# Number System & Simplification

Natural Numbers

These are the numbers (1, 2, 3, etc.) that are used for counting.
It is denoted by N.
There are infinite natural numbers and the smallest natural number is one (1).

Even numbers

Natural numbers which are divisible by 2 are even numbers.
It is denoted by E.
E = 2, 4, 6, 8,...
Smallest even number is 2. There is no largest even number.

Odd numbers

Natural numbers which are not divisible by 2 are odd numbers.
It is denoted by O.
O = 1, 3, 5, 7, ...
Smallest odd number is 1.
There is no largest odd number.

Based on divisibility, there could be two types of natural numbers : Prime and Composite.
Prime Numbers

Natural numbers which have exactly two factors,
i.e., 1 and the number itself are called prime numbers.
The lowest prime number is 2.
2 is also the only even prime number.

Composite Numbers

It is a natural number that has atleast one divisor different from unity and itself.
Every composite number can be factorised into its prime factors.
For Example : 24 = 2 × 2 × 2 × 3. Hence, 24 is a composite number.
The smallest composite number is 4.

Whole Numbers

The natural numbers along with zero (0), form the system of whole numbers.
It is denoted by W.
There is no largest whole number and The smallest whole number is 0.

Integers

The number system consisting of natural numbers,their negative and zero is called integers.
It is denoted by Z or I.
The smallest and the largest integers cannot be determined.

The number line

The number line is a straight line between negative infinity on the left to positive infinity on the right.

Real Numbers

All numbers that can be represented on the number line are called real numbers.
It is denoted by R.
R+:Positive real numbers and
R–:Negative real numbers.
Real numbers = Rational numbers + Irrational numbers.

(A) Rational numbers

Any number that can be put in the form of p/q, where p and q are integers and q≠0, is called a rational number.
It is denoted by Q.
Every integer is a rational number.
Zero (0) is also a rational number.The smallest and largest rational numbers cannot be determined. Every fraction (and decimal fraction) is a rational number.

$Q = \frac{p}{q}\frac{(Numerator)}{(Denominator)}$

If x and y are two rational numbers, then (x + y)/2 is also a rational number and its value lies between the given two rational numbers x and y.

An infinite number of rational numbers can be determined between any two rational numbers.

Example 1

1. Find three rational numbers between 3 and 5.

Sol. 1st rational number = (3 + 5)/2 = 8/2 = 4

2nd rational number (i.e., between 3 and 4)

= (3 + 4)/2 = 7/2

3rd rational number (i.e., between 4 and 5)

= (4 + 5)/2 = 9/2

(B) Irrational numbers

The numbers which are not rational or which cannot be put in the form of p/q, where p and q are integers and q≠0,is called irrational number.
It is denoted by Q' or qc

$\sqrt{2},\sqrt{3},\sqrt{5},2 + \sqrt{3} , 3 - \sqrt{5},3\sqrt{3}\; are \; irratioanal\; numbers$.

Note

(i) Every positive irrational number has a negative irrational number corresponding to it.

(ii) $\sqrt{2} + \sqrt{3} \neq \sqrt{5}$

=> $\sqrt{5} ? \sqrt{3} \neq \sqrt{2}$

=> $\sqrt{3} \times \sqrt{2} = \sqrt{3 \times 2} = \sqrt{6}$

=> $\sqrt{6} \div \sqrt{2} = \sqrt{\frac{6}{2}} = \sqrt{3}$

(iii) Some times, product of two irrational numbers is a rational number.

For example : $\sqrt{2} \times \sqrt{2} = \sqrt{2 \times 2} = 2$

=> $(\sqrt{2} + \sqrt{3}) \times (\sqrt{2} ? \sqrt{3}) = (2)^{2} ? (\sqrt{3})^{2} = 4 ? 3 = 1$

Both rational and irrational numbers can be represented in number line. Thus real numbers is the set of the union of rational and irrational numbers.
R = Q ∪ Q'
Every real number is either rational or irrational.

Fraction

A fraction is a quantity which expresses a part of the whole.

Example 2

2. Write a fraction whose numerator is 22 + 1 and denominator is 32 – 1.

Sol. Numerator = 22 + 1 = 4 + 1 = 5
Denominator = 32 – 1= 9 – 1 = 8

$\therefore \; Fraction = \frac{Numerator}{Denominator}$
TYPES OF FRACTIONS :

1.Proper fraction : If numerator is less than its denominator,then it is a proper fraction.

For example : 2/5 , 6/18

2.Improper fraction : If numerator is greater than or equal to its denominator, then it is a improper fraction.

