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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.

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, ….}

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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.

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, …...}

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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

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.

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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.

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

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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.

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

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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