Last updated on May 26th, 2025
In mathematics, the square root is the inverse operation of squaring a number. For negative numbers, square roots involve imaginary numbers. Here, we will explore the square root of -145.
The square root of a negative number involves the imaginary unit 'i', where i is defined as √-1. Therefore, the square root of -145 can be expressed in the form of √145 * i. The square root of 145 is approximately 12.04159, so the square root of -145 is approximately 12.04159i.
For negative numbers, the concept of square roots extends to the complex plane. The square root of a negative number is not a real number but an imaginary one. Here are the steps to find the square root of -145:
1. Recognize that √-145 = √145 * √-1.
2. Calculate √145 using approximation methods such as the long division method or a calculator, obtaining approximately 12.04159.
3. Combine this with the imaginary unit: √-145 = 12.04159i.
The approximation method involves finding the square root of the positive part of the number non-imaginarily and then combining it with 'i'.
Step 1: Find the square root of 145, which is approximately 12.04159.
Step 2: Multiply this by √-1, giving us 12.04159i as the square root of -145.
Complex numbers allow us to handle square roots of negative numbers. The imaginary unit 'i' is crucial here.
1. Express -145 as -1 * 145.
2. Find √145, which is approximately 12.04159.
3. Combine this with 'i', so √-145 = 12.04159i.
Understanding the properties of the square root of -145 involves knowing about imaginary numbers.
1. The square root of -145 is not a real number.
2. It can be expressed as a complex number: 0 + 12.04159i.
3. Square roots of negative numbers like -145 are essential in many fields, including engineering and physics, where complex numbers are used to solve equations that have no real solutions.
Mistakes often arise when dealing with square roots of negative numbers due to misunderstanding imaginary numbers and complex arithmetic. Here are some common errors and how to avoid them.
What is the square root of -145 expressed as a complex number?
The square root of -145 is approximately 12.04159i.
The square root of -145 involves the imaginary unit 'i'.
First, find the square root of 145, which is approximately 12.04159.
Then, multiply by 'i' to express it as 12.04159i.
If a side of a square is given as √-145, what is the perimeter of the square?
The perimeter is 48.16636i units.
The perimeter of a square is 4 times the side length.
Given the side length is √-145 = 12.04159i, the perimeter is 4 × 12.04159i = 48.16636i units.
Calculate (√-145)².
The result is -145.
When you square a square root of a number, you return to the original number. (√-145)² = -145 because (12.04159i)² = 145 * (-1) = -145.
What is the sum of √-145 and 5i?
The sum is approximately 17.04159i.
The square root of -145 is 12.04159i.
Adding 5i gives 12.04159i + 5i = 17.04159i.
Express the square root of -145 in polar form.
The polar form is 12.04159 ∠ 90°.
The polar form of a complex number is given as r∠θ, where r is the magnitude and θ is the angle. Here, r = 12.04159 and the angle for a purely imaginary number is 90°.
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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