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 concept extends into complex numbers when dealing with negative numbers. Here, we will discuss the square root of -169.
The square root is the inverse of the square of the number. Since -169 is negative, its square root is not a real number. Instead, it is an imaginary number. The square root of -169 is expressed as √(-169) = √(169) × √(-1) = 13i, where i is the imaginary unit defined as √(-1).
The square root of a negative number involves the imaginary unit 'i'. Here, we find the square root by separating the negative part from the positive square root: 1. Separate the negative and positive part: √(-169) = √(169) × √(-1). 2. Calculate the positive square root: √169 = 13. 3. Combine with the imaginary unit: 13 × i = 13i. Hence, the square root of -169 is 13i.
Imaginary numbers are used to represent the square roots of negative numbers. The imaginary unit 'i' is defined as √(-1). For -169, we can express the square root as:
1. Identify the real square root of the absolute value: √169 = 13.
2. Combine with the imaginary unit: √(-169) = 13i. This shows that the square root of -169 is 13 times the imaginary unit, meaning it is 13i.
Imaginary numbers, including square roots of negative numbers, are used in various fields:
1. Electrical Engineering: Used in alternating current (AC) circuit analysis.
2. Control Theory: Utilized in the design and stability analysis of control systems.
3. Quantum Physics: Complex numbers are fundamental in quantum mechanics equations.
4. Signal Processing: Applied in the analysis and manipulation of signals. Understanding imaginary numbers extends the capacity to solve real-world problems where real numbers are insufficient.
When working with imaginary numbers, common mistakes can occur:
1. Misunderstanding 'i': Remember, i² = -1, not 1.
2. Incorrect Simplification: Ensure correct use of 'i' in expressions.
3. Ignoring 'i' in Calculations: Do not treat 'i' as a variable; it has specific properties.
4. Forgetting Negative Signs: When taking square roots of negative numbers, the 'i' must be included. By avoiding these mistakes, calculations involving imaginary numbers can be accurate and meaningful.
Students often make mistakes with imaginary numbers, such as ignoring the imaginary unit or simplifying incorrectly. Let's explore these errors and how to avoid them.
What is the result of multiplying √(-169) by 2?
The result is 26i.
First, find the square root of -169, which is 13i.
Then multiply by 2: 13i × 2 = 26i.
If the side of a square is represented by √(-169), what would be the perimeter of the square?
The perimeter would be 52i units.
The side length is 13i.
Perimeter of a square is 4 times the side length: 4 × 13i = 52i units.
Calculate (√(-169))².
The result is -169.
(√(-169))² = (13i)² = 169 × i² = 169 × (-1) = -169.
If z = √(-169), what is z + z?
The sum is 26i.
If z = 13i, then z + z = 13i + 13i = 26i.
What is the modulus of the complex number 13i?
The modulus is 13.
The modulus of a complex number a + bi is √(a² + b²).
Here, a = 0 and b = 13, so modulus = √(0² + 13²) = √169 = 13.
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.