Video Transcript
If the complex numbers four plus five 𝑖 and 𝑎 plus five 𝑖 are equal, what is the value of 𝑎?
First, we recall that A complex number is a number of the form 𝑎 plus 𝑏𝑖, where both 𝑎 and 𝑏 are real numbers and 𝑖 is the square root of negative one. For a complex number 𝑧 is equal to 𝑎 plus 𝑏𝑖, we define the real part of 𝑧 to be 𝑎 and the imaginary part to be 𝑏. Two complex numbers 𝑧 one equal to 𝑎 plus 𝑏𝑖 and 𝑧 two equal to 𝑐 plus 𝑑𝑖 are equal if, and only if, 𝑎 is equal to 𝑐 and 𝑏 is equal to 𝑑. In other words, if two complex numbers are equal, then their real parts are equal to one another and their imaginary parts are also equal to one another.
We’ve been given the complex numbers four plus five 𝑖 and 𝑎 plus five 𝑖 and told that they are equal. We can see that the imaginary part of these complex numbers is the same; in each case, the imaginary part is equal to five. If they are equal, then the real parts must also be equal, which tells us that 𝑎 must be equal to four.
So by recalling that two complex numbers are equal if, and only if, the real parts of the two complex numbers are the same and the imaginary parts of the two complex numbers are the same, we’ve found that the value of 𝑎 is four.
