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 vehicle design, finance, etc. Here, we will discuss the square root of -245.
The square root is the inverse of the square of a number. Since -245 is a negative number, its square root is not a real number. The square root of -245 is expressed in terms of imaginary numbers. In radical form, it is expressed as √(-245) = √245 * i, where i is the imaginary unit (i = √-1). In the exponential form, it is expressed as (245)^(1/2) * i. √245 ≈ 15.6525, so the square root of -245 is approximately 15.6525i, which is an imaginary number.
For negative numbers, the square root involves imaginary numbers. Here, we are looking for a number that, when squared, gives -245. This number is represented using the imaginary unit i. Let us consider the following methods used for calculating the square roots of positive numbers and modify them for our purpose:
The prime factorization method involves expressing a number as a product of its prime factors. Let's see how we can express 245:
Step 1: Finding the prime factors of 245 Breaking it down, we get 5 × 7 × 7 = 5 × 7².
Step 2: With these prime factors, the square root of 245 in terms of real numbers is √245 = √(5 × 7²) = 7√5.
Since -245 is negative, we include the imaginary unit: √(-245) = 7√5 * i.
The long division method is typically used for non-perfect square numbers. However, since -245 is negative, we have to consider the imaginary unit.
Step 1: Calculate the square root of the positive part, which is 245, using long division to get approximately 15.6525.
Step 2: Since the original number is negative, the square root of -245 is 15.6525i.
The approximation method involves finding the square root of the positive part and then multiplying by the imaginary unit.
Step 1: Identify the closest perfect squares around 245, which are 225 (15²) and 256 (16²). So √245 is between 15 and 16.
Step 2: Use the formula: (Given number - smaller perfect square) / (larger perfect square - smaller perfect square). For approximation: (245 - 225) / (256 - 225) = 20/31 ≈ 0.645.
Thus, √245 ≈ 15 + 0.645 = 15.645 and therefore, √(-245) ≈ 15.645i.
Students often make mistakes while finding the square root, especially with negative numbers where imaginary numbers come into play. Let’s look at some common mistakes and how to avoid them.
Can you help Max find the area of a square box if its side length is given as √(-138)?
The area cannot be determined as a real number because the side length is imaginary.
The area of a square = side².
Since the side length is √(-138), it involves imaginary numbers, and squaring it results in a negative area, which is not meaningful in real terms.
A square-shaped building measuring -245 square feet is claimed to exist; what can you infer about this claim?
The claim is invalid in real-world terms.
Area cannot be negative in reality.
The mention of -245 square feet likely involves an error or a misunderstanding, as square footage is always a positive measure.
Calculate √(-245) × 5.
Approximately 78.2625i
First, calculate √(-245) ≈ 15.6525i.
Then, multiply by 5: 15.6525i × 5 = 78.2625i.
What will be the square root of (-138 + 6)?
The square root is approximately 11.4455i
First, calculate the expression: -138 + 6 = -132.
Then, √(-132) = √132 * i ≈ 11.4455i.
Find the perimeter of the rectangle if its length ‘l’ is √(-138) units and the width ‘w’ is 38 units.
The perimeter cannot be determined as a real number.
Perimeter = 2 × (length + width). With length √(-138), the calculation involves imaginary numbers, making the perimeter not meaningful in real terms.
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.