less than a

multiple of 4, N cannot be a square.

2. Perfect squares end in 00, e1, e4, 25, d6 or e9 ford = odd digit and e = even digit.

3. A square ending in 25 must end in 125, 225 or 625.

4. A square ending in e1 or e9 with e divisible by 4must have an odd thousands digit:

661 is not a square, 361 may be a square789 is not a square, 589 may be a square

Averaging: A good method to approximate a non-integer square root.

For an estimate r, a better estimate is the average of r and N/r:

√51: Estimate 7Better estimate is (7 + 51/7)/2 = 100/14 = 50/7 = 7.1429

For an estimate of a fraction s/t, this average can be re-written as:

√51: Estimate s/t = 50/7Better estimate is (2500 + 51x49)/ 2x50x7 = 4999/700 = 7.14143

Quartering the Difference: A more accurate approximation but more difficult.

For an estimate r, a better estimate is the weighted average of r and d = N – r2:

√51: Estimate 7Better estimate is 7 x (49 + ¾ x 2) / (49 + ¼ x 2) = 7 x 50.5/49.5

= 7 x 101/99 = 707/99 = 7.14141which is better than averaging once but poorer than averaging twice

Digit-by-Digit Extraction:Step 1: Move the decimal point in N left two places at a time until only one ortwo digits are left of the decimal point. Find the closest digit A whose square willbe less than this new N. Find the remainder R0 = N-A2 using only the integer portion of N.

Square root of 5163: N 51.63, A = 7, R0 = 51 – 72 = 2 Estimate: 7 R0=2

Step 2: Find 10 x R0 / 2 and add ½ the next digit of N if it exists. Divide by A to get the next digit B and remainder R1. The current square root is now A.B

B = (10x2/2 + 6/2) / 7 = 1 remainder 6 Estimate: 7.1 R1=6

Step 3: Find 10 x R1 and add ½ the next digit of N if it exists. Subtract B2/2. If the result is negative, reduce B by 1 and add A to R1 and try again. Otherwise, divide by A to get the next positive digit C and remainder R2. The current square root is now A.BC

C = (10x6 + 3/2 – 12 /2) / 7 = 8 remainder 5 Estimate: 7.18 R2=5

Later Steps: Repeat for more digits. In each step, subtract multiplications of digits pairing them inward to the middle, then if there is a digit left over we subtract half its square. See the pattern:

A.B subtract B2/2 A.BC subtract BxCA.BCD subtract (BxD + C2/2) A.BCDE subtract (BxE + CxD)A.BCDEF subtract (BxF + CxE + D2/2)

If a negative value occurs, back up and reduce the previous digit by 1, add A to the previous R, and start again from there.

D = (10x5 + 0/2 – 1x8) / 7 = 6 remainder 0 Estimate: 7.186 R4=0

E = (10x0 + 0/2 – 1x6 – 82/2) / 7 is negative, so previous estimate 7.185 with R4 = 0+7 = 7then E = (10x7 + 0/2 – 1x5 – 82/2) / 7 = 4 remainder 5 Estimate: 7.1854 R5=5

F = (10x5 + 0/2 – 1x4 – 8x5) / 7 = 0 remainder 6 Estimate: 7.18540

Final Step: Move the decimal point of the answer to the right once for each time it was moved left in Step 1.

7.18540 71.8540

AP

RIL

06 07

13 14

20 21

27 28

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29 30

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17 18

24 25

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12

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26

01 02

May 2011

S M T W T F S1 2 3 4 5 6 78 9 10 11 12 13 14

15 16 17 18 19 20 2122 23 24 25 26 27 2829 30 31

March 2011

S M T W T F S1 2 3 4 5

6 7 8 9 10 11 1213 14 15 16 17 18 1920 21 22 23 24 25 2627 28 29 30 31

32 33 27 06 28 29 30

17

10

03

01

24

2.2360679776.0

8.124038405

Sep 52018

(07)

2.4494897436.0827625308.185352772

May 61937

(16)

2.645751316.16441400

8.246211251

Apr 72072

(16)

3.4641016156.5574385248.544003745

Oct 121783

(15)

3.605551286.63324958

8.602325267

Aug 131919

(08)

3.741657396.70820393

8.660254038

Nov 141883

(16)

2.05.9160797838.062257748

Nov 41922

(11)

3.3166247906.4807406988.485281374

Sep 111800

(11)

4.2426406877.0

8.888194417

Dec 182082

(14)

1.7320508085.830951895

8.0

Mar 31773

(27)

3.1622776606.4031242378.426149773

Mar 102037

(10)

4.1231056266.9282032308.831760866

Apr 171818

(05)

4.3588989447.0710678128.944271910

Oct 191836

(11)

4.8989794867.4161984879.219544457

Nov 241910

(07)

5.07.4833147749.273618495

Oct 251793

(25)

4.582575697.21110255

9.055385138

Feb 211885

(12)

4.690415767.280109899.110433579

Oct 221758

(14)

4.795831527.348469239.165151390

Apr 231961

(13)

3.872983356.782329988.717797887

Feb 151799

(08)

4.06.855654608.774964387

Mar 162045

(06)

1.05.6568542497.8740078740

Feb 12003

(14)

1.4142135625.7445626477.937253933

Sep 21810

(16)

2.828427126.244998008.306623863

June 81933

(32)

3.06.324555328.366600265

Jan 91786

(01)

4.472135957.14142843

9.0

Apr 201939

(11)

5.0990195147.5498344359.327379053

Mar 261859

(07)

5.196152427.61577311

9.380831520

Jan 271754

(09)

5.291502627.68114575

9.433981132

Sep 281790

(73)

5.385164817.745966699.486832981

July 291848

(04)

5.477225587.810249689.539392014

Nov 301963

(12)


2x3-Digit Multiplication

12

06

12 x 21243 x 34373 x 373

Jan 121918

(12)

06 x 60637 x 73767 x 767

Nov 61798

(14)

LegendThe boxes contain products of the connected dates.


Then attach the original last digit of the leftmost date to the start of the rightmost date and multiply.



2. Right date: Blue end 2 x 06 = 12. Attach the last digit 2 from the leftmost date (12) to the front 212

3. 24 x 212 = 5088 (answer in box)

Single-Date Rules

Add 31: 12 + 31 = 4343 x 343 = 14749

Add 61: 12 + 61 = 7373 x 373 = 27229

12 x 212 = 2544


12 + 12 = 24, 2 + 4 = 6Jan 12, 1918, was a Saturday

14 + 06 = 202 + 0 = 2

Nov 6, 1798, was a

Tuesday

1 Mon

2 Tue

3 Wed

4 Thu

5 Fri

6 Sat

9 Sun

Two-Date Rules

Multiplication of larger numbers can be performed using the same methods we use for multiplying smaller numbers.

Remember the Good Neighbor Method? Multiply by a nearby round number and adjust for the difference:

29x342 = 30x342 − 342 = 991842x543 = 40x543 + 2x543 = 22806

To find 30x342 here, we would multiply from left to right:30x300 + 30x40 + 30x2 = 10260

Then to subtract 342 we can subtract 350 and add 8.

We are not yet squaring 3-digit numbers, just multiplying a 2-digit number by a 3-digit number. The August calendar covers 3x3-digit multiplication, and there we will extend our previous rules for 2-digit squares to 3-digit squares. If you can’t wait, do a time leap to August now!

Remember Cross Multiplication?

73x342:1: 3x2 = 6 , or 6 with a carry of 02: 7x2 + 3x4 + 0 = 26 , or 6 with a carry of 2

3: 7x4 + 3x3 + 2 = 39 , or 9 with a carry of 34: 7x3 + 3 = 24 Answer: 24966

Remember Partial Products?

24x342 = 20x300 + 20x40 + 20x2+ 4x300 + 4x40 + 4x2

= 8208

Remember the Anchor Method? Anchor one multiplier at the round number a, and then string out the differences (plus or minus) of the original numbers from this anchor to find the other multiplier.

If the differences have the same sign, the correction at the end will be added, and if one is positive and the other is negative the correction will be subtracted. For a 2x3-digit multiplication, the numbers are generally far apart, but the 3-digit number might be broken into a multiple of 10 plus an offset:

43x412 = 43x410 + 43x2= (40x44 + 3x1)x10 + 43x12= 17630 + 86 = 17716

87x985 = 87x980 + 87x5= (100x85 + 3x2)x10 + 87x5= 85060 + 870/2 = 85695

43x624 = 43x620 + 43x4= (50x55 – 7x12)x10 + 43x4= (2750 – 84)x10 + (4x40 + 4x3) = 26832

(where we subtract 84 by adding 100 and subtracting 16)

The calculating prodigy Truman Henry Safford (1836-1901) grew to become the director of the Hopkins Observatory at Williams College .

Remember the Midpoint Method? This technique converts the multiplication of two different numbers into the square of the average minus the square of the distance to the average

where a is the average of the two numbers, (a+c) is one of the numbers, and (a−c) is the other number.

Of course, this applies to larger numbers. Here the midpoint of 24 and 38 is 41 at a distance of 7 from each number:

24x376 = 24x380 – 24x4 = (412 − 72)x10 – 96 = 9024

Here we can find 412 as 40x42 + 1 using the Midpoint Method again, or 402 + 40 + 41 or 42 || (2x4) || 1 = 1681 by the Neighbors of Squares Method (see February). Then we subtract 72 by subtracting 50 and adding 1, attach a 0 to the end, and subtract 96 by subtracting 100 and adding 4.

Very useful!

MA

Y

11 12

18 19

25 26

13 14

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08 09

15 16

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31

04 05 06 0701 02 03

June 2011

S M T W T F S1 2 3 4

5 6 7 8 9 10 1112 13 14 15 16 17 1819 20 21 22 23 24 2526 27 28 29 30

April 2011

S M T W T F S1 2

3 4 5 6 7 8 910 11 12 13 14 15 1617 18 19 20 21 22 2324 25 26 27 28 29 30

31 32 33

04 26 27 28

22

15

08

12

32

100578112141

June 101959

(11)

9091475629696

Apr 31831

(15)

16161872536725

May 41788

(23)

25252289643956

July 51981

(13)

12211016419584

Aug 111880

(19)

25441474927229

Jan 121918

(12)

121894070468484

Apr 171935

(13)

147244650177341

Nov 181840

(03)

1746125006400

Feb 192028

(05)

818116004900

Feb 92027

(11)

98563510959829

May 162073

(04)

74292451640656

Oct 232019

(07)

64643662166861

Dec 81770

(07)

77252971651376

Jan 151771(05)

48841870931789

Dec 221816

(05)

101763052549725

May 241946

(08)

2694136008100

Mar 292058

(03)

900982117381

May 301844

(10)

162764314968469

July 261990

(05)

196294976478144

Oct 271822

(27)

231845658188021

Dec 281796

(02)

400770114661

Nov 201878

(10)

25411310423124

Mar 211802

(15)

36362726951389

Nov 61798

(14)

49493184459024

May 72015

(15)

40691953635076

Sep 131805

(10)

57962452543125

Dec 141792

(09)

131253673658996

Oct 251786

(05)

40611624426864

Aug 311997

(14)

101742416244

July 11867

(00)

4041098922869

Nov 21907

(22)


Cubes and Cube Roots

15

09

Legend

The boxes contain cube roots of the connected dates separated by a decimal point.





3. (30.18)1/3 = 3.113434583 (answer in box)

Single-Date Rules

Add 31: 15 + 31 = 4646^3 = 97336

Add 61: 15 + 61 = 7676^3 = 438976

15^3 = 3375



09 + 06 = 151 + 5 = 6


1 Mon

2 Tue

3 Wed

4 Thu

5 Fri

6 Sat

9 Sun

Two-Date Rules

9^340^370^3

Nov 91799

(06)

15^346^376^3

Jan 151925

(16)

Binomial Expansion for Cubes: As we did for squares, we can express the number to be cubed as the sum of two other numbers that are more easily cubed:

343 = (30+4)3 = 303 + 43 + 3x30x4x34= 27000 + 64 + 90x136 = 27064 + (13600 – 1360) = 39304

693 = (70−1)3 = 703 – 13 – 3x70x1x69= 343000 – 1 – 3x70x69 = 342999 – 3x702 + 3x70 = 328509

Very useful for larger numbers

Cubes are often asked of lightning calculators, but non-integer cube roots are very rarely encountered.

Two-Step Approximation: An improvement to one weighted average.

For an initial estimate r,

(119)1/3 : Estimate 5add b = (119 – 125) / 75 = -0.08 to get 4.92subtract c = 0.082 / 5 = 0.00128 to get 4.91872 Actual Value: 4.918685

Our answer of 4.91872 is better than weighted averaging once

Thirding the Difference: Our most accurate approximation but more difficult.

For an estimate r, a better estimate is the weighted average of r and d = N – r3:

(119)1/3 : Estimate 5Better estimate is 5 x [125 + (2/3)(-6)] / [125 + (1/3)(-6)]

= 5 x 121/123 = 605/123 = 4.918699 Actual Value: 4.918685

A.C. Aitken (1895-1967) excelled in non-integer root extraction . The “Thirding” method here was pioneered by Aitken.

Square and Multiply: Square, then multiply againusing Partial Products or Cross Multiplication:

342 = 1156so 343 = 34 x 1156 = 34000 + 3400 + 1700 + 204 = 39304

or 34 x 1156:

1: 4x6 = 124, or 4 with a carry of 22: 3x6 + 4x5 + 2 = 40 , or 0 with a carry of 43: 3x5 + 4x1 + 4 = 23 , or 3 with a carry of 24: 3x1 + 4x1 + 2 = 9 , or 9 with a carry of 05: 3x1 + 0 = 3 Answer: 39304

Weighted Averaging: A good method to approximate a non-integer cube root.

For an estimate r, a better estimate is the weighted average of r and N/r:

(119)1/3 : Estimate 5Better estimate is (2x5 + 119/25)/3 = (10 + 476/100)/3 = 14.76/3 = 4.92

For an estimate expressed as a fraction s/t, this average can be re-written as:

(119)1/3 : Estimate s/t = 49/10Better estimate is ≈ (235300 + 119x1000)/ 3x2401x10 = 354300/72030

≈ 354.3/72.03 = 354.3/[72(1+0.03/72)]≈ (354.3/72)x(1 – 1/2400)= 4.9188 Actual Value: 4.918685

JU

NE

08 09

15 16

22 23

29 30

10 11

17 18

24 25

05 06

12 13

19 20

26 27

07

14

21

28

01 02 03 04

July 2011

S M T W T F S1 2

3 4 5 6 7 8 910 11 12 13 14 15 1617 18 19 20 21 22 2324 25 26 27 28 29 3031

27 28 29 01 02

03 04

19

12

05

0128

31

34354872

314432

Aug 71887

(11)

132768

238328

May 11886

(14)

835937

250047

Aug 21918

(11)

51259319

328509

Oct 81937

(06)

72964000

343000

Aug 91791

(11)

274491125

421875

June 141956

(08)

337597336

438976

May 151827

(05)

4096103823456533

Sep 161810

(11)

21650653300763

Mar 61794

(16)

219785184405224

Mar 132021

(11)

8000132651531441

Mar 201839

(10)

12546656287496

Nov 51857

(17)

172879507389017

Dec 121767

(21)

6859125000512000

May 191929

(08)

9261140608551368

Dec 212012

(11)

17576185193658503

May 262039

(14)

19683195112681472

Aug 271884

(03)

12167157464592704

Dec 231972

(10)

13824166375614125

Aug 242017

(07)

15625175616636056

June 251913

(05)

4913110592474552

July 171985

(13)

5832117649493039

Jan 181815

(03)

2739304262144

June 31787

(15)

6442875274625

Apr 41968

(27)

100068921357911

Feb 101876

(21)

133174088373248

June 111988

(13)

10648148877571787

July 221841

(09)

21952205379704969

June 281991

(04)

24389216000729000

Apr 291948

(11)

27000226981753571

July 301803

(12)

May 2011

S M T W T F S1 2 3 4 5 6 78 9 10 11 12 13 14

15 16 17 18 19 20 2122 23 24 25 26 27 2829 30 31


Factoring and Integer Roots

15

09

x3

y5

z7

Jan 151925

(16)

x3

y5

z7

Nov 91799

(06)

Legend

The boxes contain prime factors of the connected dates

Blue End: Add 31 to date Red End: Add 61 to date


1. Left date: Blue end 15 + 31 = 46

2. Right date: Blue end 09 + 31 = 40

3. 4640 = 2x2x2x2x2x5x29 or 2^5x5x29 (answer in box)

Find y = integer fifth root

Find z = integer seventh root

Find x = integer cube root



09 + 06 = 151 + 5 = 6


1 Mon

2 Tue

3 Wed

4 Thu

5 Fri

6 Sat

9 Sun

Two-Date Rules

Factoring and finding roots of perfect powers is a fun and challenging pursuit.

As a child, Zerah Colburn(1804-1839) factored 4294967297into 641 x 6700417

Sieve: Eliminate small prime factors of N, since 82% of random 4-digit numbers (such as this month’s exercises) have prime factors <= 11.

1. Factor of 2 if N is even; 4 if 4 divides last two digits; 8 if 8 divides last three digits2011 not even, 4 does not divide 11, and 8 does not divide 011

2. Factor of 5 if N ends in 52011 does not end in 5

3. Factors of 3 or 9 if they divide the 9-Remainder (Add all digits, repeat until < 10)2 + 0 + 1 + 1 = 4 is not a multiple of 3 or 9

4. Factors of 11 if 11 divides the 11-Remainder (odd-place digits minus even-place digits, repeat until less than 11, add 11’s if needed to get result >0)

1 - 1 + 0 – 2 = -2 so add 11 to get 9, which is not a multiple of 11

The year 2011 is prime!

Is the year 2011 prime?

Difference/Sum of Powers: Is N one of these forms?

N = (a2 – b2) has factors (a+b) and (a-b) 1596 = 402 – 22 = 42 x 38N = (a3 – b3) has a factor (a-b) 973 = 103 – 33 has factor 7N = (a3 + b3) has a factor (a+b) 1001 = 103 + 13 has factor 11

2011 is not!

We’ve tested all primes < 100 (the range for the 4-digit numbers in this calendar) except 29, 59, 61, 71, 73, 79, 83, and 97. The only one less than 44 is 29, and 2011/29 is not an integer, so….

Other Divisor Tests: Simplify very large numbers before testing some primes by mental division. Repeat until N is too small for these to be useful.

Remove m = last 3 digits of N (not useful for 4-digit N in this calendar unless m is small)Find (N-m) for 7, 11, 13 2 – 011 = -9 not divisible by 7, 11, 13Find (N+m) for 37 2 + 011 = 13 not divisible by 37Find (N-2m) for 23, 29 2 – 022 = -20 not divisible by 23Find (N+4m) for 31, 43 2 + 088 = 90 not divisible by 31,43

Remove k = last 2 digits of N (useful for most 4-digit N in this calendar )Find (N+4k) for 19 20 + 44 = 64 not divisible by 19Find (N+8k) for 17, 47 20 + 88 = 108 not divisible by 17,47Find (N+16k) for 41 20 + 176 = 196 not divisible by 41Find (N-2k) for 67 20 – 22 = -2 not divisible by 67Find (N-8k) for 89 20 – 88 = -66 not divisible by 89Find (N-9k) for 53 20 – 99 = -77 not divisible by 53

We don’t need to test for primes above √2011 = 44

Integer Roots: For a 3-digit integer cube root, fifth root or seventh root, as in the exercises in this month’s calendar:

Seventh root of N = 41,998,948,952,729,733:

1. The one’s digit is unique for all these roots. It is the same as in the power for a root of order of the form 4k+1 like the fifth root. The third and seventh root differ only in the blue rows of the table here. For a seventh power ending in 3, the root is nn7.

2. The hundreds digit is found from the ranges of the powers. Here 2007 = 128x1014

and 3007 = 2187x1014 so the fifth root here would be in the 200s. The root is 2n7.

3. The tens digit is found by matching the 11-Remainders for cube or seventh roots, or the 13-Remainders for fifth or seventh roots, since these produce unique results.

11-Remainders (Remainders after division by 11)Subtract even digits from odd digits in N,repeat until <11, adding 11’s if needed until >0.Check table for the matching result for the root.Then do the 11-Remainder on the root and deduce n:11-Remainder on 41,998,948,952,729,733 = -3 8,so 11-Remainder on 2n7 = 6, so n must equal 3and so the root = 237.

Or use 13-Remainders:We can reduce N using“Other Divisor Tests”or simply divide the entire Nby 13 to get a remainder of 3.From the 13-test table, the13-Remainder on 2n7 = 3,and by trial we find root = 237.

One-Digit Endings

PowerCube Root

Fifth Root

Seventh Root

0 0 0 01 1 1 12 8 2 83 7 3 74 4 4 45 5 5 56 6 6 67 3 7 38 2 8 2

9 9 9 9

Power11’s Test 13’s Test

Cube Root

Seventh Root

Fifth Root

Seventh Root

0 0 0 0 01 1 1 1 12 7 8 6 113 9 5 9 34 5 9 10 45 3 4 5 86 8 7 2 77 6 2 11 68 2 6 8 59 4 3 3 9

10 10 10 4 1011 - - 7 212 - - 12 12

No answers are provided for these (use the Calculator on your PC)

If the letter “P” appears anywhere in the factor list the number is actually prime!

JU

LY

August 2011

S M T W T F S1 2 3 4 5 6

7 8 9 10 11 12 1314 15 16 17 18 19 2021 22 23 24 25 26 2728 29 30 31

38

June 2011

S M T W T F S1 2 3 4

5 6 7 8 9 10 1112 13 14 15 16 17 1819 20 21 22 23 24 2526 27 28 29 30

31

06 07

13 14

20 21

27 28

08 09

15 16

22 23

29 30

03 04

10 11

17 18

24 25

05

12

19

26

01 0201

03

10

17

24

37363506343332

136519191947195170207

1231171548132409344

Oct 92028

(01)

8615125551473077343

570044943920267264

Sep 81830

(13)

1404928178689902368

294313621587817417

Feb 71795

(08)

11764900028247524900000

178832839131982314368

Apr 161806

(14)

14142076117254995508224

11140385236461026176

Nov 151895

(08)

7840275211727599043051

28321829503567564201

Aug 141960

(22)

569722789290572941207901

724475329537064921875

Mar 232030

(10)

4855876561921517976995993401580794509246464

Sep 221859

(09)

4286610641271422659408325618606965795036053

Jan 211934

(15)

411830784140126044921875

170980716079866232953

Jan 301777

(01)

6539203103244904396875

61243167054566186591

Dec 291864

(11)

4943086387658651299862419310159666381286089

Aug 281963

(02)

99700299916850581551

8300513205665792

Nov 271898

(09)

799178752593777798104501

34048254470000000

Apr 261986

(07)

700227072587216781904499

57698133708111872

Sep 251901

(05)

30482121711592740743

266001988046875

Feb 201765

(10)

25363613763955671886657

366790143213462347

Nov 191863

(12)

20020162543016596437893

245526784064140625

Apr 181779

(18)

7044499761917364224

698260569735168

Jan 131889

(24)

486271255963102065799

17419031429960369

May 122043

(08)

254121842956466552832

96733929416521923

Oct 111881

(09)

2685619234849287168

213647747443112099

Feb 61952

(15)

19248832164130859375

84744015603303808

June 51874

(18)

358157755730836701

13254776280841401

July 41776

(18)

618470208408348897330176

581746347858515625

Mar 241832

(09)

15922008839883798828125

160388518974780383

Dec 171903

(05)

349657831057227821024

753669927250029952

Sep 102007

(00)

186086719254145824

5231047633534976

Oct 31990

(18)

17779581995009990004999

314636844829229056

Mar 311923

(11)

133100037129300000

532875860165503

June 21946

(25)

103030114025517307

131593177923584

Mar 11839

(22)


3-Digit Multiplication

Add 31: 12 + 31 = 43343^2 = 117649

12

06

212^2343^2373^2

Jan 121918

(12)

606^2737^2767^2

Nov 61798

(14)

Legend

The boxes contain products using digits of the connected dates.


1. Attach the original first digit of the rightmost date (without

doubling or tripling) to the end of the leftmost date

1. Attach the original last digit of the leftmost date (without

doubling or tripling) to the start of the rightmost date

1. Multiply the two numbers

Example for the second box from the bottom:


Attach 0 from 06 to end 240

1. Right date: Red end 3 x 06 = 18

Attach 2 from 12 to start 218

3. 240 x 218 = 52320 (answer in box)

Single-Date Rules

Add 61: 12 + 61 = 73373^2 = 139129

212^2 = 44944



14 + 06 = 202 + 0 = 2

Nov 6, 1798, was a

Tuesday

1 Mon

2 Tue

3 Wed

4 Thu

5 Fri

6 Sat

9 Sun

Two-Date Rules

Midpoint Method: Square the average and subtract the square of the distance to that average, or algebraically,

where a is the average of the two numbers, (a+c) is one of the numbers, and (a−c) is the other number. Here the midpoint of 244 and 382 is 313, a distance of 69 from each number. Then we usethe Reverse Midpoint Method to simplify 3132 :

244x382 = 3132 − 692 = (300x326 + 132) − 692

Multiplication of 3x3-digit numbers extends our earlier techniques (January/February/May)

Good Neighbor Method: Multiply by a nearby round number and adjust for the difference:

294x327 = 300x327 − 6x327 = 96138

411x543 = 400x543 + 11x543= 223173

To find 300x342, we can multiply from left to right as (3x300 + 3x40 + 3x2)x100 = 102600. Then 6x327 = 1962, so we subtract 2000 and add 38.

The abilities of Jacques Inaudi (1867-1950) were investigated and written about by Alfred Benet and others.

Cross Multiplication:

873x342:1: 3x2 = 6 , or 6 with a carry of 02: 7x2 + 3x4 + 0 = 26 , or 6 with a carry of 23: 8x2 + 7x4 + 3x3 + 2 = 55 , or 5 with a carry of 54: 8x4 + 7x3 + 5 = 58 , or 8 with a carry of 55: 8x3 + 5 = 29 Answer: 298566

Anchor Method: Anchor one multiplier at a nearby round number a, and then string out the differences (plus or minus) of the original numbers from this anchor to find the other multiplier. Then add the product of the differences. Algebraically this is represented as

If the differences have the same sign, the correction at the end will be added, and if one is positive and the other is negative the correction will be subtracted.

443x412 = 400x455 + 43x12= (4x400 + 4x50 + 4x5)x100 + 43x10 + 43x2 = 182516

922x985 = 1000x907 + 78x15= 907000 + 78x10 + ½(78x10) = 908170

463x536 = 500x499 – 37x36= 499000/2 – (40x33 + 3x4) = 249500 – 1332 = 248168

(where we might subtract 1332 by subtracting 1500 and adding 168)

Partial Products:

247x342 = 200x300 + 200x40 + 200x2+ 40x300 + 40x40 + 40x2+ 7x300 + 7x40 + 7x2

= 84474

Special Neighbors: (the comma indicates a 3-digit group, so a fourth digit is added to the group on the left)

(250+a)2 = (62 + a/2) , 500+a2

(500+a)2 = (250 + a) , a2

(750+a)2 = (562 + 3a/2) , 500+a2

2682 = (62 + 9) , 500+182 = 71,8245132 = (250 + 13) , 132 = 263,1694292 = (250 - 71) , 712 = 179 , 5041 = 184,0417572 = (562 + 21/2) , 500+72 = 572.5 , 549

= 572 , 1049 = 573,049

Cool Facts: For numbers ending in 25 or 75,

(n25)2 = (n2 + n/2) | 06 | 25(n75)2 = (n2 + n + n/2) | 56 | 25

3252 = (32 + 1.5) | 06 | 25 = 10.5 | 06 | 25 = 1056256752 = (62 + 6 + 3) | 56 | 25 = 455625

Advanced: Can you separate out a product that is easy to multiply by the rest?

67 x 3 = 20123 x 13 = 29919 x 21 = 39917 x 47 = 79989 x 9 = 801

53 x 17 = 90137 x 27 = 999

7 x 11 x 13 = 100123 x 29 x 3 = 200131 x 43 x 3 = 3999

159 x 172= (3x53)x170 + 2x159= 3x9010 + 318= 27348

Reverse Midpoint Method: Split the square into the product of two numbers equidistant from the original number, and add the square of that distance (a type of Anchor Method):

6172 = 600x634 + 172

= (6x600 + 6x30 + 6x4)x100 + (14x20 + 32) = 380689

When spreading a square to the product of two numbers you add the correction, and when collapsing two multipliers to a square you subtract the correction.

Binomial Expansion for Squares: Separate the number into the sum of two other numbers that are more easily squared:

3422 = (300+42)2 = 3002 + 2x300x42 + 422

= 116964 3842 = (400-16)2 = 4002 – 2x400x16 + 162

= 147456

AU

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ST

10 11

17 18

24 25

31

12 13

19 20

26 27

07 08

14 15

21 22

28 29

09

16

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03 04 05 0601 02

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03

21

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07

01

033332

04 05 06

32

82628116004900

July 91765

(11)

40804110889131769

July 21892

(22)

91809188356215296

Jan 31755

(11)

163216286225319225

Oct 41957

(19)

1001988129241

Jan 101879

(13)

123215856473984

Apr 112014

(21)

379456558009603729

Jan 162036

(14)

514089719104770884

Nov 171943

(04)

669124900601958441

Sep 181756

(06)

652864881721938961

May 81945

(12)

265225417316456976

Dec 151846

(05)

49284124609146689

Nov 221963

(10)

499849702244753424

Oct 71860

(29)

171396297025330625

Sep 141757

(07)

146416350479524

Aug 212087

(10)

104329206116234256

Jan 231936

(08)

685584919681978121

Feb 281914

(05)

86304136008100

Aug 291801

(13)

275625430336470596

Feb 251940

(20)

391876573049619369

June 262019

(04)

528529736164788544

Sep 271765

(14)

84456125006400

July 191867

(13)

4002280132761

May 202015

(10)

255025404496443556

Aug 51931

(16)

367236543169588289

June 61944

(05)

44944117649139129

Nov 121760

(09)

97969197136224676

Dec 131873

(11)

179776308025342225

June 241761

(06)

9002592136481

Apr 302038

(11)

171616864485264

Oct 311895

(09)

102015382468644

Dec 11932

(12)

July 2011

S M T W T F S1 2

3 4 5 6 7 8 910 11 12 13 14 15 1617 18 19 20 21 22 2324 25 26 27 28 29 3031


4-Digit Multiplication

15

09

LegendThe boxes contain products using digits of the connected dates.


1. Form a number from the leftmost date followed by the rightmost date. Double or triple if blue or red end on left date.

2. Form a number from the rightmost date followed by the leftmost date. Double or triple if blue or red end on right date.

3. Multiply the two numbers

Example for the second box from the bottom:


2. Right date: Red end 3 x 0915 = 2745

3. 3018 x 2745 = 8284410 (answer in box)

Single-Date Rules



09 + 06 = 151 + 5 = 6


1 Mon

2 Tue

3 Wed

4 Thu

5 Fri

6 Sat

9 Sun

Two-Date Rules 9009^2

0440^20770^2

Nov 91799

(06)

5115^26446^26776^2

Jan 151925

(16)

Add 31: 15 + 31 = 466446^2 = 41550916

Add 61: 15 + 61 = 766776^2 = 45914176

Add reversed date to front and square

5115^2 = 26163225

Multiplication of 4x4-digit numbers can benefit from our earlier techniques for special cases, but cross-multiplication is typically the most realistic option. We also introduce a new group technique here.

Johann Dase (1824-1861) once multiplied 79532853 by 93758479 in 54 seconds; two 20-digit numbers in 6 minutes; two 40-digit numbers in 40 minutes; and two 100-digit numbers in 8 hours 45 minutes.

Cross Multiplication:

5873x6342:1: 3x2 = 6 , or 6 with a carry of 02: 7x2 + 3x4 + 0 = 26 , or 6 with a carry of 23: 8x2 + 7x4 + 3x3 + 2 = 55 , or 5 with a carry of 54: 5x2 + 8x4 + 7x3 + 3x6 + 5 = 86, or 6 with a carry of 85: 5x4 + 8x3 + 7x6 + 8 = 94 , or 4 with a carry of 96: 5x3 + 8x6 + 9 = 72 , or 2 with a carry of 77: 5x6 + 7 = 37 Answer: 37246566

2x2 Cross Multiplication:

5873x6342:1: 73x42 = 3066 , or 66 with a carry of 302: 58x42 + 73x63 + 30 = 7065 , or 65 with a carry of 703: 58x63 + 70 = 3724 Answer: 37246566

It helps if you are skilled at 2x2 multiplications!

Group Notation: It is convenient to treat hundreds groupings as separate blocks for multiplication. The notation "|n" represents a two-digit number string. If more than two digits exist in n, they are merged (or added) to the digits to the left of the "|" sign. The hundreds groups can carry or borrow as needed from neighboring groups to make each group positive and less than 100.

45 | 3 = 45033 | 129 = 4 | 29 = 42925 | -125 = 23 | (200-125) = 2375

Do you see that 34|145|16|-248 = 35451252 ?

4x4 Group Multiplication:

6143 x 2839:

Represent this as 61|43 x 28|39 or A|B x C|D

1. Find AxC = 61x28 = 1708. Find BxD = 43x39 = 1677

1708 | 1677 = 17 | 24 | 77

2. Add the last two numbers, 24+77, and subtract (A-B) x (C-D)

24 + 77 - (61-43)x(28-39) = 101 + 11x18 = 299

3. Previous Results: 17 | 24 | 77 299

Answer: 17 | (17 + 24) | 299 | 77

Merge terms left to right as we go to get 17439977

5873x6342:

1. AxC = 58x63 = 3654 BxD = 73x42 = 3066

3654 | 3066 = 36 | 84 | 66

2. Add the last two numbers, 84+66,and subtract (A-B) x (C-D)

84 + 66 - (58-73)x(63-42) = 150 + 15x21 = 465

3. Answer: 36 | (36 + 84) | 465 | 66 = 37246566

1. Requires onlythree 2x2 digit multiplications

2. Naturally produces a left-to-right answer

3. Mechanical process is simple when learned

4. Ideal when A and B or C and D are close in value!

SE

PT

EM

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R

07 08

14 15

21 22

28 29

09 10

16 17

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30

04 05

11 12

18 19

25 26

06

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27

01 02 03

October 2011

S M T W T F S1

2 3 4 5 6 7 89 10 11 12 13 14 15

16 17 18 19 20 21 2223 24 25 26 27 28 2930 31

32 33 28 29 30

31

18

11

04

01

31

27

360720365383156958782889

July 61817

(12)

490980496952224475134224

Mar 71908

(08)

641280648721692193489561

Aug 81824

(19)

96907691974913622791076

Apr 132029

(19)

169249962964802533350625

July 141789

(19)

261632254155091645914176

Apr 151912

(85)

250500254014489644435556

Dec 52001

(07)

44605441185424914235529

July 121938

(08)

83156161302500774400

Dec 191773

(17)

160320162846222532092225

Jun 41989

(14)

123432159633647683984

Jan 111753

(20)

659019248928360195628841

Mar 181828

(02)

4840024056013538161

Sep 201815

(10)

273006254298113647416996

July 251821

(05)

387630765710824962204769

Feb 261944

(07)

49372841262380915077689

Aug 221788

(10)

103877292073891623853456

June 232011

(08)

178421763085802534633225

Jan 241838

(06)

374054565545780960481729

Feb 161843

(15)

506516897136870477053284

June 171927

(14)

40080041110888913417569

Apr 21853

(13)

90180091878355621752896

Sep 32099

(19)

81162081193600592900

Mar 91838

(14)

1210020764813136441

May 102032

(00)

149084165127048305924

May 211927

(12)

522295297323936478996544

June 271762

(18)

676999849137448197792321

July 281870

(12)

85174441435600980100

May 291953

(12)

10890027589213964081

Sep 301957

(70)

100200154382247086244

Nov 11980

(23)

August 2011

S M T W T F S1 2 3 4 5 6

7 8 9 10 11 12 1314 15 16 17 18 19 2021 22 23 24 25 26 2728 29 30 31


Trigonometric Functions

12

06

Legend

Single-Date Rules

cos 12 ° = 0.97815

tan 12 ° = 0.21256

sin 12 ° = 0.20791



14 + 06 = 202 + 0 = 2

Nov 6, 1798, was a Tuesday

1 Mon

2 Tue

3 Wed

4 Thu

5 Fri

6 Sat

9 Sun

Two-Date Rules

sin 6 °cos 6 °tan 6 °

Nov 61798

(14)

sin 12 °cos 12 °tan 12 °

Jan 121918

(12)

