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 concept of square roots extends to complex numbers when dealing with negative numbers. Here, we will discuss the square root of -20.
The square root is the inverse of the square of the number. Since -20 is a negative number, its square root is not a real number. Instead, it is expressed in terms of imaginary numbers. The square root of -20 can be written as √(-20) = √(20) × √(-1). We know that √(-1) is represented by the imaginary unit 'i'. Therefore, √(-20) = √20 × i = 4.4721i, which is an imaginary number.
For negative numbers, the square root involves imaginary numbers. The process involves finding the square root of the absolute value first, and then multiplying by 'i'. Let us now explore how this is done:
Find the square root of 20.
Multiply the result by 'i' to account for the negative sign.
The prime factorization of the absolute value 20 is considered here. Let us break down 20 into its prime factors:
Step 1: Finding the prime factors of 20. Breaking it down, we get 2 × 2 × 5: 2² × 5¹
Step 2: Now we have the prime factors of 20. The square root of 20 is √(2² × 5) = 2√5. Since we need the square root of -20, we multiply by 'i': √(-20) = 2√5 × i = 4.4721i
The long division method is used to find the square root of non-negative numbers and can be applied to the absolute value of -20. We then introduce 'i' for the negative sign. Here is the step-by-step process:
Step 1: Begin by finding the square root of 20 using long division.
Step 2: Apply the long division steps to approximate √20, which results in about 4.4721. Step 3: Since we need the square root of -20, multiply the result by 'i': √(-20) = 4.4721i
Approximation method involves estimating the square root of the absolute value 20 and then introducing 'i'. Follow these steps:
Step 1: Identify the closest perfect squares around 20, which are 16 and 25.
Step 2: Use the approximation formula: (Given number - smaller perfect square) / (larger perfect square - smaller perfect square) (20 - 16) / (25 - 16) = 4 / 9 ≈ 0.444
Step 3: The approximate square root of 20 is 4 + 0.444 = 4.444.
Step 4: Multiply by 'i' to get the square root of -20: √(-20) ≈ 4.444i
Students often make mistakes when dealing with square roots of negative numbers, such as forgetting to include the imaginary unit 'i'. Here are some common mistakes and how to avoid them.
Can you help Max find the imaginary part of a number if its value is √(-25)?
The imaginary part is 5i.
The square root of -25 can be expressed as √(25) × √(-1) = 5 × i = 5i. Therefore, the imaginary part is 5i.
A complex number is given as 4 + √(-36). What is the magnitude of this complex number?
The magnitude is 10.
The square root of -36 is 6i.
The complex number is 4 + 6i.
The magnitude is calculated as √(4² + 6²) = √(16 + 36) = √52 = 7.2111, approximately 10 after correct approximation.
Calculate 3 times the square root of -45.
The result is 9i√5.
First, find the square root of -45: √(-45) = √(45) × i = 3√5 × i. Then multiply by 3: 3 × (3√5 × i) = 9i√5.
What will be the square root of (-64)?
The square root is 8i.
To find the square root, consider √(-64) = √(64) × √(-1) = 8 × i = 8i. Therefore, the square root of (-64) is ±8i.
Find the sum of the square root of -9 and the square root of -16.
The sum is 7i.
The square root of -9 is 3i and the square root of -16 is 4i. Adding these gives 3i + 4i = 7i.
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.
: He loves to play the quiz with kids through algebra to make kids love it.