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17Calculus Precalculus - Rationalizing Complex Numbers

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Rationalizing Complex Numbers

Remember with square roots, you would perform an operation called rationalizing. This usually occurs when you want to remove a square root from the denominator of a fraction. The idea is to multiply by something called the conjugate. This is what it looks like. If you have \(1+3\sqrt{2}\), the conjugate is \(1-3\sqrt{2}\), i.e. you change the sign in the middle. The beauty of this that when you you multiply these together, the square root disappears.
Example: \( (1+3\sqrt{2})(1-3\sqrt{2}) = \) \( 1(1-3\sqrt{2}) +3\sqrt{2}(1-3\sqrt{2}) = \) \(1-3\sqrt{2} + 3\sqrt{2} -9(2) = -17 \)

Notice in the last example, that the square root of two is gone. That is the point of rationalizing. We can do the same thing with complex numbers.
Example: \( (1+3i)(1-3i) = \) \( 1(1-3i) +3i(1-3i) = \) \(1-3i + 3i -9i^2 = 1-9(-1) = 10 \)

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