Sales Toll Free No: 1-855-666-7440

# Elementary Number Theory

In number system, we use ten symbols 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 called digits to represent any number. This is the decimal system where we use the numbers 0 to 9. 0 is called insignificant digit whereas 1, 2, 3, 4, 5, 6, 7, 8, 9 are called significant digits.

A group of figures, denoting a number is called a numeral. For a given numeral, we start from extreme right as Unit’s place. Ten’s place, Hundred’s place and so on.

## Number System

Natural Numbers

Counting numbers1, 2, 3, 4, 5, …. are known as natural numbers.

The set of all natural numbers can be represented by

N = {1, 2, 3, 4, 5, ….}

## Whole Numbers

If we include 0 among the natural numbers, then the numbers 0, 1, 2, 3, 4, 5, …. are called whole number.

The set of whole numbers can be represented by

W = {0,1, 2, 3, 4, 5, …..}

Clearly, every natural number is a whole number but 0 is a whole number which is not a natural number.

## Integers

All counting numbers ad their negatives including zero are known as integers.

The set of integers can be represented by

Z or I = {…. – 4, – 3, – 2, – 1, 0, 1, 2, 3, 4, …...}

## Positive Integers

The set I+ = {1, 2, 3, 4, …...} is the set f all positive integers. Clearly, positive integers and natural numbers are synonyms

## Negative Integers and Non-negative Integers

The set I = {– 1, – 2, – 3, ….} is the set of all negative integers.

Remember

0 is neither positive nor negative

Non-Negative Integers

The set {0, 1, 2, 3, …..} is the set of all non-negative integers

## Rational Numbers

The numbers of the form p/q, where p and q are integers and q does not belong 0, are known as rational numbers, e.g., 4/7, 3/2, - 5/8, 0/1, - 2/3, etc.

The set of all rational numbers is denoted by Q.

i.e., Q = {x : x = p/q; p, q ? I, q does not belong 0}

Note:- Since every natural number a can be written as a/1, every natural number is a rational number. Since 0 can be written as 0/1 and every non-zero integer a can be written as a/1, every integer is a rational number.

## Important

Every rational number has a peculiar characteristic that when expressed in decimal form is expressible either in terminating decimals or in non-terminating repeating decimals.

For example, 1/5 = 0.2, 1/3 = 0.333 …. , 22/7 = 3.1428714287, 8/44 = 0.181818 …. , etc.

The recurring decimals have been given a short notation as

_ _ _ _

0.333… = 0. 3, 4.1555…. = 4.05, 0.323232…… = 0.3 2

## Irrational Numbers

Those numbers which when expressed in decimal form are neither terminating nor repeating decimals are known as irrational numbers, e.g., ?2, ?3, ?5, ?.

Caution:

The exact value of ? is not 22/7, 22/7 is rational while ? is irrational number. 22/7 is approximate value of ?. Similarly, 3.14 is not an exact value of it.

## Fractions

A fraction is a part of an integer. Learn How to do Fractions step by step from our online tutor, Here is an example of Fractions, 1/3 means one-third of the whole. A fraction includes tow parts, numerator and denominator.

Key Points

• Any integer can be expressed as a fraction with any denominator, as long as the same denominator is multiplied with it.

For example, 15 = 15 × 7/7 = 105/7

• The value of a fraction remains unaltered by multiplying or dividing the numerator and the denominator by the same non-zero number.

For example, 2/3 = 2/3 × 9/9 = 18/27

Also, 12/15 = 12/15 ÷ 3/3 = 4/5

## Proper Fraction and Improper Fraction

A fraction I which numerator is less than the denominator is called a proper fraction e.g., 4/9

Improper Fraction

a fraction in which numerator is more than the denominator is called an improper fraction e.g., 5/3.

## Mixed Fraction

A mixed fraction consists of integral as well as the fractional part e.g., 3,4/5

Key Point

• A mixed fraction consists of integral as well as the fractional can be expressed as an improper fraction and vice-versa. e.g., 3,4/7 = 3 + 4/7 = 25/7;

An improper fraction

Also, 5/2 = 4 +1/ 2 = 4/2 + ½ = 2 + 1/ 2 = 2,1/2, a mixed fraction

Get a free session with our online tutor and learn how to add fractions and get other math help free.

## Vulgar Fraction and Decimal Fraction

A fraction in which the denominator is an integer but other than 10 or a multiple of 10 is called a vulgar fraction e.g., 2/5, 5/2, 3/7

Decimal Fraction

A fraction with denominator 10 or a multiple of 10 is called a decimal fraction e.g., 3/10, 4/100, 7/1000.

## Real Numbers

The rational and irrational numbers combined together are called real numbers, e.g., 13/21, 2/5, - 3/7, ?3, 4 + ?2, are real numbers

The set of all real numbers is denoted by R.

Note:- The sum, difference or product of a rational and irrational number is irrational, e.g., 3 + ?2, 4 - ?3, 2/3 - ?5, 4?3, - 7?5 are all irrational

## Even Numbers

All those numbers which are exactly divisible by 2 are called even numbers, e.g., 2, 6, 8, 10, are even number.

Note:- Even numbers are expressible in the form 2n, where n is an integer. Thus, – 2, – 4 etc., are also even numbers.

## Odd Numbers

All those numbers which are not exactly divisible by 2 are called odd numbers, e.g., 1, 3, 5, 7 are odd numbers.

Note:- Odd numbers are expressible in the form (2n + 1) where n is an integer. Thus, – 1, – 3, – 5 etc, are also odd numbers

## Prime Number

A natural number other than 1, is a prime number if it is divisible by 1 and itself only.

For example, each of the number 2,3, 5, 7 are prime number.

## Composite Numbers

Natural numbers greater than 1 which are not prime, are known as composite numbers.

For example, each of the numbers 4, 6, 8, 9, 12 are composite number

## Perfect Number

If the sum of the divisors of n excluding itself is equal to n, then n is called a perfect number

For example, 6 = 1 + 2 + 3, where 1, 2 and 3 are the divisors of 6. Therefore, 6 is a perfect number.

Also, 28 = 1 + 2 + 4 + 7 + 14, therefore 28 is also a perfect umber.

Problem Solving Trick

The sum of the reciprocals of the divisors of a perfect number including that of its own is always equal to 2

For example, for the perfect number 6

1/1 + 1/2 + 1/3 + 1/6 = 2