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Basic Arithmetic
Number Sets
Author: Raghu M.D.
Contents
NUMBER SETS ........................................................................................................... 2
SQUARES AND SQUARE NUMBERS.................................................................................................... 2
SQUARE ROOTS .................................................................................................................................. 8
CUBES................................................................................................................................................ 13
CUBE ROOTS ..................................................................................................................................... 16
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NUMBER SETS
Introduction:
Numbers can be classified as Integers, Whole numbers and Natural number sets. These sets
of numbers are discrete, meaning each number in a set is more or one less than the next
number in the set. Quotients are ratios of any two discrete numbers and are elements of
another set. Number sets can be represented by a line, generally called a number line. A
number line has all the other numbers sets as subsets.
I
N
-5
0
5
W
Fig 1.1 An integer line showing
(a) Natural numbers
(b) Whole numbers
(c) Integers
N
W
I
1, 2, 3……..
0, 1, 2, 3………
-5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5………….
Quotients can be any point on the line examples
3 7
−4
……..
, ,.........
7 5
2
SQUARES AND SQUARE NUMBERS
An integer n is said to be a square number if it is equal to the product of an integer multiplied
by itself.
∴n = m × m = m 2 (n and m are integers)
A square root is the inverse of a square
If n = m2
then n = m
If m is an integer then n is a square number
1, 4, 9, 16, 25……….are all square numbers or perfect squares.
Properties of square numbers:
1. Unit digit of a square number can be 0, 1, 4, 5, 6 or 9, in other words, a square number
can only end in 0, 1, 4, 5, 6 or 9 and not in 2, 3, 7, 8.
2. A square number can not be negative
3. If m2 = n, where n is a square number and if p is a prime number then p×n is not a
square number. If p×n = m2, m = p × n and p cannot be exactly square rooted as it
has no factors.
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4. Product of two square numbers is also a square number. For example, 4 and 9 are
square numbers. Hence 4 × 9 = 36, where 36 is also a square number having a square
root of 6.
5. A square number or a perfect square when divided by 3 leaves a reminder of 1 or 0
For example: Given 49 = 72
16
We have 3) 49
48
r1
6. A square number can be expressed as a set of dots arranged in equal number of rows
and columns. Every column or row has the same number of dots
For example 9 which is 32 can be drawn as
a pattern or 9 dots arranged in 3 rows and 3
columns with 3 dots in each row or columns
7. Square of an even number is even and an odd number is odd. For example
62 = 36
92 = 81
Column method:
A simpler method of finding the square of a number is to use the formula
(a+b)2 = a2 + 2ab + b2
For example 272 = (20 + 7)2 = 202 + 2 × 20 × 7 + 72
= 400 + 280 + 49
= 729
This method can be improved to find 312
302
2 × 30 × 1
12
900
62
1
312 = 963
Square diagonal method
This method is useful for finding squares of numbers with 3 or more digits. In this method a
square table is drawn. This square is divided into square cells further divided into two parts
by a diagonal. Each part represents a digit. For example 4562 is represented below.
4
4
5
6
5
1
2
6
2
2
0
2
0
2
4
3
5
3
4
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6
0
3
0
6
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4562 is calculated as shown below add numbers enclosed by diagonals as shown.
1, 2+6+2, 2+0+2+0+2, 4+3+5+3+4, 0+3+0, 6
Adding these numbers within a pair of diagonals or diagonal and a corner is shown below.
1, 10, 6, 19, 3, 6
Please note that number 10 is written as 1 followed by 0 underlined to facilitate easy addition.
1
+1
2
0 6
0 1
0 7
9 3 6
0 0 0
9 3 6
∴ 4562 = 207936
Add and separate the unit digit as shown by an underline, write the tens digits one space
moved to the left in the second line. Add the two lines of numbers to get the answer as square
of 456 which is equal to 207936.
