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Is Every Real Number a Complex Number?

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  • Last Updated : 29 Aug, 2022
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A complex number is referred to as the sum of a real number and an imaginary number. It is generally expressed as “z” and is written in the form of a + ib, where a and b are real numbers and i = √(-1). Here, “a” is a real part that is represented as Re(z) and “ib” is an imaginary part that is represented as Im(z). Some examples of complex numbers are 2 + 3i, 5–7i, 3 + i√4, etc. The imaginary number is generally expressed either as “i” or “j”, whose value is equal to √(-1). Hence, the square of an imaginary number gives us a negative value. The square root of negative numbers can be calculated using complex numbers. Some applications of complex numbers are in signal processing, fluid dynamics, quantum mechanics, electromagnetism, vibration analysis, and also many scientific research areas.

complex number

 

Real numbers are referred to as the union of the set of rational numbers and the set of irrational numbers, i.e., positive numbers, whole numbers, integers, rational numbers, irrational numbers, etc. are real numbers. Some examples of real numbers are -4, -7/11, 0, 9, √6, 3.8, etc.

A number that gives a negative value when squared is called an imaginary number. It is the product of a non-zero real number and the imaginary unit “i”, whose value is √(-1). An imaginary number can also be defined as the square root of negative numbers. Some examples of imaginary numbers are -2i, √5i, 3i, etc.

Is Every Real Number a Complex Number?

Answer:

A complex number generally has two parts, i.e., real and imaginary parts. It is written in the form of a + ib, where “a” is a real number and “ib” is an imaginary number. Let us assume that the imaginary part is 0. Now, we will only have the real number part. So, every real number is a complex number when the imaginary part is zero. But remember that every complex number is not a real number.

For example, we can write 5 as 5+0×(i).

Hence, Every Real Number is a Complex Number.

Power of i

The alphabet “i” represents the imaginary part of a complex number, also known as “iota”. The value of “i” is √(-1), i.e., i2 = -1. Iota aids in calculating the square root of negative numbers. For example, √-9 = √(i2 × 9) = +3i.

  • i = √(-1)
  • i2 = -1
  • i3 = i2 × i = -i
  • i4 = i2 × i2 = (-1) × (-1) = 1
  • i4n = 1
  • i(4n+1) = i
  • i(4n+2) = -1
  • i(4n+3) = -i

Graphical Representation of complex numbers

Representation of a Complex Number

Representation of a Complex Number

A complex number of the form z = a + ib can be represented on the argand plane by considering its coordinates as (Re(z), Im(z)) = (a, ib). An Argand plane or a complex plane is a Euclidean plane concerning complex numbers where the real part of a complex number “a” is represented on the X-axis and the imaginary part “ib” is represented on the Y-axis. The modulus of the complex number (r) is the distance of the complex number represented as a point in the argand plane (a, ib), i.e., the linear distance between the origin (0, 0) and the point (a, ib). 

r = √(a2 + b2)

The argument of the complex number is the angle in the anticlockwise direction made by the line joining the geometric representation of the complex number and the origin, with the positive x-axis.

Argz (θ) = tan−1(b/a)

Polar Form of a Complex Number

A complex number can also be represented and identified on the argand plane by using its polar form. To represent the complex number on the argand plane, the polar form makes use of the modulus and argument of the complex number. A complex number z = a + ib is expressed as z = r(cosθ + isinθ) in its polar form, where r is the modulus and θ is the argument of a complex number. Here, r is equal to √(a2 + b2), whereas θ is equal to tan-1(b/a).

Conjugate of a Complex Number

The conjugate of a complex number is another complex number that has the same real part as the original complex number, and the magnitude of the imaginary part is also the same but with the opposite sign. Two complex numbers are said to be each other’s conjugates if their sum and product are real numbers. 