The boxes contain sines and cosines of the connected dates separated by a decimal point, in degrees





3. sin 24.12° = .4086 and cos 24.12° = .9127

(concatenated answer in box is .4086.9127)

Cosine without Memorization: For an angle in degrees d ≤ 36°,

cos 28.4°: 1000 cos d ≈ 1000 – 28.4x29.4 / 7 = 1000 – 28.4x4.2 = 880.7cos d = 0.8807 Actual Value: 0.8796

Cosine with Memorization (~3 digit accuracy): For an angle in degrees d ≤ 36°, split it into two parts d = a + b, where cos a has been memorized:

cos 28.4°: 1000 cos d ≈ 1000 cos 30 – (-1.6)(30 + 28.4 + 1) / 7= 866.0 + 1.6x8.5= 879.6

cos d = 0.8796 Actual Value: 0.8796

Degrees/Radians Conversion: d (degrees) ≈ 401/7 x d (radians) d (radians) ≈ 7/401 x d (degrees) ≈ (7/400 – 1/2000) x d (degrees)

Tangent by Definition:

tan 28.4° = 4756 / 8796 ≈ 4864/9000 = 0.5404 Actual Value: 0.5407

Tangent with Memorization (~3 digit accuracy): For angle d ≤ 45°,split it into two parts d = a + b, where tan a has been memorized

tan 28.4°: 1000 tan d ≈ 1000 tan 30 + (-1.6/10)(174 + 30x28.4/20) = 541.2tan d = 0.5412 Actual Value: 0.5407

Sine without Memorization: For an angle in degrees d ≤ 54°,

sin 28.4°: 1000 sin d ≈ (28.4/10)[174.4 – 28.4x29.4/120] = 475.5sin d = 0.4755 Actual Value: 0.4756

Sine with Memorization (~3 digit accuracy): For an angle in degrees d ≤ 54°, split it into two parts d = a + b, where sin a has been memorized:

sin 28.4°: 1000 sin d ≈ 1000 sin 30° + (-1.6/10)[174 – 30x28.4/40]= 500.0 – 0.16 (174 – 30x0.71) = 475.6

sin d = 0.4756 Actual Value: 0.4756

Tangent with Memorization (~4 digit accuracy):1. If d > 45° (or 0.7854), replace d with (90°−d) or (1.571−d).

2. Find the angle or sum or difference of a pair of angles in Table 1that is nearest to d. For a single angle, the fraction in column 3 is our first estimate. If a pair, use the equation below to add orsubtract their fractions, where N and D are the numerator anddenominator of each fraction. Swap signs to subtract a fraction.

3. If the remaining difference is < ±0.3° (or .005 radians), skip to Step 4. Otherwise, convert it to radians if in degrees (multiply by the simpler ratio 7/400). Find a simple fraction that approximates it and use the equation in Step 2 to add this fraction to the result of Step 2.

4. Flip over the fraction if d was originally >45° (or 0.7854). Divide fraction to 4 digits.

5. Convert the remaining angle difference b to radians if still in degrees,again using 7/400. Then calculate c below to 4 digits, where tan a isour current estimate from Step 4. Subtract c if d was originally >45°,otherwise add c.

The mathematician Johann Carl Friedrich Gauss (1777-1855) possessed enormous skill in lightning calculation.

tan 28.4°: Doubling 14.04° in the table gives 28.08°. Add N1/D1 = ¼ and N2/D2 = ¼ using our formula:

(1x4 + 1x4) / (4x4 – 1x1) = 8/15Since b = 28.4 – 28.08 = 0.32 is just over 0.3°, we might skip Step 3 and divide 8/15 to get 0.5333. Convert b to radians and add c:

0.32 x 7/400 = 0.08 x 0.07 = 0.0056c = 0.0056 (1 + 0.53332) ≈ 0.0056 (1 + 0.52) = 0.00700.5333 + 0.0070 = 0.5403 Actual Value: 0.5407

If we do not skip Step 3, approximate the difference 0.33 with 2/7 = 0.286 to simplify the radian conversion: 2/7 x 7/400 = 1/200. Then

(8x200 + 1x15) / (15x200 – 8x1) = 1615/2992 = 0.5398Final difference 0.034 x 7/400 = 0.0006

c = 0.0006 (1 + 0.53982) = 0.0008, and 0.5398 + 0.0008 = 0.5404

For angle d ≥ 36°, calculate the sine:cos d sin (90°–d)

For angle d ≥ 54°, calculate the cosine:sin d cos (90°– d)

For angle d ≥ 45°, calculate as:

tan d 1 / tan (90°– d)

OC

TO

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RNovember 2011

S M T W T F S1 2 3 4 5

6 7 8 9 10 11 1213 14 15 16 17 18 1920 21 22 23 24 25 2627 28 29 30

September 2011

S M T W T F S1 2 3

4 5 6 7 8 9 1011 12 13 14 15 16 1718 19 20 21 22 23 2425 26 27 28 29 30

03

0.190810.981630.19438

Feb 112034

(22)

0.207910.978150.21256

Dec 121902

(11)

0.325570.945520.34433

Mar 192004

(13)

0.309020.951060.32492

Oct 181766

(15)

0.156430.987690.15838

Nov 91989

(13)

0.173650.984810.17633

Nov 101864

(12)

0.292370.956300.30573

Mar 171970

(03)

0.275640.961260.28675

July 161837

(11)

0.390730.920500.42447

May 231754

(08)

0.406740.913550.44523

Sep 241999

(08)

0.515040.857170.60086

May 311917(09)

0.50.866030.57735

Dec 301899

(12)

0.069760.997560.06993

Aug 41782

(14)

0.087160.996190.08749

Oct 52040

(27)

0.034900.999390.03492

Oct 21913

(29)

0.052340.998630.05241

May 31865

(18)

0.453990.891010.50953

May 271853

(14)

0.469470.882950.53171

Mar 281954

(17)

0.422620.906310.46631

Aug 252067

(15)

0.438370.898790.48773

Sep 261958

(06)

0.121870.992550.12278

Nov 71917

(14)

0.139170.990270.14054

July 82074

(19)

0.104530.994520.10510

Aug 61945

(04)

0.241920.970300.24933

Apr 141865

(09)

0.258820.965930.26795

May 151855

(05)

0.224950.974370.23087

June 131761

(29)

0.358370.933580.38386

July 211969

(79)

0.374610.927180.40403

Apr 221798

(14)

0.342020.939690.36397

Jan 201835

(00)

0.484810.874620.55431

June 291809

(11)

0.017450.999850.01746

Jan 11901

(19)

05 06

12 13

19 20

26 27

07 08

14 15

21 22

28 29

02 03

09 10

16 17

23 24

04

11

18

25

01

30 3104 05 05 08

23

16

09

02

01

0706


Logarithms

16

10

Legend

The boxes contain common (base 10) logarithms of the connected dates separated by a decimal point.





3. log 32.20 = 1.50786 (answer in box)

Single-Date Rules

Add 31: 16 + 31 = 47log 47 = 1.67210

Add 61: 16 + 61 = 77log 77 = 1.88649


16 + 07 = 23, 2 + 3 = 5Jan 16, 1925, was a Friday

10 + 17 = 272 + 7 = 9


1 Mon

2 Tue

3 Wed

4 Thu

5 Fri

6 Sat

9 Sun

Two-Date Rules

log 10log 41log 71

Nov 101799

(17)

log 16log 47log 77

Jan 161925

(07)

log 16 = 1.20412

Logarithms appear quite often in mathematics and engineering and calculating them is an impressive skill. When combined with exponentials, they can be used to calculate high-order roots.

