Last updated on May 26th, 2025
If a number is multiplied by the same number, the result is a square. The inverse of the square is a square root. The square root is used in various fields, including engineering and physics. Here, we will discuss the square root of -2.
The square root is the inverse of the square of a number. Since -2 is a negative number, its square root cannot be expressed as a real number. Instead, the square root of -2 is expressed in terms of imaginary numbers. In its simplest form, it can be represented as √(-2) = i√2, where i is the imaginary unit, defined as √(-1).
We cannot use the usual methods like prime factorization, long division, or approximation for non-perfect square numbers when dealing with negative numbers. Instead, we use the concept of imaginary numbers. Here, we will explain the concept:
Imaginary unit (i)
Expressing negative square roots
Understanding complex numbers
The imaginary unit, denoted as i, is defined by the property that i² = -1. Therefore, the square root of any negative number can be expressed using the imaginary unit. For -2, we express the square root as: √(-2) = √(2) × √(-1) = √2 × i This expression shows that the square root of -2 is an imaginary number.
Complex numbers combine real and imaginary parts and are written in the form a + bi, where a and b are real numbers. The square root of -2 can be involved in complex numbers as shown:
Example: 3 + √(-2) = 3 + √2i Here, 3 is the real part, and √2i is the imaginary part.
Imaginary and complex numbers are used in various fields such as electrical engineering, control theory, and signal processing. They help in solving equations that do not have real solutions and in representing phenomena like AC circuits where phase angles are important.
Students often make mistakes when dealing with square roots of negative numbers, such as ignoring the imaginary unit or mishandling operations involving complex numbers. Here are some common mistakes and how to avoid them.
Calculate the square of the imaginary number √(-2).
-2
The square of the imaginary number √(-2) is calculated as (i√2)² = i²(√2)² = -1 × 2 = -2.
Find the expression for adding 3 and the square root of -2.
3 + i√2
The expression for adding 3 and the square root of -2 is written as a complex number: 3 + i√2, where 3 is the real part and i√2 is the imaginary part.
Multiply 2 by the square root of -2.
2i√2
To multiply 2 by the square root of -2, we express it as 2 × i√2 = 2i√2.
What is the square root of the expression (-2)²?
2
The expression (-2)² equals 4.
The square root of 4 is 2, thus the square root of the expression (-2)² is 2.
Express the product of (3 + i) and √(-2).
3i√2 - √2
To find the product, distribute: (3 + i) × i√2 = 3i√2 + i²√2 = 3i√2 - √2, using the fact that i² = -1.
Jaskaran Singh Saluja is a math wizard with nearly three years of experience as a math teacher. His expertise is in algebra, so he can make algebra classes interesting by turning tricky equations into simple puzzles.
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