What is Z Bar in Complex Numbers?

Answer: We refer to or the complex number obtained by altering the sign of the imaginary part (changing positive to negative or vice versa) as the conjugate of z.

A complex number is defined as the addition of a real number and an imaginary number. It is represented as "z" and is written in its standard form as (a + ib), where a and b are real numbers and i is an imaginary unit whose value is √(-1). The real part of the complex number is represented as Re (z), and its imaginary part is represented as Im(z). Some examples of complex numbers are 1 + √2i, 6–4i, 5 + 7i, etc. The imaginary unit is called "iota," which is either represented as "i" or "j". Complex numbers aid in calculating the square root of negative numbers. In addition to this, complex numbers also play a major role in signal processing, fluid dynamics, AC circuit analysis, electromagnetics, quantum mechanics, etc.

Complex Conjugate

The conjugate of a complex number is also a complex number obtained by changing the sign between the real and imaginary parts of the original complex number. The magnitude of a complex number and its conjugate is the same. The conjugate of a complex number z is denoted by z̅ or z*. If z = a + ib is a complex number, where a is the real part and ib is the imaginary part, then its conjugate is z* or z̅ = a – ib. In an argand plane, the complex conjugate (a – ib) is the mirror image of the complex number (a + ib) about the real axis.

Properties of a Complex conjugate

• The real part of the complex conjugate is equal to the real part of the complex number, while the imaginary part of the complex conjugate is equal to the negative of the imaginary part of the complex number.

Re(z̅) = Re(z)

Im(z̅) = −Im(z) 

• The sum of a complex number z and its complex conjugate z* is a real number.

z + z̅ = (a + ib) + (a − ib) = 2a = 2Re(z)

• The difference between a complex number z and its complex conjugate z* is an imaginary number.

z − z̅ = (a + ib) − (a − ib) = 2ib = 2Im(z)

• The product of the complex number z and its complex conjugate z* is a real number.

z × z̅ = (a + ib)×(a − ib)= a²+b²

• If z and w are two complex numbers, then the complex conjugate of their product is equal to the product of their complex conjugates.

• If z and w are two complex numbers, then the complex conjugate of their quotient is equal to the quotient of their complex conjugates.

• If z and w are two complex numbers, then the complex conjugate of their sum is equal to the sum of their complex conjugates.

• If z and w are two complex numbers, then the complex conjugate of their difference is equal to the difference between their complex conjugates.

Graph of a complex number and its conjugate

A complex number z = a + ib can be represented as a point on the Euclidean plane coordinates as (Re(z), Im(z)). The Euclidean plane that represents complex numbers as points where X and Y axes represent the real and imaginary parts of a complex number is called an argand plane or a complex plane. From the below graph, we can observe that the conjugate of a complex number is the reflection of a complex number about the real axis (X-axis). 

• The polar form of a complex number z = a + ib is z = re^ix = r(cosθ + isinθ), where r is the modulus and θ is the argument of a complex number. The polar form of a complex number helps to represent and identify it on an argand plane. The polar form of the complex conjugate is z̅ =  re^−ix = r(cosθ − isinθ).
• The modulus of a complex number is defined as the distance of the complex number z = a + ib from the origin, in an argand plane. It is denoted by |z| and its value is r = √(a² + b²). 
• The angle made in the anticlockwise direction by the line that joins the origin and the point that represents a complex number with the positive X-axis is called the argument of the complex number.

Argz (θ) = tan⁻¹(b/a)

Arithmetic Operations on Complex Numbers

The arithmetic operations like addition, subtraction, multiplication, and division can be performed on complex numbers just as we can do on natural numbers. Make a note that, while performing arithmetic operations on complex numbers, we have to combine the like terms, i.e., the real parts are combined separately and the imaginary parts are combined separately. Let z₁ = a + ib and z₂ = c + id be two complex numbers. Now, the arithmetic rules of complex numbers are as follows:

Addition of Complex Numbers

If two complex numbers z1 and z2 are added, then the real parts of z1 and z2 are added separately and the imaginary parts of z1 and z2 are added separately.

z₁ + z₂ = (a + ib) + (c + id) = (a + c) + i(b + d)

Subtraction of Complex Numbers

If a complex number z₁ is subtracted from z₂, then the real part of z₁ is subtracted from z₂, and the imaginary part of z₁ is subtracted from z₂.

z₂ – z₁ = (c + id) – (a + ib) = (c – a) + i(d – b)

Multiplication of Complex Numbers

The process of multiplying two complex numbers is identical to multiplying two binomials. i² = -1 formula is used while performing the multiplication of two complex numbers.

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

Division of Complex Numbers

If a complex number z₁ is divided by z₂, then the result is equal to the product of z₁ and the reciprocal of z₂.

The reciprocal formula of a complex z = a + ib is z⁻¹=1/(a + ib) =(a − ib)/(a²+b²)

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

What is  in complex numbers?

