Last updated on May 26th, 2025
The square of a number is the result of multiplying the number by itself. The inverse operation is finding the square root. Square roots are essential in various fields, including complex number analysis and engineering. Here, we will discuss the square root of -289.
The square root is the inverse operation of squaring a number. Since -289 is negative, its square root is not a real number. Instead, it is expressed using the imaginary unit. The square root of -289 can be written as √-289, which simplifies to 17i, where i is the imaginary unit defined by i² = -1.
To understand the square root of a negative number, we use the concept of imaginary numbers. The imaginary unit i is defined such that i² = -1. Therefore, the square root of -289 is computed using the relationship: √-289 = √(289 × -1) = √289 × √-1 = 17i.
The properties of square roots of negative numbers rely on the imaginary unit i. Key points include:
1. The square root of a negative number always involves the imaginary unit i.
2. (√-a)² = -a for any positive real number a.
3. Imaginary numbers are used in complex number calculations, where a complex number is of the form a + bi.
Imaginary numbers have applications in various fields:
1. Engineering: Used in signal processing and control systems.
2. Physics: Appear in quantum mechanics and electrical engineering.
3. Mathematics: Essential for solving certain polynomial equations.
4. Computer Science: Used in algorithms dealing with complex numbers.
When dealing with square roots of negative numbers, common mistakes include:
1. Assuming the square root of a negative number is real.
2. Forgetting the imaginary unit i in calculations.
3. Misinterpreting i² = -1 as a real number operation.
Students often make mistakes when working with square roots of negative numbers. It's important to correctly apply the concept of imaginary numbers. Let's explore some common errors and how to avoid them.
What is the product of the square root of -289 and 3?
The product is 51i.
The square root of -289 is 17i.
Multiply this by 3: 17i × 3 = 51i.
If z = 5 + √-289, what is the conjugate of z?
The conjugate of z is 5 - 17i.
The conjugate of a complex number a + bi is a - bi.
Given z = 5 + 17i, its conjugate is 5 - 17i.
How do you express the square of the square root of -289?
The expression is -289.
The square of the square root of -289 is (√-289)² = (17i)² = 17² × i² = 289 × (-1) = -289.
What is the imaginary part of the square root of -289?
The imaginary part is 17.
The square root of -289 is 17i.
The imaginary part is the coefficient of i, which is 17.
If the imaginary unit is defined as i, what is i raised to the power of 4?
i raised to the power of 4 is 1.
i is defined such that i² = -1.
Therefore, i⁴ = (i²)² = (-1)² = 1.
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.