# Question Video: Forming a Complex Number in Algebraic Form given Its Principal Argument and Modulus

Given that |𝑧| = 5 and the argument of 𝑧 is 𝜃 = 270°, find 𝑧, giving your answer in algebraic form.

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### Video Transcript

Given that the modulus of 𝑧 is five and the argument of 𝑧 is 𝜃, which is equal to 270 degrees, find 𝑧, giving your answer in algebraic form.

When we write a complex number in algebraic or rectangular form, we write it as 𝑎 plus 𝑏𝑖, where 𝑎 is the real component of the complex number and 𝑏 is the imaginary component.

We can use these conversion formulae to convert the polar coordinates with a modulus of 𝑟 and an argument 𝜃 into the corresponding rectangular form. 𝑎 is equal to 𝑟 cos 𝜃 and 𝑏 is equal to 𝑟 sin 𝜃. The modulus of our complex number is five. And 𝜃, the argument, is 270 degrees.

Often, the argument will be given in radians. But since we’re converting a complex number into rectangular form, this doesn’t really matter, as long as we remember to make sure our calculator is working in degrees.

Let’s substitute these values into the conversion formulae for 𝑎 and 𝑏. 𝑎 is equal to five multiplied by cos of 270 degrees, which is zero. And 𝑏 is equal to five multiplied by sin of 270 degrees, which is negative five.

In rectangular form, 𝑎 plus 𝑏𝑖 then, we can write our complex number as zero plus negative five 𝑖, which is simply negative five 𝑖.

We have expressed our complex number in algebraic or rectangular form. 𝑧 is equal to negative five 𝑖.