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 the field of engineering, physics, etc. Here, we will discuss the square root of -900.
The square root is the inverse of the square of the number. The square root of a negative number is not a real number; it is an imaginary number. The square root of -900 is expressed in the form of an imaginary number as √-900 = 30i, where i is the imaginary unit defined by i² = -1.
To find the square root of a negative number, we express it in terms of imaginary numbers. The steps involve separating the negative sign and finding the square root of the positive number. Let us learn the following steps:
The prime factorization method involves breaking down a number into its prime components. However, since -900 is negative, we handle it by focusing on the positive part:
Step 1: Find the prime factors of 900
Breaking it down, we get 2 × 2 × 3 × 3 × 5 × 5: 2² × 3² × 5²
Step 2: Find the square root of 900 Since 900 is a perfect square, the square root is 30.
Step 3: Combine with the imaginary unit
The square root of -900 is 30i.
The long division method is typically used for finding square roots of non-negative numbers. For -900, we focus on √900 and then apply the imaginary unit:
Step 1: Identify the positive number 900.
Step 2: Use the long division method to find the square root of 900, which is 30.
Step 3: As the original number is negative, multiply the result by i.
Thus, the square root of -900 is 30i.
Approximation is not necessary for perfect squares like 900, but we can still illustrate how it aligns with imaginary numbers:
Step 1: Recognize that 900 is a perfect square with √900 = 30.
Step 2: For -900, append the imaginary unit i to the result: 30i.
Imaginary numbers are a crucial part of complex numbers and are used in various scientific and engineering fields. The imaginary unit i is defined such that i² = -1. Thus, the square root of any negative number can be expressed using i. For example, the square root of -900 is 30i.
Students often make mistakes when dealing with negative square roots, such as ignoring the imaginary unit or misapplying methods meant for real numbers. Let's examine a few common errors in detail.
Can you help Max find the perimeter of a square if one side is √-900?
The perimeter is 120i units.
The perimeter of a square = 4 × side.
The side length is given as √-900, which is 30i.
Perimeter = 4 × 30i = 120i units.
A circular field has an area of -900π square feet. What is the radius?
The radius is 30i feet.
The area of a circle = πr².
Given area = -900π, so r² = -900.
r = √-900 = 30i.
Thus, the radius is 30i feet.
Calculate √-900 × 2.
60i
First, find the square root of -900, which is 30i.
Then, multiply 30i by 2.
So, 30i × 2 = 60i.
What is (√-900)²?
-900
(√-900)² = (30i)² = 30² × i² = 900 × (-1) = -900.
Thus, (√-900)² is -900.
If the imaginary part of a complex number is √-900, what is the number?
The complex number is 0 + 30i.
Given the imaginary part is √-900, which is 30i, the complex number with no real part is 0 + 30i.
Square root: The number that, when multiplied by itself, gives the original number. For negative numbers, this involves imaginary numbers.
Imaginary number: A number that can be expressed as a real number multiplied by the imaginary unit i, where i² = -1.
Imaginary unit: The imaginary unit i is defined by the property i² = -1, allowing extension of the real number system to complex numbers.
Complex number: A number comprising a real and an imaginary part, often expressed as a + bi.
Perfect square: A number that is the square of an integer. For positive integers, it involves real numbers; for negative integers, it involves imaginary numbers.
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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