Real and imaginary numbers
In this article we’re going to be looking at how to use real and imaginary numbers. Below is a quick summary of what will be covered in this article.
- √(-1) = I and i²
- An imaginary number is a number of the form bi where b is a real number (ℝ)
- A complex number is a number of a + bi where a ∈ ℝ and b ∈ ℝ
- For a complex number a + bi a is called the real part and b is called the imaginary part.
Trying to solve a square root of a negative number on your calculator will result in an error.
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…this is because you can’t square root a negative number. This is particularly a problem in quadratic equations. When using the quadratic equation.
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When b² – 4ac < 0 you find that the equation has no solutions because you can’t find the square root of any number less than zero. To solve the problem, we introduce another type of number called an imaginary number.
We represent the square root of a negative number √-1 with i where i is called the imaginary number. And a real number is any number that is not a square root of a negative number.
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We can add a real and an imaginary number together such as;
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…were 4+3i is known as a complex number.
Notes
- A complex number is written in the form a+bi
- You can add and subtract complex numbers.
- √(-1) = i
- An imaginary number is a number of the form bi where b is a real number (b ∈ ℝ)
We shall look at some examples below.
Note that we can’t solve √(-36) we therefore have to write it in the form of bi to do that we expand the expression involving the √-1 . We know that;
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…therefore;
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…the same applies for the square root of -90 (√(-90))
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Solving complex numbers equations
Example
Suppose we had to solve the equation x² + 36 = 0;
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We continue as we did above;
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Notes
You should know by now that;
- A complex number of the form a + bi, where a ∈ ℝ and a ∈ ℝ
- For the complex number a + bi, a is called the real part and b is called the imaginary part.
- The complete set of complex numbers is called ℂ
Example
In this example we’re going to solve;
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…there is two ways we can solve the equation. We can solve by completing the square or by using the quadratic formula.
METHOD 1: Completing the square
We first need to complete the square for;
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To complete the square we know that;
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…therefore;
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So;
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METHOD 2: Quadratic equation
We can also use the quadratic equation;
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We’re trying to solve;
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…therefore;
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- In a complex number, the real part and the imaginary part cannot be combined to form a single term
- Complex numbers can be added by adding the real parts and adding the imaginary parts separately/
Below are some examples;
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