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1. Quick Multiplication

2. Squaring:

• To learn how to find the square of a number (having any number of digits) we first learn how to find the duplex of any number.

• To find the duplex of any number we need to remember the rules as follows: 1. If Single digit is encountered; we simply find its square to get the

duplex. 2. If two digits are encountered; we first find the product of the digits,

and then double it to get the duplex. 3. If we encounter more than two digits then we decompose it into

Single and Two digits and then find the duplex by using the above rules, finally we add the all to get the duplex of the number.

• Observe the following examples:

Number Remark Duplex 2 Single Digit 4 3 Single Digit 9 9 Single Digit 81 32 Two Digit 2×(3×2)=12 45 Two Digit 2×(4×5)=40 123 Three Digit 2× (1×3)+4=10 1234 Four Digit 2× (1×4)+2× (2×3)=20 22345 Five Digit 2× (2×5)+2× (2×4)+9=45

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• Now we will see how to find the square of nay number by using the duplex.

1. Get the number 2. Group the digits of the number from left to right first in

incremented way and then in decremented way. 3. Find the duplex of each group individually. 4. Add the duplex to get the square of the number.

• Examples:

3. Cube:

• The rule which we are going to discuss for finding the cube; is applicable only in case of two digit number. With the help of this rule we can find the cube of only two digit numbers.

• Rule: 1. Obtain the number; say ab. 2. In first row write the cube of a. 3. Then write three more numbers in the same row; which will be b/a of

the previous one. 4. Now in the second row write the double of the middle numbers of the

first row; just below the numbers. 5. Now add column-wise to get the cube of ab.

• Example

5499025

25/40/46/44/25/12/4

5/45/345/2345/234/23/223452

===

536028979673214

1443621 2 =

1 6 7 6 8 2 2 2 5 5 1 5 295432819

8 2 14 6 12 12 16 10 11 5 5 4 2 =

12167

7 6 1 12

------------------------

36 24

17 18 12 8

?23

254

3

=

=

804357

7 5 3 804

---------------------------

162 486

27 81 243 729

?93

22475

3

=

=

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4. Square Roots:

5. Cube Roots:

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6. Inequalities:

7. The Standard Order of Operations (Operator Precedence):

• The order of operations, or precedence, used in mathematics is expressed

here:

1. Brackets (first “( )”, then” { }” and lastly ”[ ]”) 2. Exponents and Roots 3. Multiplication (the word ‘of ’ also means multiply) and Division � as

they appear left to right 4. Addition and Subtraction � as they appear left to right

• This means that if a mathematical expression is preceded by one operator and followed by another, the operator higher on the list should be applied

first. • This order of precedence can be remembered by the word BEODMAS,

standing for Brackets, Exponents/Roots, Of, Division, Multiplication, Addition and Subtraction.

• The basic laws governing the use of brackets and operators are shown by the following examples:

1. 2+3=3+2, i.e. the order of numbers when adding does not matter; 2. 2×3=3×2, i.e. the order of numbers when multiplying does not matter; 3. 2+(3+4)=(2+3)+4, i.e. the use of brackets when adding does not affect

the result; 4. 2×(3×4)=(2×3)×4, i.e. the use of brackets when multiplying does not

affect the result;

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5. 2×(3+4)=2(3+4)=2×3+2×4, i.e. a number placed outside of a bracket indicates that the whole contents of the bracket must be multiplied by that number;

6. (2+3)(4+5)=(5)(9)=45, i.e. adjacent brackets indicate multiplication; 7. 2[3 + (4 × 5)] = 2[3 + 20] = 2 × 23 = 46, i.e. when an expression

contains inner and outer brackets, the inner brackets are removed first.

8. Laws of indices:

• When simplifying calculations involving indices, certain basic rules or laws can be applied, called the laws of indices.

• These are given below. • When multiplying two or more numbers having the same base, the indices are

added. Thus 32×34=32+4=36

• When a number is divided by a number having the same base, the indices are subtracted. Thus

35÷32=35−2=33 • When a number which is raised to a power is raised to a further power, the

indices are multiplied. Thus (35)2=35×2=310

• When a number has an index of 0, its value is 1.Thus 30 =1

• A number raised to a negative power is the reciprocal of that number raised to a positive power. Thus

3−4 = 1/34 Similarly, 1/2−3 =23

• When a number is raised to a fractional power the denominator of the fraction is the root of the number and the numerator is the power. Thus

82/3 = 3√82 = (2)2 =4 and 251/2 = 2√251=√251 =±5 (Note that√ ≡ 2√ )

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9. Divisibility Rules:

Divisibility

By

Rule Example

2 Last digit is either 0 or even 348, 49734 3 Sum of digit is divisible by 3 123, 5673 4 Two or more zeros at the end OR

Last two digit is divisible by 4 5300, 526428

5 Number ending with either 0 or 5 1345. 3450 6 If divisible by both 2 and 3 63924, 261 7 When (first (n-1) digit – 2 × last digit) is divisible

by 7 (apply it repeatedly until we get a comparatively smaller number divisible by 7 )

112, 2961, 55272

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8 Last three digits are 0 OR Last three digit is divisible by 8

1256, 1397000

9 Sum of the digits is divisible by 9 39681, 456138 10 Ends with 0 330, 8780 11 Sum of digits at even and odd places are equal OR

differ by a number divisible by 11 3245682, 84927291658

12 If divisible by both 4 & 3 135792 13 When (first (n-1) digit + 4 × last digit) is divisible

by 13 143

14 Even number which is divisible by 7 55272 15 Divisible by both 3 and 5 16 Last 4 digit is divisible by 16 17 When (first (n-1) digit – 5 × last digit) is divisible

by 17 (apply it repeatedly until we get a comparatively smaller number divisible by 17 )

95750

18 Last digit is even (or 0) and divisible by 9 and 926568 19 When (first (n-1) digit + 2 × last digit) is divisible

by 13 149264

Note: Application of divisibility rules for the numbers 7, 13, 17 and 19 should be

avoided because it takes more time than actual checking for the divisibility by

division method.