lesson
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Every time you square a real number, the result is positive or zero — 32=9 and (−3)2=9.
So what happens if an equation demands a number whose square is negative, like x2=−1?
The Imaginary Unit
In 1777, Swiss mathematician Leonhard Euler introduced a breakthrough symbol to solve this puzzle: the imaginary unit, written as i.
We define i as the principal square root of −1, which establishes its foundational property: i2=−1.
📊Create a clean, responsive visual diagram explaining the imaginary unit. Show two main formula badges: 'i = √(-1)' and 'i² = -1' in bold with a distinct accent background (#e0f2fe border #0284c7). Below, display a simple, clear diagram showing squaring: a box with 'i' going through an arrow labeled '( )²' leading to a box with '-1'. Use dark text #1e2945, card background #ffffff, border #e6e6e6, and rounded 12px corners.
How does defining this single unit allow us to take the square root of any negative number?
Extracting the Negative
A radicand is the expression resting inside a radical symbol x.
For any positive real number b, we split the negative radicand using the multiplication property: −b=b⋅(−1)=b⋅−1=ib.
Simplifying Perfect Squares
To simplify −25, we separate the negative factor first: −25=25⋅−1.
Since 25=5 and −1=i, our final simplified form is 5i.