Complex Number Concept

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.

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.

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.

With this definition, we get an important property:

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

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

General Form

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

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

The term $bi$ 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 ($b$): $-7$

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. $1 + 2i$

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: $2$