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

Given that |𝑧| = 3 and the argument of 𝑧 is 𝜃 = 𝜋/3, find 𝑧, giving your answer in algebraic form.

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### Video Transcript

Given that the modulus of 𝑧 is three and the argument of 𝑧 is 𝜃 which is 𝜋 over three, find 𝑧, giving your answer in algebraic form.

When we write a complex number in algebraic or rectangular form, we write it as 𝑧 is equal to 𝑎 plus 𝑏𝑖. We can then use these conversion formulae to convert the polar coordinates with the modulus of 𝑟 and an argument of 𝜃 into the corresponding rectangular form. The modulus of our complex number is three. And the argument is 𝜋 over three.

So we can substitute these values into the conversion formulae for 𝑎 and 𝑏. 𝑎 is equal to three multiplied by cos 𝜋 over three, which is three over two. And 𝑏 is equal to three multiplied by sin of 𝜋 over three, which is three root three over two. This means that the rectangular or algebraic form of our complex number is three over two plus three root three over two 𝑖.