Example 1: Find the squares of 21, 411 and 78
Working:
a) 212 = 21 × 21
Answer: 212 = 441
b) 4112 = 411 × 411
411 × 411
411
411
1644
168921
Answer: 4112 = 168921
c) 782 = 78 × 78
78 × 78
624
546
6084
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Answer: 782 = 6084
Example 2: Observe the pattern and fill up the missing number
112 = 121
1112 = 12321
11112 = 1234321
111112 = ______________
Ans: 123454321
Example 3: Find 672 using (a + b)2 or column method
672 = (60+7)2
672 =
a2
+
2ab
+
b2
602
+
2x6x7
+
72
840
+
49
3600 +
= 4489
Answer: 672 = 4489
Column method
a2
2 × a × b b2
6x6 2×6×7 7×7
36
84
49
8
4
44
8
9
Note: 8 is carried over from 84 to the first column and 4 is carried over from 49 to the second
column.
Answer: 672 = 4489
Example 4: Find 352 using the square diagonal method
3
0
5
2
0, 1+9+2, 5+2+5, 5
0, 12, 12, 5
9
1
5
2
00
01
01
5
5
3
5
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12
01
3
12
00
2
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05
05
5
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Note: 1 is carried over from 12 to the first column and 1 is carried over from 12 to the second
column.
Answer: 352 = 1325
Example 5: Using the remainder method to identify the numbers which are perfect squares
a) 81
b) 7396
c) 65
d) 4226
Working:
a) Divide 81 by 3
27
3) 81
81
γ 00
Answer: Since the remainder is 0, the number 81 is a perfect square.
b) Divide 7396 by 3
213
3) 7396
6
396
3
96
9
6
6
γ0
Answer: Since the remainder is 0, the number 7396 is a perfect square.
c) Divide 65 by 3
22
3) 65
6
05
3
γ2
Answer: Since the remainder is 2, the number 65 is not a square.
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Note: For a perfect square, the remainder has to be 0 or 1.
d) Divide 4226 by 3
1408
3) 4226
3
12
12
002
0006
0026
0024
γ 2
Answer: Since the remainder is 2, the number 4226 is not a square.
EXERCISE 1.1
1.
Find the squares of the following numbers
a) 31 b) 97 c) 415 d) 100 e) 1278
2.
Using the remainder method identify square numbers from the list
a) 625 b) 97 c) 12321 d) 10,000 e) 148
3.
State True or False
a) Square of an odd number is odd number
b) Square of an even number is odd number
c) Square of a negative number is a negative number
d) Square of a prime number is a prime number
e) Square of the length of a square is equal to its area
4.
Observe the following pattern and find the missing square numbers
112 = 121
1112 = ________
11112 = 1234321
5.
Observe the following pattern and find the missing numbers.
12 + 22 + 22 = 32………………… (1×2=1 and 2+1=3)
22 + 32 + 62 = 72………………… (2×3=6 and 6+1=7)
32 + 42 + 122 =
52 + 62 + 302 =
6.
Using the column method find the squares of
a) 79 b) 31 c) 53 d) 22 e) 17
7.
List all the square numbers in between the following numbers
a) 50 to 60 b) 60 to 90 c) 90 to160 d) 160 to 250 e) 250 to 350
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8.
Using the identity (a+b)2 = a2+2ab+b2 find the square of
a) 21 b) 12 c) 32 d) 101 e) 501
9.
Using the identity (a—b)2 = a2 – 2ab + b2, find the squares of
a) 99 b) 28 c) 38 d) 998 e) 199
10. Using square diagonal method find the squares of
a) 586 b) 35 c) 21 d) 145 e) 236
Answers to EXERCISE 1.1
1.
a) 961
b) 9409
c) 172225
2.
a), c), d) are square numbers, others are not square numbers.
3.
a) True
4.
12321
5.
The missing numbers are: 132 and 312
6.
a) 6241
7.
a) Nil
b) 64, 81
8.
a) 441
b) 144
9.
a) 9801
10.
a) 343396
b) False
c) False
b) 1961
b) 784
c) 2809
d) 10000
d) False
d) 484
e) True
e) 289
c) 100, 121, 144
c) 1024
d) 10201
c) 1444
b) 1225
c) 441
e) 1633284
d) 169, 225
e) 256, 289, 324
e) 251001
d) 996004
d) 21025
e) 39601
e) 55696
SQUARE ROOTS
If n=m2, then n = m, or more precisely n = ± m , because –m×–m=m2 as well as m×m=m2.
n is called the square root of n or a radical. Only perfect squares have integers as square
roots.