Answer:

 is the complex conjugate of a complex number z = a + ib. The conjugate of a complex number is also a complex number obtained by changing the sign between the real and imaginary parts of the original complex number. 

If z = a + ib is a complex number, where a is the real part and ib is the imaginary part, then its conjugate is 

z* or z̅ = a – bi

Here, (a + ib) and (a – ib) are conjugates of each other, i.e., (a + ib) is the complex conjugate of (a – ib) and vice-versa. The magnitude of the conjugate (a + ib) is the same as the (a – ib). The sum and product of a complex number and its conjugate are real.

Solved Examples on Complex Numbers

Example 1: Simplify: (3+4i)/(1–5i).

Solution:

Given: (3+4i)/(1–5i)

To rationalize the denominator, multiply both numerator and denominator with the conjugate of (1–5i), i.e., (1+5i).

(3+4i)/(1–5i) × (1+5i)/(1+5i)

(1–5i(1+5i) = 1² – (5i)²

= 1 –(–25)     {Since, i² = –1}

= 1+25 = 26

(3+4i)/(1–5i) × (1+5i)/(1+5i)= [(3+4i)(1+5i)]/26

= (3+15i+4i+20i²)/26

= (3+19i–20)/26

= (19i–17)/26

Hence, (3+4i)/(1–5i) = (19i–17)/26.

Example 2: If z = 5 – 7i and w = 4 + i, then express the complex number z/w in the form of a + ib, where a and b are real numbers.

Solution: 

Given: z = 5 – 7i and w = 4 + i

Now, z/w = (5 – 7i)/(4 + i)

To rationalize the denominator, multiply both numerator and denominator with the conjugate of (4 + i), i.e., (4 – i).

(5 – 7i)/(4 + i) × (4 – i)/(4 – i)

(4 + i) × (4 – i) = 4² – i²

= 16 – (–1)    {Since, i² = –1}

= 16 + 1 = 17

Now, (5 – 7i)/(4 + i) × (4 – i)/(4 – i)

= [(5 – 7i)(4 – i))]/17

= (20 – 5i – 28i + 7i²)/17

= (20 – 33i + 7(–1))/17

 z/w = (13 – 33i)/17

Hence,  z/w = (13 – 33i)/17.

Example 3: Simplify:

a) (4 – 9i)(6 + 3i)

b) (5 + 11i) – (12 – 13i)

Solution:

a)(4 – 9i)(6 + 3i)

= 4(6 + 3i) – 9i(6 + 3i)

= 24 + 12i – 54i – 27i²

= 24 – 42i – 27(–1)   {Since, i² = –1}

= 24 – 42i + 27 = 51 – 42i

Therefore, (4 – 9i)(6 + 3i) = 51 – 42i.

b) (5 + 11i) – (12 – 13i)

= 5 + 11i – 12 + 13i

= (5 – 12) + (11i + 13i)

= 24i – 7

Therefore, (5 + 11i) – (12 – 13i) = (24i – 7).

Example 4: Find the values of a and b if (7 + 3i)/(2 – 3i) = a + ib, where a and b are real numbers.

Solution:

Given: (7 + 3i)/(2 – 3i) = a + ib, a, b ∈ R

To rationalize the denominator, multiply both numerator and denominator with the conjugate of (2 – 3i), i.e., (2 + 3i).

(7 + 3i)/(2 – 3i) × (2 + 3i)/(2 + 3i)

(2 – 3i) × (2 + 3i) = 2² – (3i)²

= 4 – 9i² = 4 – 9(–1)    {Since, i² = –1}

= 4 + 9 = 13 

Now, (7 + 3i)/(2 – 3i) × (2 + 3i)/(2 + 3i) = [(7 + 3i)(2 + 3i)]/13

= (14 + 21i + 6i + 9i²)/13

= (14 + 27i + 9(–1))/13

= (5 + 27i)/13

(7 + 3i)/(2 – 3i) = (5/13) + (27/13)i

a + ib = (5/13) + (27/13)i

So, a = 5/13 and b = 27/13.

Thus the values of a and b are 5/13 and 27/13.

Example 5: Find the sum, difference, and product of the complex numbers z₁ = 8 – 5i and z₂ = 1 – 3i as a complex number.

Solution:

Given: z1 = 8 – 5i

z₂ = 1 – 3i

Sum:

z₁ + z₂ = 8 – 5i + 1 – 3i

= (8 + 1) + (–5i – 3i)

z₁ + z₂ = 9 – 8i

Difference:

z₁ – z₂ = (8 – 5i) – (1 – 3i)

= 8 – 5i – 1 + 3i

= (8 – 1) + (–5i + 3i)

z₁ – z₂ = 7 – 2i

Product:

z₁ × z₂ = (8 – 5i) × (1 – 3i)

= 8(1 – 3i) – 5i(1 – 3i)

= 8 – 24i – 5i + 15i²

= 8 – 29i + 15(–1)    {Since, i² = –1}

= 8 – 15 – 29i = –(7 + 29i)

z₁ × z₂ = –(7 + 29i).