# Complex number

A complex number is a number that can be expressed in the form a + bi, where a and b are real numbers, and i is the imaginary unit, which satisfies the equation i2 = −1. In this expression, a is the real part and b is the imaginary part of the complex number. If b = 0, then the complex number is equal to the real number a.

Complex numbers extend the concept of the one-dimensional number line to the two-dimensional complex plane (also called the Argand plane) by using the horizontal axis for the real part and the vertical axis for the imaginary part. The complex numbers are thus two-dimensional vectors (represented by points) in the complex plane.

The addition of complex numbers is performed by adding the corresponding real parts and adding the corresponding imaginary parts:

(a + bi) + (c + di) = (a + c) + (b + d)i

The multiplication of complex numbers is performed by multiplying the corresponding real parts and adding the corresponding imaginary parts:

(a + bi) × (c + di) = (ac − bd) + (ad + bc)i

The conjugate of a complex number is the number with the same real part and an imaginary part equal in magnitude but opposite in sign. The conjugate of the complex number a + bi is a − bi.

The absolute value (or modulus

### Why is it called a complex number?

A complex number is any number that can be written in the form a + bi, where a and b are real numbers, and i is a imaginary number. The imaginary number i is defined as the square root of -1.

The name "complex number" comes from the fact that these numbers are not just real numbers, but are a combination of both real and imaginary numbers. This makes them more complicated than just real numbers, hence the name complex numbers.

### How do you find a complex number?

To find a complex number, you need to first understand what a complex number is. A complex number is a number that can be expressed in the form a + bi, where a and b are real numbers, and i is the imaginary unit, which satisfies the equation i2 = -1.

The real part of a complex number is the a value, and the imaginary part is the b value. So, in the complex number 3 + 4i, the real part is 3 and the imaginary part is 4.

To add or subtract complex numbers, you add or subtract the real parts together, and the imaginary parts together. So, if you wanted to add 3 + 4i and 5 – 2i, you would first add the real parts (3 + 5 = 8) and then add the imaginary parts (4 – 2 = 2) to get 8 + 2i.

To multiply complex numbers, you need to use the distributive property. So, if you wanted to multiply 3 + 4i and 5 – 2i, you would first multiply 3 and 5 to get 15, then multiply 3 and –2i to get –6i, then multiply 4i and 5 to get 20i, and then multiply 4i and –2i to get –8i2. Since i2 = –1, this simplifies to –8(–1) which is 8. So, the answer is 15 + 14i.

To What is real and complex number? A real number is a number that can be represented on a number line. A complex number is a number that can be represented in the form a + bi, where a and b are real numbers and i is the imaginary unit. Is 2 is a complex number? No, 2 is not a complex number. A complex number is a number that can be expressed in the form a + bi, where a and b are real numbers and i is the imaginary unit, which satisfies the equation i^2 = -1.

##### Is 40 a complex number?

Yes, 40 is a complex number. In fact, all numbers are complex numbers, although some are more complex than others. The term "complex number" simply refers to a number that can be expressed in the form a+bi, where a and b are real numbers and i is the imaginary unit. So, 40 can be expressed as 40+0i.