**Number **is an item that describes a magnitude or a position Numbers are classified into two types, namely cardinal and ordinal numbers.

**Cardinal numbers **are numbers which allow us to count the objects or ideas in a given collection.

Example:

1, 2, 3, 4, 5, …

**Ordinal numbers **states the position of individual objects in a sequence.

Example:

First, second, third, fourth, fifth, …

### Numerals

**Numerals **are symbols, or combination of symbols which describe a number. The most widely used numerals are the **Arabic numerals **and the **Roman numerals.**

**Arabic numerals** were simply a modification of the Hindu-Arabic number signs and are written in Arabic digits. Taken single digit: 0, 1, 2. 3, 4,** **5, 6. 7. 8, 9 and in combination: 30, 41, 62, 2014, …

**The Roman numerals** are numbers which are written in Latin alphabet.

Example:

MMXIV

The following are Roman numerals and their equivalent Arabic numbers:

I = 1 | C = 100 |

V = 5 | D = 500 |

X = 10 | M = 1000 |

L = 50 |

**To increase the number, the following are used:**

- 1. Bracket to increase by 100 times.

|X| = 1000

- 2. Bar above the number – to increase by 1000 times.

- 3. A “doorframe” above the number – to increase by 1000000 times.

### DIGIT

Digit is a specific symbol or symbols used alone or in combination to denote a number. For example, the number 24 has two digits, namely 2 and 4. In Roman numerals, the number 9 is denoted as IX. So the digits I and X were used together to denote one number and that is the number 9.

In mathematical computations or engineering applications, a system of numbers using cardinal numbers was established and widely used.

### DIAGRAM SHOWING THE SYSTEM OF NUMBERS / SET OF NUMBERS

#### COMPLEX NUMBER

Complex Number is an expression of both **real** and **imaginary** **number** combined.

It takes the form of:

**a + bi**

where a and b are real numbers. If:

**a = 0: a + bi → pure imaginary**

**b = 0: a + bi → real number**

#### REAL NUMBERS

Real Numbers are the rational and irrational numbers taken together.

**Rational Numbers** – are are numbers which can be expressed as a quotient m / n (ratio). where m and n are integers; n ≠ 0 (not equal to zero).

Example:

{ 0.4, 2.5, –6, ¼, -4, 0.333, etc. }

**Irrational Numbers** – are numbers which cannot be expressed as a quotient of two integers (m / n).

Example:

{ √2, √3, π, e, 4√2, etc. }

The numbers in the examples above can never be expressed exactly as a quotient of two integers. They are in fact, a non- terminating number with non-terminating decimal.

**Integers** – are all the natural numbers, along with their negatives and zero.

Example:

{ –4, –2, 0, 3, 5, 7, etc. }

**Natural Numbers** – are numbers, except 0, formed by one or more of the digits 1, 2, 3, 4, 5, 6, 7, 8, 9, and 0. It also known as **Positive Integers**.

Example:

{ 1, 2, 3, 4, 5, 6, 7, 8, … }

Note: The number 0 (zero) is not a natural number, but is considered as a whole number.

**Even Numbers** – are integers divisible by 2 such as 2, 4, 6, 8, … etc.

**General Form: 2n**

**Odd Numbers** – are integers not exactly divisible by 2 such as 3, 5, 7, 11, -9, … etc.

**General Form: 2n + 1**

**Prime Numbers** – are natural numbers that are divisible by 1 and itself only. According to the fundamental theorem of arithmetic, “Every positive integer greater than 1 is a prime or can be expressed as a unique product of primes and powers of primes".

Example:

{ 2, 3, 5, 7, 11, etc }

Example of unique product of power of primes:

360 = 2**3** • 3**2** •5**1**

**Twin primes** are prime numbers that appear in pair and differ by 2.

Example:

{ 3 and 5, 11 and 13, 17 and 19… }

**Composite Numbers** – are natural numbers that are neither 1 nor a prime number.

Example:

{ 4, 6, 8, 10, 12, etc, }

Note: Number 1 is neither a prime number nor a composite number.

#### IMAGINARY NUMBERS

Imaginary numbers are the square roots of negative numbers.

**√-3 = 3i → is an imaginary number**

### ABSOLUTE VALUE

The absolute value of a real number is its magnitude, or its value without any reference to its sign.

Properties of Absolute Value:

- 1. | a | ≥ 0
- 2. | –a | = | a |
- 3. | ab | = | a | | b |
- 4. | a/b | = | a | / | b |

### NUMBER DIVERSIONS

**Abundant Numbers** – are numbers whose sum of the proper factors is more than the number itself.

Example:

20 = ( 1, 2, 4, 5, 10 ) → proper factors of 20

1 + 2 + 4 + 5 + 10 = 22 is more than 20

therefore, 20 is an abundant number

**Deficient Numbers** – are numbers whose sum of the proper factors is less than the number itself.

Example:

16 = ( 1, 2, 4, 8 ) → proper factors of 16

1 + 2 + 4 + 8 = 15 is less than 16

therefore, 16 is an deficient number

**Perfect Numbers** – are number whose sum of the proper factors is equal to the number itself. There are around **30** **numbers **known today as perfect number and all of which are even numbers.

Example:

6, 28, 496..

6 = ( 1, 2, 3 ) → proper factors of 6

1 + 2 + 3 = 6 is equal to 6

therefore, 6 is an perfect number

**Amicable Numbers or Friendly Numbers** – refers to two numbers where each is the sum of the proper factors of the other. The smallest known amicable numbers are 220 and 284.

Example:

220 = ( 1, 2, 4, 5, 10, 11, 20, 22, 44, 55, 110 ) → proper factors of 220

1 + 2 + 4 + 5 + 10 + 11 + 20 + 22 + 44 + 55 + 110 = 284

284 = ( 1, 2, 4, 71, 142) → proper factors of 284

1 + 2 + 4 + 71 + 142 = 220

therefore, 220 and 284 are an amicable pair

**Automorphic Numbers** – are numbers whose last digits are unchanged after the number has been squared.

Example:

(__76__)2 = 57__76__

(__625__)2 = 390__625__

therefore, 76 and 625 are automorphic numbers

**Palindrome** – is a number which is unchanged whether it is read from left to right or right to left.

Example:

{ 66, 25, 123321, … )

therefore, 66, 25, 123321, … are all palindromic numbers

**Harshad Numbers** – are numbers which can be divided exactly by the sum of its digits.

Example:

1729,

1 + 7 + 2 + 9 = 19 → 1729 / 19 = 91 (exact)

therefor, 1729 is a harshad number

**Polite Numbers** – are numbers which can be made by adding together two or more consecutive whole numbers.

Example:

15 = (1 + 2 + 3 + 4 + 5) = (4 + 5 + 6) = (7 + 8)

therefore, 15 is a polite number with a politeness of 3

Note: This often can be done in more than one way, and the number of ways it can be done is measure of the politeness of a number.