Properties:
1) If n is not a perfect square, then its square root is not an integer. For example, 10 is
not a perfect square, hence 10 cannot be an integer. Its value can only be found
approximately.
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2) Only numbers with unit digits 0, 1, 4, 5, 6 and 9 can have integers as square roots, for
example, 25 =5 but 23 is not an integer. However all numbers not ending with 0,
1, 4, 5, 6 and 9 do not have integer square roots.
3) Square root of an even number is even and an odd number is odd.
Square roots by inspection:
Square roots of 2 or 3 digit numbers can be found by inspection or trial and error. For
example, 81 is 9, because 9×9 = 81.
Table of Squares
Number
Square
1
1
2
4
3
9
4
16
5
25
6
36
7
49
8
64
9
81
10
100
∴ 81 = 9
Square roots by progression rule:
A square root of a number n=m2 is given by the number of terms, which will be in fact m
terms, in the progression of successive odd numbers, when added is
n = m2 = 1+3+………. (2m-1) is equal to m2, where (2m-1) is the last term of the
progression.
For example 16 can be calculated by the progression 16 = 1+3+5+7 which has 4 terms
16 = m = 4
The above method can be modified for calculating the square roots of numbers only.
For example 25 can be obtained by reducing 25 by successive odd numbers till it becomes
zero. The number of times the subtraction is carried out gives the square root.
25-1=24, 24-3=21, 21-5=16, 16-7=9, 9-9=0
It has taken 5 steps to reduce 25 to 0, hence 25 = 5
Factorization method:
Any composite number can be written as a product of its prime factors.
For example 36 = 2 × 2 × 3 × 3
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Or = (2 × 3) × (2 × 3) (rearranging the factors as similar pairs)
Or = 6 × 6
Hence 36 = 6
In a composite number, if the prime factors cannot be paired, then that number is not a
perfect square.
Division method:
In this method the square root of any number can be found. Answers can be integers or
decimals. For example, consider 99856 which is split as 9 98 56 and the following steps are
carried out.
Working:
Step 1: The first digit (or pair if the number has even digits) is to be divided by a number
whose square is less than or equal to 9.
1)
3
3 99856
9
0
Step 2: Drop the next pair 98 and for the divisor, add 3 to itself to give 6 which becomes the
first digit of the divisor. The second digit is the smallest number in this case 1, because 61 × 1
is just less then 98. (Note: 61 × 2 = 122 > 98). As it is the case, use 61 to divide 98. Digit 1 is
the quotient from this operation and 37 the remainder.
316
3 99856
9
61 098
61
626 3756
3756
0000
Step 3: Drop 56 down and 3756 is the number to be divided. Add the last digit of 61 to itself,
that is, 61+1=62. Now 6 and 2 are the first two digits of the divisor. For the third digit select a
number in this case 6, such that 626 x 6 ≤ 3756. Since 626 x 6 = 3756, there is no remainder.
The digit 6 is included in the quotient. Now 316 is the square root of the number.
Answer:
99856 = 316
Example 1: Find the square roots of the following numbers by inspection or referring to the
table of squares. a) 36 b) 64 c) 144
Answers from table of squares:
∴ √36 = 6
62 = 36
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82 = 64
By inspection
122 = 12×12 = 144
∴ √64 = 8
∴ √144 = 8
Example 2: State if the square roots of the following numbers are odd or even
a) 400 b) 729 c) 169
Answers:
a) 400 is even, hence
400 is even ( 400 =20 )
b) 729 is odd, hence
729 is odd ( 729 =27 )
c) 169 is odd, hence
169 is odd ( 169 = 13 )
Example 3: Find the square root of 1089 using factorization method
3 1089
3 363
11 121
11
1089=3 × 3 × 11 × 11
1089 = √3 × 3 × 11 × 11
∴ 1089 = 3 x 11
Answer: 1089 = 33
Example 4: Using the division method find the square root of 841
29
2 841
4
49 441
441
000
Answer:
841 =29
Example 5: Find the square root of 6.25
625
100
625 25
=
100 10
6.25 =
∴ √6.25 =
Answer: √6.25 = 2.5
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EXERCISE 1.2
1.
Find by inspection, or from the table, the square roots of the following:
a) 121 b) 625 c) 1225
2.