N log N

1 0

2 .30103000

3 .47712

4 .60206 = 2 x log 2

5 .69897 = 1 – log 2

6 .77815 = log 2 + log 3

7 .84510

8 .90309 = 3 x log 2

9 .95424 = 2 x log 3

10 1

11 1.04139

13 1.11394

Integers with Low Factors: Memorize the logarithms of low primes in the table to the right . Notice how the other logarithms in the table can be calculated as simple combinations of these---you can combine these to find logarithms of many numbers according to these rules:

log ab = log a + log b log 42 = log 2 + log 3 + log 7log (a/b) = log a – log b log 5.5 = log (11/2) = log 11 – log 2log ab = b log a log 49 = log 72 = 2 log 7log a1/b = (1/b) log a log 631/2 = ½ log 63 = ½ (log 9 + log 7)log (10bN) = b + log N log 450 = 1 + log 45 = 2 + log 4.5log (10-bN) = log N – b log 0.045 = log 45 – 3 = log 4.5 - 2

Series Approximation: Factor out a nearby number whose logarithm is easier to find. Then add to the first term or two in the following series expansion for ln (1+a) (multiplied by 0.43 for log N):

1211 = 1210 x (1 + 1/1210), so ln 1211 = ln 1210 + ln (1 + 1/1210):log 1210 = log (10 x 112) = 1 + 2 log 11 = 3.08279log (1 + 1/1210) ≈ 0.43 x 1/1210 = 0.00036 first term only

so log 1211 ≈ 3.08279 + 0.00036 = 3.08315 Actual Value: 3.08314

Start off with lower accuracy before attempting 5-digit solutions!

George Parker Bidder Method: Factor out a nearbynumber whose logarithm is easy to find, then use theFollowing expression, where m is the number of placesthat a needs to be shifted to lie between 1 and 10:

log 1211 = log 1200 + (1 + 11/1200):log 1200 = 2 + log (4x3) = 2 + 2 log 2 + log 3 = 3.07918103 x 11/1200 = 11/1.2 which lies between 1 and 10, so m = 3so log (1 + 11/1200) ≈ 11/1.2 log 1.001

= 11/1.2 x 0.000434 = 0.00398so log 1211 = 3.07918 + 0.00398 = 3.08316 Actual Value: 3.08314

Note how far we were able to go from 1211 to the very convenient 1200 and still get a very accurate answer for log 1211!

To use this method we need to learn log (1+10–m) except for m=1, in which case a closer convenient number is needed for accuracy. These values approach 0.4343x10–m as m increases:

log 1.01 = 0.004321log 1.001 = 0.000434

log 1.0001 = 0.000043

We will be working here in common logarithms, or logarithms to base 10. To calculate natural logarithms we can use the relation ln N = 2.3026 log N = (log N) / 0.4343 , or for easier calculation,

Example: log 7 = 0.84510ln 7 ≈ (0.84510/0.43) x (1 – 0.01) = 1.9653 – 0.0197 = 1.9456

where -0.0197 -0.02 + .0003 Actual Value: 1.9459

Use whichever is easier for a number N

Among many other facts, Wim Klein (1912-1986) drew on his knowledge of logarithms to five places of the first 150 integers

Series Modification: Modify the first term in the series expansion, where the 0.8686 is omitted if calculating ln (N+a).

where 0.8686 = 2x0.4343 and may be calculated as 0.86 (1 + 0.01) if desired (add final result shifted right by two places). If a = 1 this will be accurate to 5 decimal places for any N>20.

log 1211 ≈ log 1210 + 0.8686 (1/2421) = 3.08279 + 0.8686 / (9 x 269) = 3.08314 Actual Value: 3.08314

log 62 ≈ log 64 - 0.8686 (2/126) for N = 64 and a = -2log 64 = log 26 = 6 (0.30103) = 1.80609–0.8686 (2/126) = – (1 + 0.01)x0.86/63

= – (1 + 0.01)x0.01365 = –0.0137so log 62 ≈ 1.80618 – 0.01379 = 1.79239 Actual Value: 1.79239

NO

VE

MB

ER

09 10

16 17

23 24

30

11 12

18 19

25 26

06 07

13 14

20 21

27 28

08

15

22

29

02 03 04 0501

December 2011

S M T W T F S1 2 3

4 5 6 7 8 9 1011 12 13 14 15 16 1718 19 20 21 22 23 2425 26 27 28 29 30 31

28 29 30 30

31 32 33

20

13

06

0128

03

0.903091.591061.83885

June 82000

(14)

0.301031.518511.79934

Mar 21918

(22)

0.477121.531481.80618

July 31811(09)

0.954241.602061.84510

Apr 91896

(22)

1.01.612781.85126

Aug 101795

(00)

1.176091.662761.88081

Sep 151854

(16)

1.204121.672101.88649

Nov 162016

(14)

1.230451.681241.89209

Feb 171846

(03)

0.845101.579781.83251

Jan 71930

(13)

1.146131.653211.87506

Oct 141905

(10)

1.322221.716001.91381

June 211922

(10)

0.778151.568201.82607

Dec 61769

(15)

1.113941.643451.86923

May 131791

(19)

1.301031.707571.90849

Aug 201851

(10)

1.342421.724281.91908

Feb 221782

(19)

1.431361.763431.94448

Dec 271913

(15)

1.447161.770851.94939

Apr 281764

(05)

1.380211.740361.92942

Dec 241757

(09)

1.397941.748191.93450

Nov 252035

(20)

1.414971.755871.93952

Aug 261971

(05)

1.255271.690201.89763

June 181952

(12)

1.278751.698971.90309

Sep 191872

(12)

0.602061.544071.81291

Oct 42026

(14)

0.698971.556301.81954

Jan 51916

(16)

1.041391.623251.85733

July 111928

(19)

1.079181.633471.86332

Feb 122009

(09)

1.361731.732391.92428

July 232023

(22)

1.462401.778151.95424

Nov 292087

(13)

1.477121.785331.95904

Oct 301847

(21)

0.01.505151.79239

Sep 11819

(11)

October 2011

S M T W T F S1

2 3 4 5 6 7 89 10 11 12 13 14 15

16 17 18 19 20 21 2223 24 25 26 27 28 2930 31


Exponentials

Add .31: 0.16 + 0.31 = 0.4710^( 0.47) = 2.95121

Add .61: 0.16 + 0.61 = 0.7710 (̂0.77) = 5.88844

10^(0.16) = 1.44544

16

10

Legend

The boxes contain exponentials (in powers of 10) for the connected dates following a decimal point.





3. 10 (̂0.3220) = 2.09894 (answer in box)

Single-Date Rules


16 + 07 = 23, 2 + 3 = 5Jan 16, 1925, was a Friday

10 + 17 = 272 + 7 = 9


1 Mon

2 Tue

3 Wed

4 Thu

5 Fri

6 Sat

9 Sun

Two-Date Rules

10^(0.10)10^(0.41)10^(0.71)

Nov 101799

(17)

10^(0.16)10^(0.47)10^(0.77)

Jan 161925

(07)

Calculating exponentials involves raising a number to a power, such as 102.85251 = 712.049… The value eN is often a solution to mathematical or physical equations, but this is a simple extension of finding 10N since eN = 10(0.4343…) N.

We can find a higher-order root r of a number N as 10log(N)/r. So to find the 11th root of 513, we find log 513 = 2.71012, divide by 11 to get 0.24637, and find 100.24637 = 1.76350

Use creativity! We may know that log 7 = 0.84510,so 102.85251 102 x 100.00741 and we are already at Step 3!

