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In mathematics, a complex number is said to be a number which can be expressed in a + bi form, where a and b are real numbers and i is the imaginary unit.

Here, a is named as the real part of the number and b is referred to as the imaginary part of a number.

The following table provides a representation of the complex numbers.

Modulus of complex number defined as | z | where if z = a + bi is a complex number. The modulus of the complex number will be defined as follows:

| Z | =a + bi

| z | =0 then it indicates a=b=0

| -z | = | z |

Imagine z_{1} and z_{2} are two complex numbers, then

| z1.z2 | = | z1 | | z2 |

| z1 + z2 | ≤ | z1 | + | z2 |

| z1/ z2 | = | z1 | / | z2 |

There seems to be a method to get a sense of how large these numbers are. We consider the conjugate complex and multiply it by the complex number specified in (1). Therefore we describe the product \[z\overline{z}\] as the square of a complex number's Absolute value or modulus. So let's write \[z\overline{z}\] = |z|2.

As per the explanation, \[z\overline{z}\] provides a calculation of the absolute value or magnitude of the complex number. When you learn about the Argand Plane, the exact explanation for that concept will become apparent.

Therefore, |z|2 = (a2 + b2) [Using (1)]

Hence, |z| = √(a2+b2) …(2)

The equation above is the modulus or absolute value of the complex number z.

The complex conjugate of a complex number is the number with the same real part and the imaginary part equal in magnitude, but are opposite in terms of their signs.

Complex conjugates are responsible for finding polynomial roots. According to the complex conjugate root theorem, if a complex number in one variable with real coefficients is a root to a polynomial, so is its conjugate.

If you are wondering how to find the conjugate of a complex number, then go through this. Each complex number has a relationship with another complex number known as its complex conjugate. You can find the conjugate complex by merely changing the symbol of the imaginary part of the Complex numbers.

For example:

We alter the sign of the imaginary part to find the complex conjugate of 4 + 7i. So the complex conjugate is 4 − 7i.

Example:

We alter the sign of the imaginary part to find the complex conjugate of 1−3i. So the complex conjugate is 1 + 3i.

Example:

We alter the sign of the imaginary component to find the complex conjugate of −4 − 3i. So the complex conjugate is −4 + 3i.

It is to be noted that the conjugate complex has a very peculiar property.

If we multiply a complex number by its complex conjugate, think about what will happen.

Let’s take this example:

Multiply (4 + 7i) by (4 − 7i): (4 + 7i)(4 − 7i) = 16 − 28i + 28i − 16 + 49i2 = 65

We find the answer to this is a strictly real number; there is no imaginary part. It often occurs when a complex number is multiplied by its conjugate, the consequence is a real number.

To add two complex numbers of the x plus iy form, we have to add the real parts and the imaginary parts individually.

Let z = a + ib reflect a complex number. Module of z , referred to as z, is defined as the real number (a2 + b2)1/2 z = (a2 + b2)1/2

Numerical: Evaluate the modulus of (3-4i)

z = (a2 + b2)1/2 = (32 + 42)1/2 = 5

Let z = a + ib reflect a complex number. Z conjugate is the complex number a - ib, i.e., = a - ib.

Z * = Z

Or Z–1 = / Z (Useful to find a complex number in reverse)

Properties of complex numbers are mentioned below:

1. In case of a and b are real numbers and a + ib = 0 then a = 0, b = 0

Proof:

According to the property,

a + ib = 0 = 0 + i ∙ 0,

Therefore, we conclude that, x = 0 and y = 0.

2. For any three the set complex numbers z1, z2 and z3 satisfies the commutative, associative and distributive laws.

(i) z1 + z2 = z2 + z1 (Commutative law for addition)

(ii) z1 ∙ z2 = z2 ∙ z1 (Commutative law for multiplication)

(iii) (z1 + z2) + z3 = z1 + (z2 + z3) (Associative law for addition)

(iv) (z1z2)z3 = z1(z2z3) (Associative law for multiplication)

(v) z1(z1 + z3) = z1z2 + z1z3 (Distributive law)

3. The sum of two complex conjugate numbers is real.

Proof:

Let, z = a + ib (a, b are real numbers) be a complex number. Then, a conjugate of z is \[\overline{z}\] = a - ib.

Now, z + \[\overline{z}\] = a + ib + a - ib = 2a, which is real.

4. The product of two complex conjugate numbers is real.

Proof:

Let, z = a + ib (a, b are real numbers) be a complex number. Then, a conjugate of z is \[\overline{z}\] = a - ib.

z ∙ \[\overline{z}\] = (a + ib)(a - ib) = a2 - i2b2 = a2 + b2, (Since i2 = -1), which is real.

Hence, \[z\overline{z}\] −−√ = a2+b2−−−−−−√

Therefore, |z| = \[\overline{z}\] −−√

Thus, the modulus of any complex number is equal to the positive square root of the product of the complex number and its conjugate complex number.

5. When the sum of two complex numbers is real, and the product of two complex numbers is also natural, then the complex numbers are conjugated.

Proof:

According to the property,

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

Therefore, b + d = 0

⇒ d = -b

And,

z1z2 = (a + ib)(c + id) = (a + ib)(c +id) = (ac– bd) + i(ad + bc) is real.

Therefore, ad + bc = 0

⇒ -ab + bc = 0, (Since, d = -b)

⇒ b(c - a) = 0

⇒ c = a (Since, b ≠ 0)

Hence, z2 = c + id = a + i(-b) = a - ib = \[\overline{z1}\]

Therefore, we conclude that z1 and z2 are conjugate to each other.

|z1 + z2| ≤ |z1| + |z2|, for two complex numbers z1 and z2.

FAQ (Frequently Asked Questions)

Q1. What is a Complex Conjugate Number?

Ans. Complex numbers are called a complex conjugate of each other when in two complex numbers, the sign of the imaginary part is differing. Complex numbers in the binomial form are depicted as (a + ib). It invites you to work with the "+" symbol. What about if we turn it to a minus sign? Let z = a + ib represent a complex number. We describe another complex number ¯z such that ¯z = a – ib. We are calling ¯ z or even the complex number acquired by altering the symbol of the imaginary part (positive to negative or vice versa), as the conjugate of z. Let's just find the product z¯z = (a + ib)×(a – ib).

Thus, z¯z = {a^{2} -i(ab) + i(ab) + b^{2} } = (a^{2} + b^{2} ) …(1)

If a and b are big numbers, then the sum in (1) becomes more significant. And one can use this equation to determine a complex number's value.