Which complex number has an absolute value of 5? This question might seem straightforward, but it actually leads to an interesting exploration of complex numbers and their properties. In this article, we will delve into the concept of absolute value in the context of complex numbers and discuss the various complex numbers that have an absolute value of 5.
Complex numbers are numbers that consist of a real part and an imaginary part. They are represented in the form a + bi, where a is the real part, b is the imaginary part, and i is the imaginary unit, which is defined as the square root of -1. The absolute value of a complex number, also known as its magnitude, is a measure of its distance from the origin in the complex plane. It is calculated using the Pythagorean theorem, which states that the magnitude of a complex number z = a + bi is given by |z| = √(a² + b²).
Now, let’s consider the complex number with an absolute value of 5. To find such a number, we need to satisfy the equation |z| = 5. Substituting the magnitude formula, we get √(a² + b²) = 5. Squaring both sides of the equation, we obtain a² + b² = 25. This equation represents a circle with radius 5 centered at the origin in the complex plane.
Since there are infinitely many points on a circle, there are infinitely many complex numbers with an absolute value of 5. To illustrate this, let’s consider a few examples:
1. z = 5: This complex number has a real part of 5 and an imaginary part of 0. It lies on the positive real axis and has an absolute value of 5.
2. z = 5i: This complex number has a real part of 0 and an imaginary part of 5. It lies on the positive imaginary axis and has an absolute value of 5.
3. z = 3 + 4i: This complex number has a real part of 3 and an imaginary part of 4. It lies in the first quadrant of the complex plane and has an absolute value of 5, as confirmed by the equation √(3² + 4²) = 5.
4. z = -3 – 4i: This complex number has a real part of -3 and an imaginary part of -4. It lies in the third quadrant of the complex plane and also has an absolute value of 5.
In conclusion, the question “Which complex number has an absolute value of 5?” has an infinite number of answers. By understanding the concept of absolute value and the properties of complex numbers, we can identify and describe these complex numbers that lie on a circle with radius 5 centered at the origin in the complex plane.