  • The conjugate of the complex number z = a + ib is z̅ = a − ib.
  • Sum of a complex number and its conjugate = z + z̅ = (a + ib) + (a − ib) = 2a
  • Product of a complex number and its conjugate = z × z̅ = (a + ib)×(a − ib)= a2+b2

Reciprocal of a Complex Number

While performing the division of two complex numbers, we make use of the reciprocal formula of a complex number. When a complex number is divided by another, the result is equal to the product of one complex number with the reciprocal of another complex number.

The value of the reciprocal of the complex number z = a + ib is 

z−1=1/(a + ib) =(a − ib)/(a2+b2)

Arithmetic Operations on Complex Numbers

We can perform various arithmetic operations such as addition, subtraction, multiplication, and division on complex numbers just as we can do on natural numbers. Here, we have to combine like terms while performing arithmetic operations; i.e., we must combine the real number with the real number and the imaginary number with the imaginary number.

Addition of Complex Numbers

When two complex numbers are added, the real part is added to the real part and the imaginary part is added to the imaginary part. Let us consider two complex numbers of the form z1 = a + ib and z2 = c + id. Now,

 z1 + z2 = (a + c) + i(b + d). 

Subtraction of Complex Numbers

When a complex number is subtracted from the other, the subtraction is separately performed for the real parts and then performed for the imaginary parts. Let us consider two complex numbers of the form z1 = a + ib and z2 = c + id. Now,

z1 – z2 = (a-c) +i(b – d)

Multiplication of Complex Numbers

The multiplication of two complex numbers is similar to the multiplication process of two binomials. Here, we use the formula i2 = -1. Let us consider two complex numbers of the form z1 = a + ib and z2 = c + id. Now,

z1 × z2= (a + ib) × (c + id) = (ac – bd) + i(ad + bc)

Division of Complex Numbers

While performing the division of two complex numbers, we make use of the reciprocal formula of a complex number. Let us consider two complex numbers of the form z1 = a + ib and z2 = c + id. Now,

z1/z2 = (a + ib) × 1/(c + id) = (a + ib) × (c – id)/(c2 + d2)

Algebraic Identities of Complex Numbers

Following are some algebraic identities of complex numbers:

  • (z1 + z2)2 = (z1)2 + 2z1z2 + (z2)2 
  • (z1 – z2)2 = (z1)2 – 2z1z2 + (z2)2 
  • (z1)2 – (z2)2 =  (z1 + z2)(z1 – z2)
  • (z1 + z2)3 = (z1)3 + 3z1z2(z1 + z2) + (z2)3
  • (z1 – z2)3 = (z1)3 – 3z1z2(z1 – z2) – (z2)3
  • (z1 + z2 + z3)2 = (z1)2 + (z2)2 + (z3)2 + 2z1z2 + 2z2z3+ 2z3z1

Solved Examples on Complex Numbers

Example 1: Simplify:

a) (3i)(2 + 5i)

b) 8i + 15i – 7i

Solution:

a) (3i)(2 + 5i)

= (3i)(2) + (3i)(5i)

= 6i + 15i2

We know that i2 = –1

= 6i + 15(–1)

(3i)(2 + 5i) = 6i – 15

b) 8i + 15i – 7i = 23i – 7i = 16i

Example 2: Solve: (2+i)/(2-i).

Solution:

Given: (2+i)/(2-i)

Now multiply both numerator and denominator with (2+i)

(2+i)/(2-i) × (2+i)/(2+i)

(2+i)(2– i) = (2)2 – i2  {Since, i2 = –1}

= 4 –(–1) = 4+1 = 5

(2+i)/(2-i) × (2+i)/(2+i) = (2+i)2/5

= (22 + 2×2×i + i2)/5

= (4 + 4i –1)/5

= (3+4i)/5

Hence, (2+i)/(2-i) = (3+4i)/5.

Example 3: Determine the sum and product of the complex numbers z1 = 2 + 3i and z2 = 1 – i as a complex number.

Solution:

Given: z1 = 2 + 3i

z2 = 1 – i 

Sum: 

z1 + z2 = 2 + 3i + 1 – i 

= (2 + 1) + (3i – i)

z1 + z2 = 3 + 2i

Product:

z1 × z2 = (2 + 3i) × (1 – i)

= 2(1 – i) + 3i(1 – i)

= 2 – 2i + 3i – 3i2

= 2 + i – 3(–1)    {Since, i2 = –1}

= 2 + i + 3

z1 × z2 = 5 + i

Example 4: Determine the difference and quotient of the complex numbers z1 = 5 – 4i and z2 = 3+ i as a complex number.

Solution:

Given: z1 = 5 – 4i

z2 = 3+ i

Difference:

z1 – z2 = 5 – 4i – (3+ i)

= (5 – 3) + (–4i – i)

z1 – z2 = 2 – 5i

Division:

z1/z2 = (5 – 4i)/(3+ i)

= (5 – 4i)/(3+ i) × (3 – i)/(3 – i)

(3 + i)(3 – i) = 32 – i2 

= 9 – (–1) = 9 +  = 10   {Since, i2 = –1}

(5 – 4i)/(3+ i) × (3 – i)/(3 – i) = [(5 – 4i)(3 – i)]/10

= (15 – 5i – 12i + 4i2)/10

= (15 –17i + 4(–1))/10    {Since, i2 = –1}

z1/z2 = (11 – 17i)/10

Example 5: Determine the modulus, conjugate, and reciprocal of the complex number 4 + 3i.

Solution:

Given: z = 4 + 3i

Modulus = √(42 + 32) = √(16 + 9) = √25 = 5

Conjugate of 4 + 3i is z̅ = 4 – 3i

Reciprocal = 1/(4 + 3i) = (4 – 3i)/(42+32)

=(4 – 3i)/(16 + 9) = (4 – 3i)/25

Reciprocal of 4 + 3i = (4 – 3i)/25

FAQs on Complex Numbers

Question 1: What is a complex number?

Answer:

A complex number is referred to as the sum of a real number and an imaginary number. It is generally expressed as “z” and is written in the form of a + ib, where a and b are real numbers and i = √(-1). Here, “a” is a real part that is represented as Re(z) and “ib” is an imaginary part that is represented as Im(z). Some examples of complex numbers are 2 + 3i, 5–7i, 3 + i√4, etc.

Question 2: What is the standard form of a Complex Number?

Answer:

A complex number is generally expressed as “z” and is written in its standard form as z = a + ib. A complex number in its standard form has two parts: a real part and an imaginary part. Here, “a” is a real part that is represented as Re(z) and “ib” is an imaginary part that is represented as Im(z). 

Question 3: How to represent a complex number on a graph?

Answer:

A complex number of the form z = a + ib can be represented on the argand plane by considering its coordinates as (Re(z), Im(z)) = (a, ib). An Argand plane or a complex plane is a Euclidean plane concerning complex numbers where the real part of a complex number “a” is represented on the X-axis and the imaginary part “ib” is represented on the Y-axis.

Question 4: What is meant by modulus and argument in complex numbers?

Answer:

The modulus of the complex number (r) is the distance of the complex number represented as a point in the argand plane (a, ib), i.e., the linear distance between the origin (0, 0) and the point (a, ib).

r = √(a2 + b2)

The argument of the complex number is the angle in the anticlockwise direction made by the line joining the geometric representation of the complex number and the origin, with the positive x-axis.

Argz (θ) = tan−1(b/a)

Question 5: What is the polar form of a complex number?

Answer:

A complex number can also be represented and identified on the argand plane by using its polar form. To represent the complex number on the argand plane, the polar form makes use of the modulus and argument of the complex number. A complex number z = a + ib is expressed as z = r(cosθ + isinθ) in its polar form, where r is the modulus and θ is the argument of a complex number. Here, r is equal to √(a2 + b2), whereas θ is equal to tan-1(b/a).


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