Given that 𝑍 equals three 𝑖, find the modulus or absolute value of 𝑍.
If we are given any complex number in the form 𝑍 equals 𝑥 plus 𝑖𝑦, where 𝑥 and 𝑦 are real, then the modulus or absolute value of 𝑍 is equal to the square root of 𝑥 squared plus 𝑦 squared. We find the sum of the squares of the real and imaginary parts and then square root the answer.
In the complex number in this question, 𝑍 equals three 𝑖, the real part of 𝑍 equals zero. The imaginary part of 𝑍 is equal to three, as this is the number multiplied by 𝑖. Substituting in these values gives us the modulus of 𝑍 is equal to the square root of zero squared plus three squared. Zero squared is equal to zero, and three squared is equal to nine, so we are left with the square root of nine. This is equal to three, so the modulus of the complex number three 𝑖 is three.