Table of Contents
Number System
A number system is a key concept in every branch of mathematics. The use and scope of the number are unlimited. The system deals with the nomenclature, use, and properties of numbers.
The post is a brief introduction of numbers and its application in different competitive questions.
The number system that we use in our everyday life is called the decimal system. This is because there are 10 digits (0, 1, 2, …, 9).
Number System Types
There are two types of number systems. And they are Real numbers and Complex numbers.
Real Number
The set of all numbers that can be represented on the number line are called real numbers.
\[eg,~~5,-7,0,9.57,2+\sqrt{3},\frac{7}{11}~~etc.\]
Real Number Line
A number line is a straight line with an arbitrary defined point zero. To the right of this point lie all positive numbers and to the left, all negative numbers.
Now, real numbers can be divided into two categories, rational numbers, and irrational numbers.
Rational Numbers
If a number can be expressed in the form of p/q, where q ≠ 0, where p and q are integers, then the number is called a rational number.
\[e.g,~~\frac{7}{11},\frac{19}{7},\frac{12}{1},-\frac{19}{51}~~etc.\]
All integers are also rational numbers. Every terminating decimal or a repeating decimal is also a rational number, e.g, 5.1, 6.131313, …, etc,
Rational numbers can be further sub-divided into two parts integers and fractions.
Integers
Integers are the set of all non-fractional numbers lying between -∞ and +∞. Hence, integers include negative as well as positive non-fractional numbers. Integer is denoted by Z or I.
\[I=\left\{ -\infty ,…,-4,-3,-2,-1,0,1,2,3,4,…,+\infty \right\}\]
Note that 0 is neither a positive nor a negative integer.
Integers can be further subdivided into negative numbers and whole numbers. Whole numbers have two sections zero (0) and positive numbers popularly called as natural numbers.
Natural Numbers
Set of natural numbers is denoted by N.
\[N=\left\{ 1,2,3,4,5,…,\infty \right\}\]
Even Numbers
All numbers that are divisible by 2 are called even numbers.
\[eg,~~\left\{ 2,4,6,8,10,12,…,\infty \right\}\]
Odd Numbers
All numbers that are not divisible by 2 are called odd numbers.
\[eg,~~\left\{ 1,3,5,7,9,11,…,\infty \right\}\]
Prime Numbers
The numbers that have only two factors 1 and the number itself, are called prime numbers.
\[eg,~~\left\{ 2,3,5,7,11,13,17,…. \right\}\]
Note that number 1 is not a prime number.
Fractions
A fraction includes two parts, numerator and denominator,
\[-\frac{3}{7},\frac{9}{5},\frac{11}{7}~~etc.\]
Fractions are primarily of five types.
Proper Fraction
A rational number in the form of p/q, where q ≠ 0, where the numerator is less than the denominator.
\[eg,~~\frac{3}{7}\]
Improper Fraction
A rational number in the form of p/q, where q ≠ 0, where the numerator is more than the denominator.
\[eg,~~\frac{7}{3}\]
Mixed Fraction
Mixed fraction consists of integral as well as the fractional part
\[eg,~~2\frac{3}{7}=2+\frac{3}{7}=\frac{17}{7}\]
It means that a mixed fraction is always an improper fraction.
Compound Fraction
A fraction of a fraction is known as compound fraction.
\[eg,~~\frac{4}{5}~~of~~\frac{9}{11}=\frac{4}{5}\times \frac{9}{11}\]
Complex Fraction
Any complicated combination of the other type of fractions.
\[eg,~~2\frac{1}{3}~~of~~\frac{3}{1+\frac{2}{3}}\]
Irrational Numbers
If a number cannot be expressed in the form of p/q, q ≠ 0, then the number is called an irrational number.
In other words, non-repeating as well as the non-terminating type of decimals are called irrational numbers.
\[eg,~~\sqrt{2},3\sqrt{4},…,4.965696…,3.14592…\]
Important Number Series
The main two number series types are Arithmetic Progression and Geometric Progression.
Arithmetic Progression
A number series which progresses in such a way that the difference between two consecutive numbers is common is called the arithmetic progression.
\[eg,~~2,7,12,17,22,27,……….\]
Geometric Progression
A progression of numbers in which every term bears a constant ratio with its preceding term, is called geometric progression.
\[eg,~~2,4,8,16,……,1024\]
Geometric Progression |
Quadratic Equation |