# Nakafa Framework: LLM
URL: /en/subject/high-school/11/mathematics/complex-number/complex-number-concept
Source: https://raw.githubusercontent.com/nakafaai/nakafa.com/refs/heads/main/packages/contents/subject/high-school/11/mathematics/complex-number/complex-number-concept/en.mdx
Output docs content for large language models.
---
export const metadata = {
title: "Complex Number Concept",
description: "Discover what complex numbers are and why they exist. Understand imaginary unit i, real vs imaginary parts, and solve equations with negative roots.",
authors: [{ name: "Nabil Akbarazzima Fatih" }],
date: "05/01/2025",
subject: "Complex Number",
};
## The Need for Complex Numbers
You've probably tried solving quadratic equations. For example, the equation . Easy, right? We can factor it into , so the solutions are or . Both are real numbers.
Now, what about the equation ? If we try to find its solution in the set of real numbers, we won't find one. Why? Because the equation leads to . There is no real number that, when squared, results in a negative number.
To overcome this problem, mathematicians introduced a new type of number called **complex numbers**.
## Imaginary Numbers
The core of complex numbers is the **imaginary unit**, denoted by . This imaginary unit is defined as the square root of -1.
With this definition, we get an important property:
With , we can now find the square root of negative numbers. For example:
Numbers like and are called **purely imaginary numbers**.
## General Form
Complex numbers are generally written in the form , where:
- is the **real part** (a real number).
- is the **imaginary part** (a real number).
- is the imaginary unit (
).
The term as a whole is called the imaginary part of the complex number.
### Examples
Let's look at some examples and identify their real and imaginary parts:
1. ****
- Real part ():
- Imaginary part ():
2. ****
This is the same as .
- Real part ():
- Imaginary part ():
3. ****
This is an ordinary real number, but it can also be considered a complex number with an imaginary part of 0. Its form is .
- Real part ():
- Imaginary part ():
4. ****
This is a purely imaginary number. Its form is .
- Real part ():
- Imaginary part ():
## Exercise
Determine the real and imaginary parts of the following complex numbers:
1.
2.
3.
4.
### Answer Key
1. .{" "}
This can be written as .
- Real part:
- Imaginary part:
2. .{" "}
This can be written as .
- Real part:
- Imaginary part:
3. .{" "}
This can be written as .
- Real part:
- Imaginary part:
4. .
This can be written as .
- Real part:
- Imaginary part: