Xmania!

What day is your birthday?

Think of the DATE you were born, but don’t say it out loud!

Card #1

1 3 5 7

9 11 13 15

17 19 21 23

25 27 29 31

Card #2

2 3 6 7

10 11 14 15

18 19 22 23

26 27 30 31Card #3

4 5 6 7

12 13 14 15

20 21 22 23

28 29 30 31Car

d #48 9 10 11

12 13 14 15

24 25 26 27

28 29 30 31

Card #5

16 17 18 19

20 21 22 23

24 25 26 27

28 29 30 31

What to expect…

• Learn some new things about our number system.

• Learn some stuff about other number systems.

• Learn some cool short-cuts that work for our number system.

• Learn how the Birthday cards work.

Let’s look at what we know:

• How many digits are there?

• How many numbers are there?

• Do we have to use 1,2,3… or can we use something else?

• Do we know any other number systems?

• When is 8 + 5 = 1?

10 digits

0א (infinitely many)

Any symbol will work.

Yes!

On a Clock!

So what is the value of --

34

Why is it not 7?

So we can count to 9 then we have to use another digit for 10.

1 2 3 4 5 6 7 8 9 10

Back in the day…

• Different groups used different symbols.

• Symbols could be a single value or different values (depending on where they were).

• Here’s some examples:

Here are a few the Egyptians used

So what’s their value?

© Mark Millmore 1997 - 2004

A Few Mayan Math Symbols

In Mayan Math

Thanks to: http://www.michielb.nl/maya/math.html

This is 1 This is 2

But this is 21

The Mayans had up and down place value!

Could we count with lights?

How?

So….

• If this is one:

• And this is two:

• Then the sum is:

O O O O xx O O O xx O

O O O x xx x

(1)

(10)

(11)

Lights, Lights, Lights!

Binary Number Light 5 Light 4 Light 3 Light 2 Light 1 (1’s and 0’s)

1. __ O O O O x ____1_______2. __ O O O x O ____10______3. __ O O O x x ____11______4. __ O O x O O ____100_____5. __ O O x O x ____101_____6. __ O O x x O ____110_____

What to remember:

1 is “on” 0 is “off”

What is the value of each 1?

1

Has a value of 1

What is the value of each 1?

Has a value of 2

One’s Place

10

What is the value of each 1?

Has a value of 4

One’s PlaceTwo’s Place

100

What is the value of each 1?

Has a value of 8

One’s PlaceTwo’s Place

Four’s Place

1000

So the value of this binary number would be

8

12

4

1111 = 8 + 4 + 2 + 1

= 15

So let’s double some numbers

101 11 111 100 1010

1010 110 1110 1000 10100

Is there a pattern?

Why does it work for doubling?

Is it similar to a pattern we use in our system?

So to double over and over…

• Add a zero each time you double

• So in our number system we would write 1 x 2 x 2 x 2 if we wanted to double the number 1 three times.

• The shortcut for that would be 1 x 23

• In binary that number would be…

• 1000 (a zero for each double!) Exponent

Try writing these answers in binary --

3 x 24

4 x 23

7 x 25

13 x 23

= 11

= 100

= 111

= 1101

3 is 11 so with four zeroes it would be…0000

00000000

000

Guess what uses the binary system?

So back to the Birthday Cards

• What is so special about the numbers on card #1?

• Look at your lights, lights, lights sheet and tell me if the numbers have something in common in binary.

• What about card #2? #3? #4? And #5?

Card #1

1 3 5 7

9 11 13 15

17 19 21 23

25 27 29 31

Card #2

2 3 6 7

10 11 14 15

18 19 22 23

26 27 30 31

Card #3

4 5 6 7

12 13 14 15

20 21 22 23

28 29 30 31

Card #4

8 9 10 11

12 13 14 15

24 25 26 27

28 29 30 31

Card #5

16 17 18 19

20 21 22 23

24 25 26 27

28 29 30 31

So our base 10 system has shortcuts too…

• If binary had a shortcut for doubling ( x 2) then our system has one for…

• x 10

• So if I want to multiply a number by ten all I have to do is _______ ?

• And if I want to multiply by ten twice or three times?

For Example

• 34 x 10 =• 723 x 104 =

• 9 x 107 =• 4,571 x 102 =• 500 x 103 =

• This is TOO easy!

340

7,230,000

90,000,000

457,100

500,000

Let’s look at a different number system --

Xmania

How do the Xmanians count?

Our Number System

Xmania

How is Xmania like our decimal system?

• Has a digit for zero.

• Uses place value (except they add digits to the left instead of the right).

• Has shortcuts for multiplying.

• _________________

• _________________

Now it’s your turn to

Your system should have:

• A name

• A digit for “zero”

• 3 or 4 digits total

• Place value• Multiplication shortcut (with explanation)

Let’s sum up!

• How are place valued number systems alike?

• What are the major differences?

• What are the shortcuts to our number system?

• Do the number shortcuts work with other number systems (like Xmania)?

Let’s sum up!

650,000

784,000

400,000

930

Here are a few for you to review:

65 x 104 =

784 x 103 =

4 x 105 =

93 x 10 =

Questions?Good-bye!