Basic Concepts of Number System

Last Updated on Nov 7, 2020

What is a Number System?

A number system is a system of writing for expressing numbers. It provides a unique representation to every number and represents the arithmetic and algebraic structure of the figures. It also allows us to operate arithmetic operations like addition, subtraction, and division.

The Natural Numbers

The natural (or counting) numbers are $1, 2, 3, 4, 5,$ etc. There are infinitely many natural numbers. The set of natural numbers, ${1, 2, 3, 4, 5, ...}$ , is sometimes written $N$ for short.

The whole numbers are the natural numbers together with $0$ .

(Note: a few textbooks disagree and say the natural numbers include $0$ .)

The sum of any two natural numbers is also a natural number (for example, $4 + 2000 = 2004$ ), and the product of any two natural numbers is a natural number ( $4 \times 2000 = 8000$ ). This is not true for subtraction and division, though.

The Integers

The integers are the set of real numbers consisting of the natural numbers, their additive inverses and zero.

${..., - 5, - 4, - 3, - 2, - 1, 0, 1, 2, 3, 4, 5, ...}$

The set of integers is sometimes written $J$ or $Z$ for short.

The sum, product, and difference of any two integers is also an integer. But this is not true for division… just try $1 \div 2$ .

The Rational Numbers

The rational numbers are those numbers which can be expressed as a ratio between two integers. For example, the fractions $\frac{1}{3}$ and $- \frac{1111}{8}$ are both rational numbers. All the integers are included in the rational numbers since any integer $z$ can be written as the ratio $\frac{z}{1}$ .

All decimals which terminate are rational numbers (since $8.27$ can be written as $\frac{827}{100}$ .) Decimals which have a repeating pattern after some point are also rationals: for example,

$0.0833333.... = \frac{1}{12}$

The set of rational numbers is closed under all four basic operations, that is, given any two rational numbers, their sum, difference, product, and quotient is also a rational number (as long as we don’t divide by $0$ ).

The Irrational Numbers

An irrational number is a number that cannot be written as a ratio (or fraction). In decimal form, it never ends or repeats. The ancient Greeks discovered that not all numbers are rational; there are equations that cannot be solved using ratios of integers.

The first such equation to be studied was $2 = x^{2}$ . What number times itself equals $2$ ?

$\sqrt{2}$ is about $1.414$ , because ${1.414}^{2} = 1.999396$ , which is close to $2$ . But you’ll never hit exactly by squaring a fraction (or terminating decimal). The square root of $2$ is an irrational number, meaning its decimal equivalent goes on forever, with no repeating pattern:

$\sqrt{2} = 1.41421356237309...$

Other famous irrational numbers are the golden ratio, a number with great importance to biology:

$\frac{1 + \sqrt{5}}{2} = 1.61803398874989...$

$π$ (pi), the ratio of the circumference of a circle to its diameter:

$π = 3.14159265358979...$

and $e$ , the most important number in calculus:

$e = 2.71828182845904...$

Irrational numbers can be further subdivided into algebraic numbers, which are the solutions of some polynomial equation (like $\sqrt{2}$ and the golden ratio), and transcendental numbers, which are not the solutions of any polynomial equation. $π$ and $e$ are both transcendental.

The Real Numbers

The real numbers are the set of numbers containing all of the rational numbers and all of the irrational numbers. The real numbers are “all the numbers” on the number line. There are infinitely many real numbers just as there are infinitely many numbers in each of the other sets of numbers. But, it can be proved that the infinity of the real numbers is a bigger infinity.

The Complex Numbers

The complex numbers are the set { $a + b i$ | $a$ and $b$ are real numbers}, where $i$ is the imaginary unit, $\sqrt{- 1}$ .

The complex numbers include the set of real numbers. The real numbers, in the complex system, are written in the form $a + 0 i = a$ . a real number.

This set is sometimes written as $C$ for short. The set of complex numbers is important because for any polynomial $p (x)$ with real number coefficients, all the solutions of $p (x) = 0$ will be in $C$ .

Summary

In this blog post, the basic concept of numbers is explained with the help of detailed examples to clarify the concept. The types of Numbers (Natural, Whole, Integers, Real, Complex, Rational, Irrational) are also explained in detail taking some examples to clarify the concept.
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