Video: Expressing the Simplified Results of Complex Numbers Expressions in Algebraic and Polar Forms

Simplify (βˆ’5 + 5√3𝑖)/(βˆ’βˆš(3) βˆ’ 𝑖), giving your answer in both algebraic and trigonometric form.

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Video Transcript

Simplify negative five plus five root three 𝑖 over negative root three minus 𝑖, giving your answer in both algebraic and trigonometric form.

To divide complex numbers in rectangular or algebraic form, that’s the general form 𝑧 equals π‘Ž plus 𝑏𝑖, we first need to find the complex conjugate of the denominator. For a complex number π‘Ž plus 𝑏𝑖, its complex conjugate is π‘Ž minus 𝑏𝑖. We denote the complex conjugate with a little bar as shown. Once we have this, we can multiply both the numerator and the denominator by the complex conjugate. The denominator of our fraction is negative root three minus 𝑖. So its complex conjugate is negative root three plus 𝑖. We need to multiply both the numerator and the denominator of the fraction by negative root three plus 𝑖.

We multiply these just like we multiply any normal pair of brackets. Negative five multiplied by negative root three is five root three. Negative five multiplied by 𝑖 is negative five 𝑖. Five root three 𝑖 multiplied by negative root three is negative five root three squared multiplied by 𝑖. And five root three 𝑖 multiplied by 𝑖 is five root three 𝑖 squared.

Remember, 𝑖 squared is simply negative one. So five root three multiplied by 𝑖 squared is negative five root three. Root three squared is three. So negative five multiplied by root three squared multiplied by 𝑖 is negative 15𝑖. And this means that the numerator of this fraction simplifies to negative 20𝑖.

Let’s now multiply the denominator. Negative root three multiplied by negative root three is root three squared. Negative root three multiplied by 𝑖 is negative root three 𝑖. Negative 𝑖 multiplied by negative root three is positive root three 𝑖. And negative 𝑖 multiplied by 𝑖 is negative 𝑖 squared. And this simplifies to four. So our fraction becomes negative 20𝑖 over four, which is negative five 𝑖.

In rectangular or algebraic form, the complex number is 𝑧 is equal to negative five 𝑖. Comparing this to the general algebraic form for a complex number, we can see that π‘Ž must be equal to zero. And 𝑏 must be equal to negative five for our complex number. When we write a complex number in trigonometric or polar form, we write it as 𝑧 is equal to π‘Ÿ multiplied by cos πœƒ plus 𝑖 sin πœƒ, where π‘Ÿ is called the modulus of the complex number and πœƒ is the argument.

So we need to find a way to represent the real and complex components of our number in terms of π‘Ÿ and πœƒ. In fact, we do have some formulae we can use. The modulus is the square root of π‘Ž squared plus 𝑏 squared. This comes from the Pythagorean theorem. And to find πœƒ, we use tan πœƒ is equal to 𝑏 over π‘Ž. Let’s substitute what we know about our complex number into these formulae.

π‘Ÿ is equal to the square root of zero squared plus negative five squared, which is five. And tan πœƒ is equal to negative five divided by zero. Now, this is actually undefined. However, we know that the tangent function is undefined at negative πœ‹ over two and at intervals of πœ‹. So πœƒ must be negative πœ‹ over two for this to be true.

Notice how we chose a negative πœ‹ over two as opposed to positive πœ‹ over two since the complex component of the number in rectangular form was negative five. This means that the number would sit on the axes between the third and fourth quadrant on the complex coordinate plane. Since we measure this in an anticlockwise direction from zero to negative πœ‹, this coordinate has to be negative πœ‹ over two.

The trigonometric form of our complex number is five cos of negative πœ‹ over two plus 𝑖 sin of negative πœ‹ over two.

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