For example : 5/2 , 18/7 , 13/13

Note
If in a fraction, its numerator and denominator are ofequal value then fraction is equal to unity i.e. 1.

3.Mixed fraction : It consists of an integer and a proper fraction.

For example : 1(1/2) , 3(2/3) , 7(5/9)

Note
Mixed fraction can always be changed into improper fraction and vice versa.

For example : $7\tfrac{5}{9} = \frac{7 \times 9 + 5}{9} = \frac{63 + 5}{9} = \frac{68}{9}$

And $\frac{19}{2} = \frac{2 \times 9 + 1}{2} = 9 + \frac{1}{2} = 9\frac{1}{2}$

4.Equivalent fractions/Equal fractions : Fractions with same value.

For example : 2/3 , 4/6 , 6/9 , 8/12(=2/3).

Note
Value of fraction is not changed by multiplying or dividing the numerator or denominator by the same number.
For example :

(i) $\frac{2}{5} = \frac{2 \times 5}{5 \times 5} = \frac{10}{25}$ so, $\frac{2}{5} = \frac{10}{25}$

(ii) $\frac{36}{16} = \frac{36 \div 4}{16 \div 4} = \frac{9}{4}$ so, $\frac{36}{16} = \frac{9}{4}$

5.Like fractions: Fractions with same denominators.

For example : 2/7 , 3/7 , 9/7 , 11/7

6.Unlike fractions : Fractions with different denominators.

For example : 2/5 , 4/7 , 9/8 , 9/2

Note
Unlike fractions can be converted into like fractions.

For example : 3/5 and 4/7

$\frac{3}{5} \times \frac{7}{7} = \frac{21}{35} \; and \; \frac{4}{7} \times \frac{5}{5} = \frac{20}{35}$

7.Simple fraction : Numerator and denominator are integers.

For example : 3/7 and 2/5.

8.Complex fraction : Numerator or denominator or both are fractional numbers.

For example : $\frac{2}{\frac{5}{7}} , \frac{2\tfrac{1}{3}}{5\tfrac{2}{3}}, \frac{2 + \frac{1 + \frac{2}{7}}{3}}{2}$

9.Decimal fraction : Denominator with the powers of 10.

For example : 2/10 = (0.2), 9/100 = (0.09)

10.Vulgar fraction : Denominators are not the power of 10.

For example : 3/7 , 9/2 , 5/193.

Example 3

3. Write 2.73 as a fraction.

Sol. 2.73 = 273/100

Example 4

4. Express 2/5 as a decimal fraction.

Sol. $\frac{2}{5} = \frac{2 \times 2}{5 \times 2} = \frac{4}{10}$

Example 5

5. After doing 3/5 of the Biology homework on Monday night, Sanjay did 1/3 of the remaining homework on Tuesday night. What fraction of the original homework would Sanjay have to do on Wednesday night to complete the Biology assignment ?

(a) 1/15
(b) 2/15
(c) 4/15
(d) 2/5

Sol.(c) Remaining homework on Monday night

= 1 – (3/5) = 2/5

Work done on Tuesday night

= 1/3 of 2/5 = 2/15

Remaining homework to complete the biology assignment

= 2/5 – 2/15 = (6 – 2)/15 = 4/15

Rounding off (Approximation) of Decimals

There are some decimals in which numbers are found upto large number of decimal places.

For example : 3.4578, 21.358940789.
But many times we require decimal numbers upto a certain number of decimal places. Therefore,
If the digit of the decimal place is five or more than five, then the digit in the preceding decimal place is increased by one and if the digit in the last place is less than five, then the digit in the precedence place remains unchanged.

Example 6

6.
(a) Write 21.3751 upto two places of decimal.
(b)Write 3.27645 upto three places of decimal.

Sol.
(a)21.3751 = 21.38
(b)3.27645 = 3.276

REAL NUMBERS AND THEIR DECIMAL EXPANSIONS

(a) Terminating(or finite decimal fractions) :

For example : 7/8 = 0.875 , 21/5 = 4.2.

(b) Non-terminating decimal fractions : There are two types of Non-terminating decimal fractions :

(i) Non-terminating periodic fractions or non-terminating recurring (repeating) decimal fractions :

Form : x.a1a2a3 ... a1a2a3 ... a1a2a3

For example : 10/3 = 3.333... = 3.3
1/7 = 0.142857142857... = $\overline{0.142857}$

(ii) Non-terminating non-periodic fraction or non-terminating non-recurring fractions :

Form : x.a1a2a3 ... b1b2b3 ... c1c2c3

1.The decimal expansion of a rational number is either terminating or non-terminating recurring. More over , a number whose decimal expansion is terminating or non-terminating recurring is rational.

2.The decimal expansion of an irrational number is non-terminating non recurring. Moreover, a number whose decimal expansion is non-terminating non recurring is irrational.

For example : √2 = 1.41421356237309504880...
π = 3.1415926535897932384626433...

We often take 22/7 as an approximate value of π, but π ≠ 22/7.

Example 7

7. Find an irrational number between 1/7 and 2/7.

Sol. We find by dividing, 1/7 = $\overline{0.142857}$ and 2/7 = $\overline{0.285714}$.

To find an irrational number between 1/7 and 2/7, we find a number which is non-terminating non-recurring lying between them.

So, 0.1501500150000... is an irrational number between 1/7 and 2/7.

Operations

The following operations of addition, subtraction,multiplication and division are valid for real numbers.

(a) Commutative property of addition :
a + b = b + a
(b) Associative property of addition :
(a + b) + c = a + (b + c)
(c) Commutative property of multiplication :
a * b = b * a
(d) Associative property of multiplication :
(a * b) * c = a * (b * c)
(e) Distributive property of multiplication with respect to addition :
(a + b) * c = a * c + b * c

Complex numbers

A number of the form a + bi, where a and b are real number and i = √-1 (imaginary number) is called a complex number. It is denoted by C.

For Example : 5i (a =0 and b = 5), √5 + 3i(a = √5 and b = 3)

Note
i = $\sqrt{-1}$ , i2 = -1 , i3 = -i , i4 = 1
DIVISIBILITY RULES

Divisibility by 2 : A number is divisible by 2 if its unit’s digit is even or 0.

Divisibility by 3 : A number is divisible by 3 if the sum of its digits are divisible by 3.

Divisibility by 4 : A number is divisible by 4 if the last 2 digits are divisible by 4, or if the last two digits are 0’s.

Divisibility by 5 : A number is divisible by 5 if its unit’s digit is 5 or 0.

Divisibility by 6 : A number is divisible by 6 if it is simultaneously divisible by 2 and 3.

Divisibility by 7 : A number is divisible by 7 if unit’s place digit is multiplied by 2 and subtracted from the remaining digits and the number obtained is divisible by 7.

Divisibility by 8 : A number is divisible by 8 if the last 3 digits of the number are divisible by 8, or if the last three digits of a number are zeros.

Divisibility by 9 : A number is divisible by 9 if the sum of its digits is divisible by 9.

Divisibility by 10: A number is divisible by 10 if its unit’s digit is 0.

Divisibility by 11 : A number is divisible by 11 if the sum of digits at odd and even places are equal or differ by a number divisible by 11.

Divisibility by 12 : A number is divisible by 12 if the number is divisible by both 4 and 3.

Divisibility by 13 : A number is divisible by 13 if its unit’s place digit is multiplied by 4 and added to the remaining digits and the number obtained is divisible by 13.

Divisibility by 14 : A number is divisible by 14 if the number is divisible by both 2 and 7.

Divisibility by 15 : A number is divisible by 15 if the number is divisible by both 3 and 5.

Divisibility by 16 : A number is divisible by 16 if its last 4 digits is divisible by 16 or if the last four digits are zeros.

Divisibility by 17 : A number is divisible by 17 if its unit’s place digit is multiplied by 5 and subtracted from the remaining digits and the number obtained is divisible by 17.

Divisibility by 18 : A number is divisible by 18 if the number is divisible by both 2 and 9.

Divisibility by 19 : A number is divisible by 19 if its unit’s place digit is multiplied by 2 and added to the remaining digits and the number obtained is divisible by 19.

Example 8

8. Without actual division, find which of the following numbers are divisible by 2, 3, 4, 5, 7, 9, 10, 11 :
(i) 36324
(ii) 2211
(iii) 87120

Sol. (i) 36324

It is divisible by 2 because 4 (unit’s digit) is divisible by 2. It is divisible by 3 because 3 + 6 + 3 + 2 + 4 = 18 is divisible by 3. It is divisible by 4 because 24 is divisible by 4.
It is not divisible by 5.
It is not divisible by 7.
It is divisible by 9 because 3 + 6 + 3 + 2 + 4 = 18 is divisible by 9.
It is not divisible by 10.It is not divisible by 11.

(ii) 2211
It is not divisible by 2.
It is divisible by 3 because 2 + 2 + 1 + 1 = 6 is divisible by 3.
It is not divisible by 4, 5, 7, 8, 10.
It is divisible by 11 because 2211 → (2 + 1) – (2 + 1) = 3 – 3= 0.