Bemer Method

McIntosh-DoerflerMethod

Here you only need to know log 3 and log 2!

log M M

.04139 11/10

.08715 11/9

.13830 11/8

.19629 11/7

.26324 11/6

.34242 11/5

.43933 11/4

log M M

.011 40/39

.03 15/14

.05 46/41

.06 31/27

.07 47/40

.15 89/63

.20 84/53

.27 54/29

.30103 2

.34 35/16

.45 31/11

.47712 3

Good at memorization? You can extract logarithms of conveniently multiplied and divided fractions from these tables.

Step 2: Add or subtract up to two copies of log 3 = 0.47712 to make thenumber close to a multiple of 0.1. You can determine the right number of copies by looking at the second and third digits of the number, where adding log 3 is like subtracting 23 in those digits, and vice versa.

–0.14749 + 2 log 3 = 0.80675 where 47 is nearly 2x23

Step 3: Add or subtract up to five copies of log 2 = .30103 to make the decimal part close to zero, and separate out the integer part. Notice how easy it is to multiply by log 2 since log 2 = 0.3010300!

0.80675 + 4 log 2 = 2 + 0.01087 since 4 (0.30103) = 1.20412

Step 4: Calculate the following for our remainder b, where 3 digits are sufficient for the last term (omit it completely for less accuracy):

1 + 2.3b (1 + 0.001) + (2.3b)2/2

1 + 2.3(0.01087)(1 + 0.001) + (2.3x0.011)2/2 = 1 + 0.02503 + 0.00032 = 1.02535

Step 5: Undo Steps 2 and 3. Divide if we added, multiply if we subtracted. Perform divisions last. We added 2 log 3, so we divide by 32. We added 4 log 2 so we divide by 24 = 16. Then shift the decimal point based on the integers we extracted along the way:

102.85251 ≈ 103+2 x 1.02535 / (9x16) = 105 x 0.0071205= 712.049 Actual Value: 712.049

Step 2: Add or subtract memorized logarithms of lownumbers or convenient fractions from the remaining exponent to reduce it to near zero (within ±0.019 if 5-digit accuracy is desired). Subtracting a logarithm will flip the multiplying number or fraction.

10-.14749 = 7/11 x 100.04880 since -0.14749 + 0.19629 = 0.04880= 7/11 x 11/10 x 100.00741 since 0.04880 - 0.04139 = 0.00741

Using the fractions in the 11/n table often cancels terms, as here:

10-.14749 = 7/10 x 100.00741

Step 3: Calculate the following for our remainder b, where 3 digits are sufficient for the last term (omit it completely for less accuracy):

1 + 2.3b (1 + 0.001) + (2.3b)2/2

1 + 2.3(0.00741)(1 + 0.001) + (2.3x0.0074)2/2= 1 + (0.01704 + 0.000017) + 0.00014 = 1.01721

Step 4: Multiply the results:

102.85251 ≈ 103 x 7/10 x 1.01721 = 102 x 7 x 1.01721= 712.047 Actual Value: 712.049

Step 1: Extract powers of 10 from N (by subtracting or adding integers) to leave the smallest difference from zero:

102.85251 103 x 10-0.14749

In the calendar, all exponents lie between 0 and 1, so 101 is extracted if greater than 0.5.

George Parker Bidder (1806-1878) was a calculating prodigy who in 1856 described his method of mentally calculating compound interest.

DE

CE

MB

ER

07 08

14 15

21 22

28

09 10

16 17

23 24

04 05

11 12

18 19

25 26

06

13

20

27 31

01 02 03

29 30

January 2012

S M T W T F S1 2 3 4 5 6 78 9 10 11 12 13 14

15 16 17 18 19 20 2122 23 24 25 26 27 2829 30 31

26 27 28 28 29 30 31

18

11

04

01

25

1.148152.344234.67735

Oct 61777(04)

1.023292.089304.16869

Apr 11914

(11)

1.174902.398834.78630

Dec 71941

(29)

1.202262.454714.89779

Feb 81801

(19)

1.348962.754235.49541

Oct 131785

(09)

1.380382.818385.62341

Mar 142008

(18)

1.412542.884035.75440

Aug 151947

(08)

1.122022.290874.57088

May 52055

(16)

1.318262.691535.37032

Aug 121852

(19)

1.548823.162286.30957

June 191955

(08)

1.096482.238724.46684

Sep 41856

(18)

1.288252.630275.24807

Nov 111900

(16)

1.513563.090306.16595

Feb 181770

(18)

1.584893.235946.45654

Oct 202023

(21)

1.778283.630787.24436

Dec 251800

(06)

1.819703.715357.41310

Jan 262048

(19)

1.659593.388446.76083

Jan 221902

(08)

1.698243.467376.91831

Sep 231790

(08)

1.737803.548137.07946

July 242064

(07)

1.445442.951215.88844

Dec 161823

(04)

1.479113.019956.02560

Oct 172040

(13)

1.047132.137964.26580

Jan 21890

(29)

1.071522.187764.36516

Aug 32015

(07)

1.230272.511895.01187

May 92031

(23)

1.258932.570405.12861

July 101883

(10)

1.621813.311316.60693

Apr 211849

(12)

1.862093.801897.58578

July 271859

(03)

1.905463.890457.76247

Jan 281774

(13)

1.949843.981077.94328

Sep 291950

(03)

1.995264.073808.12831

Mar 301833

(21)

2.041744.168698.31764

Dec 312099

(09)

November 2011

S M T W T F S1 2 3 4 5

6 7 8 9 10 11 1213 14 15 16 17 18 1920 21 22 23 24 25 2627 28 29 30


For additional details on advanced topics of mental calculation, please see

“Lightning Calculators I -III” essays at http://www.myreckonings.com/wordpress

Benjamin, Arthur. Secrets of Mental Math: The Mathemagician's Guide toLightning Calculation and Amazing Math Tricks(2006).

Doerfler, Ronald. Dead Reckoning: Calculating Without Instruments(1993).

Lane, George. Mind Games: Amazing Mental Arithmetic Tricks Made Easy (2004).

Lightning calculators possess startling abilities to mentally compute products, quotients, powers, roots, and sometimes functions such as logarithms and exponentials. This calendar presents methods used by these individuals, along with daily exercises for fun and practice.

Lightning Calculators

The study of lightning calculators of the past and present is fascinating from more than a mathematical aspect. Many presentations, particularly in the popular media, ascribe abilities in these areas to mysterious machinations in the minds of remote geniuses, which makes for a good story but can be discouraging. In fact, these individuals through talent and training acquired a knack for racing headlong through calculations that can often seem mysterious to the uninformed.

Other than rough estimation, techniques ofmental calculation are not being taught in ourschools today. Yet presentations on even themost basic methods of mental calculation aremet with incredible interest among people.This calendar attempts to address that need.

Mental calculation can be a highly creative and satisfying endeavor offering a variety of interesting strategies, many more than most people realize. It is a skill that engages both children and adults, and one that naturally leads to a real familiarity with the properties and relationships of numbers. It provides a useful and fun approach for developing a number sense and generating a true appreciation for the elegance of elementary mathematics. It is an art as fundamentally important as other areas of mathematics.

For information on obtaining this calendar, including a free PDF download to create it on your own printer, visit http://www.myreckonings.com/wordpress

© 2010 Ron Doerfler All Rights Reserved Lightning front cover photo by John A. CobbLightning back cover photo by Erica Burrell

2011 Lightning Calculation Calendar

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