A number that, when squared, results in a negative value.
A number of the form \$a + bi\$, where \$a\$ and \$b\$ are real numbers, and \$i\$ is the imaginary unit with \$i^2 = -1\$.
The imaginary unit \$i\$ is defined such that \$i^2 = -1\$.
Simplifying Imaginary Numbers
To simplify the square root of a negative number, we use the property of the imaginary unit \$i\$:
- Factor the number under the square root into a positive and negative part.
- Replace the square root of the negative part with \$i\$.
Simplify \$\sqrt{-9}\$.
- Factor: \$\sqrt{-9} = \sqrt{9 \cdot (-1)}\$
- Replace: \$\sqrt{9} \cdot \sqrt{-1} = 3i\$
If the number inside the square root is not a perfect square, leave it under the radical:
\$\sqrt{-12} = \sqrt{12} \cdot i = 2\sqrt{3}i\$
Adding and Subtracting Imaginary Numbers
Imaginary numbers can be added or subtracted like like terms in algebra.
Simplify \$5i + 2i\$.
- Combine like terms: \$5i + 2i = 7i\$
Simplify \$5i - 2i\$.
- Combine like terms: \$5i - 2i = 3i\$
Multiplying Imaginary Numbers
Multiplying imaginary numbers follows the same rules as multiplying variables, with the additional property that \$i^2 = -1\$.
Multiply \$5i \cdot 2i\$.
- Multiply the coefficients: \$5 \cdot 2 = 10\$
- Multiply the imaginary units: \$i \cdot i = i^2\$
- Simplify using \$i^2 = -1\$: \$10i^2 = 10(-1) = -10\$
- Simplify \$\sqrt{-16}\$.
- Add \$3i + 4i\$.
- Multiply \$2i \cdot 3i\$.
How do imaginary numbers challenge our understanding of what it means for a number to be "real"?
