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What is the difference between Real and Complex Numbers?

  • Last Updated : 03 Sep, 2021

A number system is a way of showing numbers by writing, which is a mathematical way of representing the numbers of a given set, by using the numbers or symbols in a mathematical manner. The writing system for denoting numbers using digits or symbols in a logical manner is defined as the Number system. The numeral system,

  • which Represents a useful set of numbers
  • also Reflects the arithmetic and algebraic structure of a number
  • and Provides standard representation.

The digits from 0 to 9 can be used to form all other numbers. With the use of these digits, an infinite set of numbers can be created. For example, 156,3907, 3456, 1298, 784859 etc.

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Real Numbers 

All the negative and positive integers, decimal and fractional numbers without imaginary numbers are called real numbers. Real numbers are represented by the “R” symbol. Real numbers can be explained as the union of both rational and irrational numbers. They can be both negative or positive and are denoted by the symbol “R”. All the decimals, natural numbers, and fractions come under this category. The examples below show the classification of real numerals.



Rational  ⇢   – {5/3 , 0 .63 , -6/5 O.7116 ….}    

Irrational numbers  ⇢  -{√3, √5, √11, √21, π(Pi)}                                             

Integers  ⇢   – {-3, -2,-1,0,1,2 , 3….}

Whole numbers  ⇢   -{ 0,1,2,3,4..}

Natural numbers  ⇢   –  {1,2,3,4….}

There are different sets of real numbers such as natural and whole numbers, integers, rational and irrational numbers. here below all these are defined by examples,

Natural numbers are which Contains all numbers start from 1

N = {1, 2, 3, 4,…} All NUMBERS such as 1, 2, 3, 4, 5…. and so on.



Whole numbers are defined as the set of natural number and zero 

W = { 0, 1, 2, 3…} such as 0,1, 2, 3, 4, 5…

Integers are the collection of all negative natural numbers and whole numbers are called integers.

such as : – infinity(∞),… -5 , -4 , -3 ,-2 , -1 , 0  , 1 , 2 , 3 , 4 , 5… +∞

Rational numbers are all numbers which we can write in form of a/b, where b ≠ 0.

such as : 2/4 , -3/5 , 0.768 , 0,50 …

Irrational numbers are the numbers that we cannot write in form of a/b and numbers that are not rational are called irrational numbers. Such as √6, √8 …

Complex Numbers

The sum of a real number and an imaginary number is defined as a complex number, and the numbers which are not real numbers are called imaginary numbers. The number can be written in form of b+ic, where b and c are real numbers and ic is an imaginary number, and ” i” is an imaginary part which is called iota. hence here the value of i is (√-1) . so i2=-1  

The symbol “i” is referred to as iota and represents the imaginary part of the complex number. Further the iota(i) is very useful to find the square root of negative numbers. For example, 5+6i is a complex number, so here 5 is a real number and 6i is an imaginary number. Hence, a complex number is a representation of the addition of two numbers, one is a real number and the second is an imaginary number. One part of its purely real and the second part is purely imaginary.

Note The combination of both Imaginary number and the Real number is called the Complex number and represented by “C”. This can be written as b+ic, which is mostly represented by z=b+ic.



Difference between the Complex number and Real number

From the above definitions, there are few differences that can be easily taken out. The real numbers are a subset of Complex numbers and the complex numbers are the superset of real numbers. Let’s take a look at the differences more clearly,

  • All the negative and positive integers, decimal and fractional numbers without imaginary numbers are called real numbers. Real numbers are represented by the “R” symbol. Whereas the sum of a real number and an imaginary number is called a complex number represented by C. The numbers which are not real numbers are called imaginary numbers. The number which we can write in form of b+ic, where b and c are real numbers and ic is an imaginary number, and ” i” is an imaginary part which is called iota. hence here the value of i is (√-1). So i2=-1
  • Another important point is that real numbers can be plotted on the number line, whereas complex numbers cannot be plotted on a number line.
  • All real numbers are also complex numbers with zero for the imaginary part, whereas all imaginary numbers are also complex numbers with zero for the real part.
  • Real numbers include all decimal fractional, negative, and positive integers, whereas the Complex number can be written as the sum or difference of a real number and imaginary number, include numbers like 4 – 2i or 6+√6i.

Given the table below contain examples that show how real numbers are a part of complex numbers, complex numbers are shown in two parts, one real and the other imaginary.

Complex numberReal number Imaginary number
-3 + 2i-32i
8 – 9i8-9i
-5i0-5i (purely imaginary)
550i  (purely real)

Sample Problems

Question 1: Do the addition on the two complex numbers 4 + 2i and 4 + 7i.

Solution:

First add the real number and 

Add the imaginary numbers

(4 + 2i )  + (4 + 7i)

= 4 + 4 + (2i+7i)

=  8 +  (2 + 7)i

= 8 + 9i



Question 2: Add the complex numbers 4 + 5i and  7− 3i.

Solution:

Again first add the real number and 

Add the imaginary number 

( 4 + 5i ) +  (7− 3i)

= 4 + 7 + (5i – 3i)

=  11  +  (5 – 3)i

=  11 + 2i

Question 3: Multiply the two Complex numbers, (5 + 2i)(1 + 7i)

Solution:           

Every each part of the first complex number gets multiplied by each part of the second complex number 

(5 + 2i)(1 + 7i) = 5 × 1 + 5 × 7i + 2i × 1 + 2i × 7i

=  5 + 35i + 2i + 14i2

=  5 + 35i + 2i + 14(-1)  (because i2 = −1)

=  5 + 37i – 14

= -9 + 37i

Question 4: Identify the real numbers among the following numbers: √6, -3, 3.15, -1/2, √-5, 2.

Answer:

Among the given numbers, √-5 is a complex number. Imaginary numbers like √-5 can’t be a real number. The other numbers are either rational or irrational. Thus, they are real numbers. Therefore, the real numbers from the list are √6, -3,  3.15, and -1/2, 2.




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