Number Theory
Mathematics
Topic Four

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Types of Numbers

Natural Numbers

Natural numbers are defined as the set of counting numbers.

N = {1, 2, 3, 4, 5, 6, 7, 8, 9,.….}

Zero is not a natural number.

The set of even and odd numbers are two types of natural numbers. Even numbers are those which are exactly divisible by 2.

That is, the set of even numbers = {2, 4, 6, 8, 10, 12, 14, 16, 18…}

Odd numbers are those which are not exactly divisible by 2.

That is, the set of odd numbers = {1, 3, 5, 7, 9, 11, 13, 15, 17, 19…}

Whole Numbers

Whole numbers are defined as the set of natural numbers plus zero.

That is, the set of whole numbers, W = {0, 1 , 2, 3, 4, 5….}.

Integers

Integers are defined as the set of whole numbers and the negatives.

That is, the set of integers, I = {…-6 -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5, 6…}

Zero is neither positive nor negative.

Prime numbers are numbers which have only two factors (1 and itself).

That is, the set of prime numbers = {2, 3, 5, 7, 11, 13, 17, 19…}.

Note: 1 is not a prime number since it only has one factor (itself).

Composite numbers are numbers which have more than two factors (can be divided by other numbers apart from 1 and itself).

That is, the set of composite numbers = {4, 6, 8, 9, 10, 12, 14, 15, 16}.

Rational Numbers

Rational numbers are defined as the set of numbers (positive or negative) which can be written as fractions. A fraction is a number (1/2) where 1 is the numerator and 2 is the denominator. 

Irrational Numbers

Irrational numbers are defined as numbers which cannot be expressed as fractions.

Examples of irrational numbers: √3 and π=3.14159

Real Numbers

Real numbers are defined as rational and irrational numbers.

Relationship between the types of numbers

There is a relationship between the various types of numbers which are:

  • The set of natural numbers is a subset of the set of whole numbers;
  • The set of whole numbers is a subset of the set of integers;
  • The set of integers is a subset of the set of rational numbers;
  • The set of rational numbers is a subset of the set of real numbers.

Lowest Common Multiple (L.C.M.)

The lowest common multiple (L.C.M.) of a group of numbers is the lowest number that can be divided by each number in the group without resulting in a remainder.

Example

What is the L.C.M. of the numbers 3, 4 and 6?

Multiples of 3 = (3,6,9,12,15,18,21,24,27,30)

Multiples of 4 = (4,8,12,16,20,24,28,32,36,40)

Multiples of 6 = (6,12,18,24,30,36,42,48)

The LCM is the lowest number that is common to 3, 4 and 6 which is 12.

Highest Common Factor (H.C.F.)

The highest common factor (H.C.F.) of a group of numbers is the largest natural number which divides into each number exactly without leaving a remainder.  We need to look for the factors of the numbers up to each number and not exceed the numbers as with the case of the LCM.  With HCF, we look for the factors and not the multiples as in the case of LCM. 

Example

What is the H.C.F. of the numbers 12 and 24?

Factors of 12 = (1,2,3,4,6,12)

Factors of 24 = (1,2,3,4,6,8,12, 24)

The highest factor common to 12 and 24 is 12

So the HCF = 12

The Identity of Operations

The identity of an operation is defined as an action which results in the number being manipulated but remaining unchanged.  The identity for an addition and a subtraction is zero. If zero is added to or subtracted from a number, then the sum/difference obtained is that number.

Example

What is the sum:

             4 + 0 = 4

          -15 + 0 = -15

What is the difference:

14 – 0  = 14

 -8 – 0 = -8

The identity for a multiplication and a division is one. If a number is multiplied or divided by one, the product/quotient obtained is that number.

Example

  1. 9 x 1 = 9
  2. -9 x 1 = -9
  3. 15/ 1 = 15
  4. -12/ 1 = -12

The Inverse of Operations

As mentioned above, the identity of a number under addition is zero. The inverse of a number x under addition, is a number which when added to x results in zero being the sum.

Example

The inverse of 4 and – 6 are follows:

Solution

The inverse of 4:

4 + x = 0 and isolating or solving for x gives -4.

Likewise the inverse of -6:

-6 + x = 0 and isolating or solving for x gives 6. 

Likewise, with multiplication, the inverse multiplication is one. The inverse of a number x under multi of:

2 = ½

4 = ¼

1/8 = 8.

Associative Law

The associative law addresses the grouping of numbers, and states that the sum/product obtained in an addition/multiplication is not dependent on how the numbers are grouped. The following is the case for additions:

(a + b) + c = a + (b + c)

For multiplication:

(a x b) x c = a x (b x c)

Commutative Law

The commutative law deals with the order in which an operation is completed and states that numbers can be shifted in the order and the sum/product remains the same in an addition/multiplication.

The following is for additions:

a + b = b + a

The following is for multiplication:

a x b = b x a

Distributive Law

The distributive law states that multiplying a number by a group of numbers added together is the same as multiplying each separately

The distributive law is sumarised by the identity below:

(a + b) x c = a x c + b x c

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