If the unit digit of a number is 5, what will be the unit digit of its square root?
3.
Using the progression method (1+3+……… (2m-1)) = m2 = n find the square roots of
a) 49 b) 144
4.
Using prime factorization method find the smallest number that divides 2700 to give a
quotient that has an integer as the square root.
5.
If the product of 16 and 49 is 784, find the square root of 784.
6.
Area of a square is 49m2. (Area a = l2, where l is the length of side). Find the length
of its side.
7.
Find the square root of the following numbers using division method:
a) 1849 b) 2209 c) 361
8.
Find the square roots of following rational numbers:
49
17
a)
b) 9.61 c) 1
36
64
9.
Find the nearest perfect square to the following numbers:
a) 1448
b) 126
c) 3481
(For example, 9 is the nearest perfect square to 10)
10.
Find the square roots of the following numbers correct to two decimal places:
a) 10 b) 2 c) 312
Answers to EXERCISE 1.2
1.
a) 11
2.
5
3.
a) 7
4.
3
5.
28
6.
7m
7.
a) 43
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b) 25
c) 35
b) 12
b) 47
c) 19
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8.
a)
b) 3.1
9.
a) 4
b) 5
10.
a) 3.16
c)
c) 59
b) 1.41
c) 17.66
CUBES
If n = m³ = m × m × m then n is said to be the cube of m. In fact m3 is the volume of a cube
of edge length m.
The cube of a number can be simply obtained by multiplying the number by itself twice.
For example 63 = (6 x 6) x 6 = 36 x 6 = 216
Properties
1: Cube of an even number is even and odd number is odd.
2: Cube of a negative number is negative and positive number is positive.
3: Cube of a number is the product of cubes of its factors.
For example 63 = (2 x 3)3 = 23 x 33 = 8 x 27 = 216
Identity Method
Cube of a two-digit number can be found by using the identity:
(a+b)3 = a3 + 3a2b + 3ab2 + b3
For example: 313
Working:
(31)3 = (30 +1)3
= 303 + 3 x 302 x 1 + 3 x 30 x 12 + 13
= 27000 + 2700 + 90 + 1
= 29791
Column Method
For example: 563
Number 56 can be split as 50 + 6, t=50 and u=6
Working:
t3
3 t2 u
3 t u2
u3
503
3×502×6
3×50×62
63
125000
45000
5400
216
Hence (56)3 = 125000 + 45000 + 5400 + 216 = 175616
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Example 1: Write the unit digit of 2173
Unit digit of 2173 is the last digit of 7 × 7 × 7 = 343
Answer: 3
Example 2: Find 113 by multiplication
113 = 11 × 11 × 11
= (11 × 11) × 11
= 121 × 11
= 1331
Answer: 113 = 1331
Example 3: Find 423 using identity method.
423
= (40+2)2, a=40 and b=2
(a + b) 3 = a3 + 3a2b + 3ab2 + b3
(40 + 2)3 = 403 + 3 × 402 × 2 + 3 x 40 x 22 +23
= (40 × 40 × 40) + 3 × (40 × 40) × 2 + 3 × 40 × (2 × 2) + (2 × 2 × 2)
= 64000 + 9600 + 480 + 8
Answer: 423 = 74088
Example 4: Find 683 using column method
Here t=60, u=8
t3
3 t2 u
3 t u2
u3
603
3×602×8
3×60×82
83
216000
86400
11520
512
Hence (68)3 = 216000 + 86400 + 11520 + 512 = 314432
Answer: 682=314432
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Exercise 1.3
1.
Find the cubes of the following numbers by multiplication.
a) 8
b) 13 c) 20
2.
Find the cubes of the following numbers using the identity:
(a+b) 3 =a 3 +3a 2 b + 3ab 2 + b 3
a) 101
b) 72
c) 23
3.
Find the cubes of the following numbers using column method
a) 31 b) 42 c) 53
4.
A cube of ice has length of its edge = 8cms. Find the volume of the ice cube.
5.
State true or false
a) 343 is perfect cube
b) Cube of a number with 2 as the unit digit is 6
c) Cube of a negative number is positive
d) 73 > 63 + 13
6.
Given 63 =216 and 33 = 27, find 183
7.
 11 
Find  
5
3
8.
