Roots of Negative Numbers
Roots of negative numbers is a Grade 7 math topic from Yoshiwara Intermediate Algebra introducing the concept that even-indexed roots of negative numbers are not real. For example, √(-4) is not a real number, leading to the introduction of the imaginary unit i where i^2 = -1.
Key Concepts
Property 1. Every positive number has two real valued roots, one positive and one negative, if the index is even. 2. A negative number has no real valued root if the index is even. 3. Every real number, positive, negative, or zero, has exactly one real valued root if the index is odd. The symbol $\sqrt[n]{b}$ refers to the principal (positive) root when $n$ is even.
Examples $\sqrt[3]{ 64} = 4$ because $( 4)^3 = 64$. This is an odd root of a negative number.
$\sqrt[4]{ 81}$ is not a real number because the index (4) is even and the radicand ( 81) is negative.
Common Questions
What is the square root of a negative number?
The square root of a negative number is not a real number. For example, √(-4) = 2i, where i is the imaginary unit with i^2 = -1.
What is the imaginary unit i?
The imaginary unit i is defined so that i^2 = -1. It allows representation of square roots of negative numbers.
What is the difference between even and odd roots of negative numbers?
Even roots (square root, 4th root) of negative numbers are imaginary. Odd roots (cube root, 5th root) of negative numbers are real — for example, ∛(-8) = -2.
How do you simplify √(-9)?
√(-9) = √(9 × (-1)) = √9 × √(-1) = 3i.