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The i calculator will introduce you to complex numbers. Keep reading to learn:

  • What are complex numbers;
  • The value of i in complex numbers: the complex root; and
  • How to calculate i in complex numbers.

What are complex numbers?

Complex numbers (or imaginary numbers) are an extension of real numbers, where it's allowed to operate with square roots with negative arguments. Complex numbers are a fundamental part of mathematics: they appear in many areas of analysis. They are at the basis of versatile frameworks for the description of physical phenomena as oscillatory ones.

Complex numbers are "bidimensional numbers": in one dimension, we find a "normal" real number, while on the other, perpendicular one, we find imaginary numbers. How do we identify both of them?

The role of i in complex numbers: what is the value of i in complex numbers?

By definition, an imaginary number is a number that contains the complex root ii. ii is the result of an apparently impossible mathematical operation. To introduce it, we must accept that the result is not numerical but complex. Here is the formula used in math to calculate i:

i=1i = \sqrt{-1}

Take your time to let this equation set it. By "allowing" this result to be valid, we opened the door to complex numbers. We can use ii to multiply a number and make it imaginary.

Noticing that we can identify two distinct types of numbers:

  • Real numbers: 4.563644.56364, 11, etc.; and
  • Imaginary numbers: 4i4i, ii, 13.351351i13.351351i;

we can combine them to create an extended set of numbers in the form a+bia+bi: these are the properly called complex numbers.

Calculate complex numbers: i

We have multiple options to calculate i in complex numbers depending on the form in which the complex number is presented.

Calculate i in the a+bia+bi form

This is the simplest case: simply take the imaginary part of the number a+bia+bi, in our case bibi, and strip it of ii. bb is the coefficient of the imaginary part.

Calculate i in the polar form zeiφ|z| e^{i\varphi}

In this case, we need to follow a small set of steps to calculate the i part of this number.

  1. Compute the sine of the phase angle φ\varphi: sin(φ)\sin(\varphi).
  2. Multiply the result of the previous step by the magnitude z|z| of the complex number: the result is the imaginary part of our number: Im=zsin(φ)\mathrm{Im} = |z|\cdot\sin(\varphi).


How do I calculate i of complex numbers?

To calculate i in complex numbers, you can start by finding the imaginary part. To do so, follow the next steps.

  1. Identify the form of your complex number:
    a. If the number is in the form a + bi, go to step 2.
    b. If the number is in the form |z|e, go to step 3.
  2. Take the imaginary part bi and remove the complex root i. The result is b.
  3. Follow the next two steps:
    • Compute the sine of the phase φ, sin(φ).
    • Multiply it by the modulo |z|: the result is |z|sin(φ).

What is i on a calculator?

i on a calculator is the complex root of complex numbers; the value of i allows you to execute calculations with this kind of number.
By multiplying a number by i, you are making it imaginary; that is, multiply it by the seemingly impossible number i = √-1. Summing an imaginary number to a real number gives you the most commonly used form of a complex number, a + bi, where a is the real part, and b is the imaginary part.

How do I calculate the imaginary part of the number 4exp(i 45°)?

To calculate the imaginary part of the number 4ei 45°, follow these easy steps:

  1. Compute the sine of the phase angle 45°: sin(45°) = √2/2 = 0.7071.
  2. Multiply the result of step 1 by the modulo of the complex number 4:
    4 × 0.7071 = 2.8284
  3. It's done: the number you found equals b in the a + bi representation of the complex numbers.
Davide Borchia
Related calculators
First number: z₁ = a + bi
Real part (a)
Imaginary part (b)
Magnitude |z₁|
Phase (φ₁)
Second number: z₂ = c + di
Real part (c)
Imaginary part (d)
Magnitude |z₂|
Phase (φ₂)
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