lesson
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Try solving the simple equation x2+1=0. If you subtract 1 from both sides, you get x2=−1.
In the real number system, this is impossible because squaring any positive number gives a positive, and squaring any negative number also gives a positive.
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What if, instead of stopping at an impossible equation, mathematicians simply invented a number to solve it?
Defining the Imaginary Unit
In 1777, Swiss mathematician Leonhard Euler introduced the symbol i, calling it the imaginary unit, specifically to represent −1.
Because the square root and squaring operations undo each other, squaring this new unit gives its core definition: i2=−1.
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How does this single definition unlock square roots for every other negative number?
Simplifying Square Roots of Negative Numbers
The radicand is the number under the radical symbol. For any positive real number b, we split a negative radicand into positive b and −1 using −b=b⋅(−1)=b⋅−1=ib.
For example, to simplify −25, factor out the −1 to get 25⋅−1=5i.
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