# Basic facts the number system

The Number System forms the basis of mathematics. When we measure distance from a particular point, or quantity of liquid in a flask, or even the age of a person, we represent it in numbers. These are called Real Numbers. So all numbers that we deal with in everyday life are real numbers.

**Integers**: Whole numbers are called Integers. They include both negative and positive whole numbers.

**Even integers **are those which are divisible by 2. Integers which are not divisible by 2 are odd integers. The set of even integers are {...-– 4, –2, 0, 2, 4, 6....} and the set of odd integers is {....-3, -1, 1, 3, 5, 7....}

**Sum and product of integers: **If there are two integers and one of them is even, then their product is EVEN but their sum will be ODD. For example, if we have two integers 7 and 8, then:

Product (Odd × even) 7 × 8 = 56 (Even)

Sum (Odd + Even) 7 + 8 = 15 (Odd)

Difference (Odd – Even) 8 – 7 = 1 (Odd)

If two integers are both even, their product, sum and difference is EVEN. Let us take two integers 16 and 8, both even. Then

Product (Even × Even) 16 ×× 8 = 128 (Even)

Sum (Even + Even) 16 + 8 = 24 (Even)

Difference (Even – Even) 16 - 8 = 8 (Even)

But if two integers are odd, their sum and difference is EVEN but their product is ODD. Let us take two odd integers, 13 and 7

Product (Odd × Odd) 13 × 7 = 91 (Odd)

Sum (Odd + Odd) 13 + 7 = 20 (Even)

Difference (Odd – Odd) 13 – 7 = 6 (Even)

**Consecutive integers **are –2, –1, 0, 1, 2, 3, 4 and can be represented by n, n + 1, n + 2, n + 3 ... where n is an integer. Consecutive even integers are 0, 2, 4, 6 and 8.... and can be represented by 2n, 2n + 2, 2n + 4, and consecutive odd integers are 1, 3, 5, 7, 9, ... and can be represented by 2n + 1, 2n + 3, 2n +5 ... where n is an integer.

# Some Definitions

**Natural Numbers**

The numbers 1, 2, 3, 4... are called natural numbers.

**Whole Numbers**

The numbers 0, 1, 2, 3 ... are called whole numbers. Whole numbers include “0”.

**Integers**

**The numbers ... -3, -2, -1, 0, 1, 2, 3, are called integers.
**

**Negative Integers**

**The numbers -1, -2, -3, are called negative integers.
**

**Positive Fractions**

**The numbers 1/2, 3/4, 5/6... are called positive fractions.
**

**Negative Fractions**

**The numbers -3/4, -5/6, -7/8.... are called negative fractions
Real numbers can be divided into two main groups.**

**Rational Numbers**

Any number which is a positive or negative integer or fraction, or zero is called a rational number. A rational number is one which can be expressed in the following format ⇒ a/b. where b≠0 and a and b are positive or negative integers. Examples: 1⁄2, 4/7, 21/27, –5/18, etc.

**Irrational Numbers**

An infinite non-recurring decimal number is known as an irrational number. These numbers cannot be expressed in the form of a proper fraction a/b where b ≠0. Examples of irrational numbers are: √2, √5. These are infinite non-recurring decimals, and hence irrational. Any Real Number that cannot be expressed as a fraction is therefore irrational.

** Note that **π

**The reason is that 22/7 is an approximation of the value of pi (π) and it cannot be expressed as a definite fraction. It is therefore irrational.**

*is an irrational number even though it can be put in the form a/b*.*Surds***: **Any root of a number which cannot be exactly found is called a surd. All surds are irrational numbers. e.g. √2, √5, √7, etc.
Surds of the form x + √y, x - √y are called binomial quadratic surds, where x + √y and x – √y are called conjugate surds, each being the

conjugate of the other. The product of two conjugate surds is always a rational number.

*Prime Numbers**: *Prime numbers are integers that have exactly two divisors, 1 and itself. Prime numbers are 2, 3, 5, 7, 11, and 13.... but 15 is not, since 15 has four divisors: 1, 3, 5 and 15. The *number 1 is not a prime number*, since it has only one positive divisor.

# Test whether a number is a prime number

If a number has no factor equal to or less than its square root, then that number is a prime number. All prime numbers with the exception of 2 are odd numbers — as all even numbers are divisible by 2 and hence will have 2 as a factor other than the number itself and 1. To test whether a number is prime, take out its square root, then test whether the number is divisible by any of the prime numbers less than the square root. If any of the prime number exactly divides the number, it is not prime.

Hence, if none of the prime numbers upto its square root happens to be its factors, then the number is a prime number.

**Illustration 1:**

**Is 113 prime?**

**Solution:**

The square of 10 is 100 and that of 11 is 121. Therefore, we have to test if any of the prime numbers less than 11 is a factor of 113. The prime numbers, 2, 3, 5, 7, 11 are not factors of 113 and hence is a prime number.

**Time Saver****
***It is useful to learn the prime numbers up to 100 and have an idea of prime numbers beyond 100. Prime numbers up to 100 are:*

*2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71,*

*73, 79, 83, 89, 97*

# How this is helpful

*There are lottery tickets numbered serially from 1 to 50. If a person gets a prime number, he wins a prize. What is the probability of a person winning a prize?*

In the above sum, the number of prime numbers up to 50 is required. If you start counting them in the exam, it will take some time. So learning the above facts helps in sums like these.

Two numbers are considered to be prime to each other when their HCF is 1. e.g 5 and 21 are prime to each other because they do not have any common factor.

**Factors: **If x and y are integers and x ≠≠0, x is a divisor (factor) of y, provided that y = x.n for some integer n. Then y is said to be divisible by x or to be a multiple of x. For example, 5 is a divisor or factor of 25 and 6 is not a divisor of 25 since there is no integer such that 25 = 6n.

We can also say that Dividend = Divisor × quotient + remainder. In the above, 25 = 6n + 1 where n = 4.

**Composite Numbers**: A number which has factors other than itself and 1 is called a composite number. e.g. 4, 6, 8, 9, 16, 25... *Note that 1 is neither a composite number nor a prime number*.

** The number of divisors of a composite number: **If D is a composite number in the form D = ap × bq × cr, where a, b, c are primes, then the no. of divisors of D, represented by ‘n’ is given by

*n *= (p+1)(*q*+1)(*r *+1). And the sum of all those divisors Sn is given by the following formula.

** Consecutive Numbers: **Numbers arranged in increasing order and differing by 1 are called consecutive numbers e.g. 4, 5, 6, 7...

*Real Numbers***:**The above sets of natural numbers, integers, whole numbers, rational numbers and irrational numbers constitute the set of real numbers.

*Complex Numbers***: **Square root of a negative number, which is undefined, is called complex, or imaginary numbers. *e.g. *(√–3 – 4), (2 + √–3) etc. The square root of a negative number is called an imaginary number - e.g. √–2, √–3, because the square root of a negative number does not exist in the real sense. Imaginary numbers are denoted by iota, or “i” which is equal to √-1. Thus

√–2 is denoted as i√2. Remember that i2 = 1 and i4 = 1.