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# Complex Number – Definition, Formula, Properties, Examples

## Definition of Complex Numbers

A complex number is a number that comprises a real part and an imaginary part. It is expressed in the form a + bi, where “a” represents the real part, “b” represents the imaginary part, and “i” represents the imaginary unit (√(-1)).  Complex numbers are an extension of the real numbers and allow for the representation of quantities involving both real and imaginary components.

## What is Complex Number?

A complex number is a number that consists of two parts: a real part and an imaginary part. It is an extension of the real number system and allows for the representation of quantities involving both real and imaginary components. A complex number is typically written in the form a + bi, where “a” represents the real part, “b” represents the imaginary part, and “i” represents the imaginary unit (√(-1)). The real part and the imaginary part can be any real numbers.

The imaginary part of a complex number is multiplied by the imaginary unit “i” to distinguish it from real numbers. The imaginary unit “i” is defined as the square root of -1, meaning that i² = -1. This allows for the inclusion of numbers that cannot be represented on the real number line.

## Formula of Complex Numbers

The general form of a complex number is a + bi, where “a” and “b” are real numbers. The real part is denoted by “Re(z)” and the imaginary part is denoted by “Im(z)”. The modulus or absolute value of a complex number is given by |z| = √(a² + b²), and the argument or phase of a complex number is given by arg(z) = tan^(-1)(b/a).

## Operations on Complex Numbers

Operations on complex numbers involve addition, subtraction, multiplication, and division. Let’s explore each operation in simple terms:

1. Addition: To add complex numbers, you add the real parts separately and the imaginary parts separately. For example: (a + bi) + (c + di) = (a + c) + (b + d)i
2. Subtraction: To subtract complex numbers, you subtract the real parts separately and the imaginary parts separately. For example: (a + bi) – (c + di) = (a – c) + (b – d)i
3. Multiplication: To multiply complex numbers, you can use the distributive property. Multiply each term in one complex number by each term in the other, and then combine like terms. For example: (a + bi) * (c + di) = ac + adi + bci + bdi² Since i² is equal to -1, you simplify the expression: (a + bi) * (c + di) = (ac – bd) + (ad + bc)i
4. Division: To divide complex numbers, you can use a technique called “rationalizing the denominator.” Multiply the numerator and denominator by the conjugate of the denominator, which is the same as the denominator but with the opposite sign for the imaginary part. For example: (a + bi) / (c + di) = [(a + bi) * (c – di)] / [(c + di) * (c – di)] Simplifying the expression using the distributive property, you get: (a + bi) / (c + di) = [(ac + bd) + (bc – ad)i] / (c² + d²)

### Properties of Complex Numbers

1. Addition: The sum of two complex numbers z₁ = a₁ + b₁i and z₂ = a₂ + b₂i is z₁ + z₂ = (a₁ + a₂) + (b₁ + b₂)i.
2. Subtraction: The difference between two complex numbers z₁ = a₁ + b₁i and z₂ = a₂ + b₂i is z₁ – z₂ = (a₁ – a₂) + (b₁ – b₂)i.
3. Multiplication: The product of two complex numbers z₁ = a₁ + b₁i and z₂ = a₂ + b₂i is z₁ * z₂ = (a₁a₂ – b₁b₂) + (a₁b₂ + a₂b₁)i.
4. Division: The division of two complex numbers z₁ = a₁ + b₁i and z₂ = a₂ + b₂i is given by z₁ / z₂ = [(a₁a₂ + b₁b₂) / (a₂² + b₂²)] + [(a₂b₁ – a₁b₂) / (a₂² + b₂²)]i.
5. Conjugate: The complex conjugate of a complex number z = a + bi is denoted as z* and is given by z* = a – bi. The conjugate of a complex number has the same real part but an opposite sign for the imaginary part.

1. 3 + 4i
2. -2 – 7i
3. 0 + 2i
4. 5 – 2i
5. -1 + 0i

## FAQs on Complex Numbers

Q: What is the difference between a real number and a complex number?
A: Real numbers include all the familiar numbers on the number line, such as integers, fractions, and decimals. Complex numbers, on the other hand, consist of a real part and an imaginary part. While real numbers can be represented on the number line, complex numbers require a two-dimensional plane called the complex plane.

Q: Can a complex number have a real part of zero?
A: Yes, a complex number can have a real part of zero. In such cases, the complex number is purely imaginary and is written as bi, where b represents the imaginary part. Examples include 0 + 3i, 0 – 2i, and 0 + 0i.

Q: What does the absolute value of a complex number represent?
A: The absolute value (or modulus) of a complex number represents its distance from the origin in the complex plane. It is calculated as the square root of the sum of the squares of the real and imaginary parts. The absolute value of a complex number is always a positive real number or zero.

##### Q: Can complex numbers be compared using inequality symbols (>, <, ≥, ≤)?

A: No, complex numbers cannot be directly compared using inequality symbols. Inequality comparisons are only valid for real numbers. However, the magnitudes of complex numbers (their absolute values) can be compared using inequality symbols.

Q: Are all real numbers considered complex numbers?
A: Yes, all real numbers can be considered complex numbers with an imaginary part of zero. Real numbers lie on the real number line, which is a subset of the complex plane.

Q: What are conjugate complex numbers?
A: The conjugate of a complex number a + bi is obtained by changing the sign of the imaginary part, resulting in a – bi. Conjugate complex numbers have the same real part but opposite imaginary parts. The product of a complex number and its conjugate always yields a real number.

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