(iii) 87120
It is divisible by 2 because its unit’s place digit is 0.
It is divisible by 3 because 8 + 7 + 1 + 2 + 0 = 18 is divisible by 3.
It is divisible by 4 because 20 is divisible by 4.
It is divisible by 5 because its unit’s place digit is 0.
It is not divisible by 7.
It is divisible by 9 because 8 + 7 + 1 + 2 + 0 = 18 is divisible by 9.
It is divisible by 10 because its unit’s place digit is 0.
It is divisible by 11 because 87120 → (8 + 1 + 0) – (7 + 2) = 9 – 9 = 0.

Example 9

9. Is 473312 divisible by 7?

Sol. 47331 – 2 × 2 = 47327
4732 – 2 × 7 = 4718
471 – 2 × 8 = 455
45 – 2 × 5 = 35
35 is divisible by 7, therefore, 473312 is divisible by 7.

Example 10

10. What is the value of M and N respectively if M39048458N is divisible by 8 and 11, where M and N are single digit integers?
(a) 7, 4
(b) 8, 6
(c) 6, 4
(d) 3, 2

Sol. (c) A number is divisible by 8 if the number formed by the last three digits is divisible by 8.
i.e., 58N is divisible by 8.
Clearly,N = 4
Again, a number is divisible by 11 if the difference between the sum of digits at even places and sum of digits at the odd places is either 0 or is divisible by 11.
i.e (M + 9 + 4 + 4 + 8) – (3 + 0 + 8 + 5 + N)
= M + 25 – (16 + N)
= M – N + 9 must be zero or it must be divisible by 11.
i.e M – N = 2
=> M = 2 + 4 = 6
Hence, M = 6 , N = 4

Example 11

11. The highest power of 9 dividing 99!completely, is:
(a)20
(b)24
(c)12
(d)1

Sol. (c) 99! = 99 × 98 × 97 × 96 × 95 × 94…..× 1
To find the highest power of 9 that divides this product, we have to find the sum of powers of all 9’s in the expression.
In the nos. from 1 to 99, all the nos. divisible by 9 are 9,18, 27, 36, 45, 54, 63, 72, 81 (9 × 9), 90, 99, i.e. 12 in no.
This clearly shows that 99! will be completely divisible by 912.

DIVISION ALGORITHM

Dividend = (Divisor × Quotient) + Remainder where, Dividend = The number which is being divided Divisor = The number which performs the division process Quotient = Greatest possible integer as a result of division Remainder = Rest part of dividend which cannot be further divided by the divisor.

Complete remainder: A complete remainder is the remainder obtained by a number by the method of successive division.Complete remainder = [I divisor × II remainder] + I remainder

Two different numbers x and y when divided by a certain divisor D leave remainder r1 and r2respectively. When the sum of them is divided by the same divisor, the remainder is r3. Then,

Method to find the number of different divisors (or factors)(including 1 and itself) of any composite number N :

STEP I : Express N as a product of prime numbers as
N = Xa × Yb × Zc

STEP II : Number of different divisors (including 1 and it self)
= (a + 1)(b + 1)(c + 1) .......

Example 12

12. Find the number of different divisors of 50,besides unity and the number itself.

Sol. If you solve this problem without knowing the rule, you will take the numbers in succession and check the divisibility.In doing so, you may miss some numbers. It will also take more time.

Different divisors of 50 are : 1, 2, 5, 10, 25, 50 If we exclude 1 and 50, the number of divisors will be 4.

By rule : 50 = 2 × 5 × 5 = 21 × 52

∴ the number of total divisors = (1 + 1) × (2 + 1) = 2 × 3= 6 or,the number of divisors excluding 1 and 50 = 6 – 2 = 4

Example 13

13. A certain number when divided by 899 leaves the remainder 63. Find the remainder when the same number is divided by 29.

(a) 5
(b) 4
(c) 1
(d) Can not be determined

Sol.(a) Number = 899Q + 63, where Q is quotient
= 31 × 29 Q + (58 + 5) = 29 [ 31Q + 2] + 5
∴ Remainder = 5

SIMPLIFICATION

FUNDAMENTAL OPERATIONS :

(a) Sum of two positive numbers is a positive number.
For example : (+ 5) + (+ 2) = +7

(b) Sum of two negative numbers is a negative number.
For example : (– 5) + (– 3) = –8

(c) Sum of a positive and a negative number is the difference between their magnitudes and give the sign of the number with greater magnitude.
For example : (– 3) + (+ 5)= 2 and (– 7) + (+ 2)= –5

2.Subtractions :
Subtraction of two numbers is same as the sum of a positive and a negative number.
For Example :
(+ 9) – (+ 2) = (+ 9) + (– 2) = 7
(– 3) – (– 5) = (– 3) + 5 = + 2

Note
In subtraction of two negative numbers, sign of second number will change and become positive.In subtraction of two negative numbers, sign of second number will change and become positive.

