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Learn what complex numbers are, why they exist, how the imaginary unit i works, and how to solve equations with negative roots.

---

## The Need for Complex Numbers

You've probably tried solving quadratic equations. For example, the equation $$x^2 - 1 = 0$$. Easy, right? We can factor it into $$(x-1)(x+1) = 0$$, so the solutions are $$x=1$$ or $$x=-1$$. Both are real numbers.

Visible text: 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 $$x^2 + 1 = 0$$? If we try to find its solution in the set of real numbers, we won't find one. Why? Because the equation leads to $$x^2 = -1$$. There is no real number that, when squared, results in a negative number.

Visible text: 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 $$i$$. This imaginary unit is defined as the square root of $$-1$$.

Visible text: The core of complex numbers is the **imaginary unit**, denoted by . This imaginary unit is defined as the square root of .

```math
i = \sqrt{-1}
```

With this definition, we get an important property:

```math
i^2 = -1
```

With $$i$$, we can now find the square root of negative numbers. For example:

Visible text: With , we can now find the square root of negative numbers. For example:

Component: MathContainer
Children:

```math
\sqrt{-4} = \sqrt{4 \times (-1)} = \sqrt{4} \times \sqrt{-1} = 2i
```

```math
\sqrt{-9} = \sqrt{9 \times (-1)} = \sqrt{9} \times \sqrt{-1} = 3i
```

Numbers like $$2i$$ and $$3i$$ are called **purely imaginary numbers**.

Visible text: Numbers like and are called **purely imaginary numbers**.

## General Form

Complex numbers are generally written in the form $$z = a + bi$$, where:

Visible text: Complex numbers are generally written in the form , where:

- $$a$$ is the **real part** (a real number).
- $$b$$ is the **imaginary part** (a real number).
- $$i$$ is the imaginary unit ($$\sqrt{-1}$$
  ).

Visible text: - is the **real part** (a real number).
- is the **imaginary part** (a real number).
- is the imaginary unit (
 ).

The term $$bi$$ as a whole is called the imaginary part of the complex number.

Visible text: 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.  **$$2 + 3i$$**

    - Real part ($$a$$): $$2$$
    - Imaginary part ($$b$$): $$3$$

2.  **$$5 - 4i$$**
    This is the same as $$5 + (-4)i$$.

    - Real part ($$a$$): $$5$$
    - Imaginary part ($$b$$): $$-4$$

3.  **$$\sqrt{2}$$**
    This is an ordinary real number, but it can also be considered a complex number with an imaginary part of $$0$$. Its form is $$\sqrt{2} + 0i$$.

    - Real part ($$a$$): $$\sqrt{2}$$
    - Imaginary part ($$b$$): $$0$$

4.  **$$-7i$$**
    This is a purely imaginary number. Its form is $$0 + (-7)i$$.
    - Real part ($$a$$): $$0$$
    - Imaginary part ($$b$$): $$-7$$

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 - Imaginary part (): 

2. ****
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 - Imaginary part (): 

3. ****
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## Exercise

Determine the real and imaginary parts of the following complex numbers:

1. $$2 + \sqrt{(-2)^2}$$
2. $$2 + i^2$$
3. $$1 + \sqrt{-9}$$
4. $$1 + 2i$$

Visible text: 1. 
2. 
3. 
4.

### Answer Key

1. $$2 + \sqrt{(-2)^2} = 2 + \sqrt{4} = 2 + 2 = 4$$.{" "}

   This can be written as $$4 + 0i$$.

   - Real part: $$4$$
   - Imaginary part: $$0$$

2. $$2 + i^2 = 2 + (-1) = 1$$.{" "}

   This can be written as $$1 + 0i$$.

   - Real part: $$1$$
   - Imaginary part: $$0$$

3. $$1 + \sqrt{-9} = 1 + \sqrt{9 \times (-1)} = 1 + 3i$$.{" "}

   This can be written as $$1 + 3i$$.

   - Real part: $$1$$
   - Imaginary part: $$3$$

4. $$1 + 2i$$.

   This can be written as $$1 + 2i$$.

   - Real part: $$1$$
   - Imaginary part: $$2$$

Visible text: 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: