120 LearnersLast updated on May 26th, 2025
Square Root of -145
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.
What is 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.
Finding the Square Root of -145
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.
Square Root of -145 by Approximation Method
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.
Square Root of -145 by Using Complex Numbers
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.
Properties of the Square Root of -145
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.
Common Mistakes and How to Avoid Them in the Square Root of -145
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.
Mistake 1
Forgetting about the Imaginary Unit
A common mistake is overlooking the 'i' when calculating the square root of a negative number. Always remember that √-1 is 'i', and thus, √-145 should be expressed as 12.04159i.
Mistake 2
Confusing Real and Imaginary Parts
Some students mistakenly separate the real and imaginary components incorrectly. The correct form is 0 (real part) + 12.04159i (imaginary part).
Mistake 3
Ignoring the Negative Sign
It's crucial not to ignore the negative sign when dealing with square roots of negative numbers. The negative sign gives rise to the imaginary component 'i'.
Mistake 4
Not Understanding Complex Numbers
Complex numbers can be confusing, especially if one is unfamiliar with 'i'. Ensure you understand that 'i' represents the square root of -1 and is a fundamental part of complex numbers.
Mistake 5
Incorrect Approximation of the Square Root
Make sure the approximation of the square root of 145 is accurate to avoid errors in expressing √-145. Use a calculator for precision if necessary.
Square Root of -145 Examples
Problem 1
What is the square root of -145 expressed as a complex number?
The square root of -145 is approximately 12.04159i.
Explanation
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.
Problem 2
If a side of a square is given as √-145, what is the perimeter of the square?
The perimeter is 48.16636i units.
Explanation
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.
Problem 3
Calculate (√-145)².
The result is -145.
Explanation
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.
Problem 4
What is the sum of √-145 and 5i?
The sum is approximately 17.04159i.
Explanation
The square root of -145 is 12.04159i.
Adding 5i gives 12.04159i + 5i = 17.04159i.
Problem 5
Express the square root of -145 in polar form.
The polar form is 12.04159 ∠ 90°.
Explanation
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°.
FAQ on Square Root of -145
1.What is √-145 in its simplest form?
2.What is the magnitude of √-145?
3.Can the square root of a negative number be real?
4.What is the square of √-145?
5.How is √-145 used in real-world applications?
6.How does learning Algebra help students in Singapore make better decisions in daily life?
7.How can cultural or local activities in Singapore support learning Algebra topics such as Square Root of -145?
8.How do technology and digital tools in Singapore support learning Algebra and Square Root of -145?
9.Does learning Algebra support future career opportunities for students in Singapore?
Important Glossaries for the Square Root of -145
- Imaginary Number: A number that when squared gives a negative result. The unit 'i' is defined as √-1.
- Complex Number: A number in the form a + bi, where a is the real part and b is the imaginary part.
- Magnitude: The magnitude of a complex number is the distance from the origin to the point in the complex plane, calculated as √(a² + b²). For pure imaginary numbers, it is simply the absolute value of the imaginary part.
- Polar Form: A way to express complex numbers using a magnitude and an angle, represented as r∠θ.
- Real Part: The component 'a' in a complex number a + bi, representing the real number part.
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Jaskaran Singh Saluja
About the Author
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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