Find the cube of the following numbers:
a) -6 b) +5 and c) -30
9.
Write the unit digits of the cube of the following numbers:
a) 137137 b) 24 c) 398
10.
1729 is called Ramanujam number. Given 1729=123+b3 and 1729=c3+93 find b and c.
Answers to EXERCISE 1.3
1.
a) 512
2.
a) 1030301
3.
a) 29791
4.
512 cm3
5.
a) true
6.
5832
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b) 2197
c) 8000
b) 343248
b) 74088
b) false
c) 12167
c) 148877
c) false
d) true
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7.
8.
a) -216
b) 125
9.
a) 3
10.
b = 1, c = 10
b) 4
c) -27000
c) 2
CUBE ROOTS
If n=m3 , then m = √
A perfect cube has an integer as its cube root.
Properties
1: Cube root of an odd number is odd and even number is even
2: Cube root of a positive number is positive and negative number is negative
3: Cube root of the product of two numbers is equal to the product of the cube roots of the
individual numbers.
= √
×
Inspection method
Cube root can be found by inspection or by trial and error. Perfect cubes are only a few.
These are only 4 perfect two digit cubes and10 perfect three digit cubes.
A table of perfect cubes can be used for obtaining the cube root.
Table of Cubes
Number
Cube
1
1
2
8
3
27
4
64
5
125
6
216
7
343
8
512
9
729
10
1000
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Prime factorization method
Any composite number can be factorized into a set of prime numbers. If the number is a
perfect cube, it will contain 3 (or multiples of 3) sets of prime numbers. The example below
illustrates this method.
For example: √216
Factorize 216
2 216
2 108
2 54
3 27
3 9
3
Hence 216 = 2 × 2 × 2 × 3 × 3 × 3
√216 = 2 × 3 = 6
Answer: √216 = 6
Example1: Find the cube root of 125
Unit digit of 125 is 5, hence 5 is the possible cube root.
(Note: This assumption may not be valid for all cubes).
Verification: 5 × 5 × 5 = 125
Answer: √125 = 5
Example 2: Find √5832 by prime factorization
2 5832
2 2966
2 1458
3 729
3 243
3 81
3 27
3 9
3
Hence 5832 = 2 × 2 × 2 × 3 × 3 × 3 × 3 × 3 × 3
√5832 = 2 × 3 × 3 = 18
Answer: √5832 = 18
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Example 3: Find √8 × 216
√8 × 216 = √8 × √216
= 2 × 6
= 12
Answer: √8 × 216 = 12
Example 4: Find the smallest number needed to divide 1024 to get an integer as the cube
root.
1024 = 2 × 512
From the table by inspection, 512 is perfect cube and 8 is its cube root.
Hence 2 is the smallest number needed to divide 1024.
Answer: 2
Example 5: Find √−0.343
√−0.343 =
√
√
−7
+ 10
= - 0.7
=
(from the table by inspection)
Answer: - 0.7
EXERCISE 1.4
1.
Find the cube roots of
a) 8
b) 729
c) 1000
2.
By prime factorization find the cube roots of
a) 343
b) 9261
c) 21952
3.
By prime factorization check if the following numbers are perfect cubes
a) 64
b) 72
c) 140
4.
Volume of a cube is 216 cm2. Find length of its edge
5.
Find the cube root of following negative numbers
a) -1728
b) -125
c) -3375
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6.
If 63 = 216 and 83=512, find the cube root of 110592
7.
Find the unit digits of the cube roots of following perfect cubes
a) 12167
b) 8
c) 6859
8.
Find the cube roots of following decimals
a) 0.125
b) 0.008 c)-0.729
9.
Find the cube roots of following Quotients
216
512
16
a)
b)
c)
343
1331
1024
10.
Given a3 – b3 = (a-b) (a2 + ab + b2), find 43 - 33
Answers to EXERCISE 1.4
1.
a) 2
b) 9
c) 10
2.
a) 7
b) 21
c) 28
3.
a) Yes
4.
6 cm
5.
a) - 12
6.
48
7.
a) 23
b) 2
c) 19
8.
a) 0.5
b) 0.2
c) -0.9
9.
a)
10.
37
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b) No
c) No
b) - 5
b)
c) - 15
c)
or
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