# Video: Identifying When Two Complex Numbers Are Equal

If the complex numbers 7 + 𝑎𝑖 and 𝑏 − 3𝑖 are equal, what are the values of 𝑎 and 𝑏?

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### Video Transcript

If the complex numbers seven plus 𝑎𝑖 and 𝑏 minus three 𝑖 are equal, what are the values of 𝑎 and 𝑏?

Remember, for two complex numbers to be equal, their real parts must be equal and their imaginary parts must also be equal. And the beauty of this fact is it takes a problem about complex numbers and makes it purely about real numbers, since both the real parts and imaginary parts of each complex number must be real numbers.

Let’s have a look at the complex numbers seven plus 𝑎𝑖 and 𝑏 minus three 𝑖 then. The real part of the first complex number is seven, and the real part of our second complex number is 𝑏. The imaginary part of our first complex number is 𝑎, and the imaginary part of our second complex number is negative three. It follows then that seven must be equal to 𝑏 and 𝑎 must be equal to negative three. Both negative three and seven are real numbers, which satisfies our criteria for the real and imaginary parts of a complex number. So for the complex numbers seven plus 𝑎𝑖 and 𝑏 minus three 𝑖 to be equal, 𝑎 must be equal to negative three and 𝑏 must be equal to seven.