Algebra and General Mathematics: System of Numbers

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.

                                          = 10000

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

                                        = 1,000,000

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 = 2³ • 3² •5¹

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 = 5776

(625)² = 390625

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.

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