3.Multiplication :

(a) Product of two positive numbers is positive.
(b) Product of two negative numbers is positive.
(c) Product of a positive number and a negative number is negative.
(d) Product of more than two numbers is positive or negative depending upon the presence of negative quantities.If the number of negative numbers is even then product is positive and if the number of negative numbers is odd then product is negative.

For Example :
(– 3) × (+ 2) = – 6
(– 5) × (– 7) = + 35
(– 2) × (– 3) × (– 5) = – 30
(– 2) × (– 3) × (+ 5) = +30

4. Division :

(a) If both the dividend and the divisor are of same sign, then quotient is always positive.
(b)If the dividend and the divisor are of different sign, then quotient is negative,
For Example :
(– 36) ÷ (+ 9) = –4
(– 35) ÷ (– 7) = +5

#### BRACKETS :

Types of brackets are :
(i) Vinculum or bar –
(ii) Parenthesis or small or common brackets : ( )
(iii) Curly or middle brackets : { }
(iv) Square or big brackets : [ ] The order for removal of brackets is(), {}, []

Note
If there is a minus (–) sign before the bracket then while removing bracket, sign of each term will change.
‘BODMAS’ Rule
Now a days it becomes ‘VBODMAS’ where,
‘V’stands for “Vinculum”
‘B’stands for “Bracket”
‘O’stands for “Of”
‘D’stands for “Division”
‘M’stands for “Multiplication”
‘S’stands for “Subtraction”
Same order of operations must be applied during simplification.
Example 1

1.
6 + 5 – 3 × 2 of 5 -(15 ÷ $\overline{7 ? 2}$)
= 6 + 5 – 3 × 2 of 5 – (15 ÷ 5) {Remove vinculum}
= 6 + 5 – 3 × 2 of 5 – 3 {Remove common bracket}
= 6 + 5 – 3 × 10 – 3 {‘Of’ is done}
= 6 + 5 – 30 – 3 {Multiplication is done}
= 11 – 33 {Addition is done}
= – 22 {Subtraction is done}.

To simplify on expression, add all the positive numbers together and all the negative numbers separately and add or subtract the resulting numbers as the case will.

Example 2

2. Simplify : 7 – 2 + 13 – 5 – 2 + 1
Sol. 7 – 2 + 13 – 5 – 2 + 1
= 7 + 13 + 1 – 2 – 5 – 2 = 21 – 9 = 12
[7 + 13 + 1 = 21 and – 2 –5 – 2 = – 9]

Example 3

3. $\mathbf{\frac{11 \times 11 ? 21}{9 \times 6 ? (2)^{2}} =}$ (a) 0
(b) 11/52
(c) 2
(d) 40

Sol. (c) $\frac{11 \times 11 ? 21}{9 \times 6 ? (2)^{2}} = \frac{121 ? 21}{54 ? 4} = \frac{100}{50}$ = 2

Example 4

4. $\mathbf{\frac{1 + 1 \times 1 ? 1 \times 1 + 1}{1 + 1 \div 1 + (1 + 1) \times (1 + 1)}}$ (a) 1/2
(b) 1/5
(c) 2/5
(d) 1/3

Sol.(d)
$\frac{1 + 1 ? 1 + 1}{1 + 1 + 2 \times 2} = \frac{3 ? 1}{2 + 4} = \frac{2}{6} = \frac{1}{3}$

Example 5

5. What is the missing figure in the expression given below?

$\frac{16}{7} \times \frac{16}{7} ? \frac{*}{7} \times \frac{9}{7} + \frac{9}{7} \times \frac{9}{7} = 1$

(a) 1
(b) 7
(c) 4.57
(d) 32

Sol.(d) Let the missing figure in the expression be x.
$\frac{16}{7} \times \frac{16}{7} ? \frac{x}{7} \times \frac{9}{7} + \frac{9}{7} \times \frac{9}{7} = 1$

=> 16 × 16 – 9x + 9 × 9= 7 × 7

=> 9x = 16 × 16 + 9 × 9 – 7 × 7
= 256 + 81 – 49 = 288
=> x = 228/9 = 32.