Question

Perform the indicated operation and write the result in standard form. $$(2-5 i)^{2}-(2+5 i)^{2}$$

Short answer

The result in standard form is \(40i\).

Step-by-step solution

Expand each complex number squared

Process first complex number square: \((2-5i)^2 = 4 - 20i + 25i^2 = 4 - 20i -25 \) (using \(i^2 = -1\)). Then process second complex number square: \((2+5i)^2 = 4 + 20i + 25i^2 = 4 + 20i - 25\).

Simplify the result after expansion

Simplify the above expressions. First results in: \(-21 - 20i\) and the second gives: \(-21 + 20i\).

Subtract

Subtract the second result from the first: \((-21 - 20i) - (-21 + 20i) = 40i.\)

Key concepts

Standard Form

In the world of complex numbers, **standard form** refers to writing the complex number as a sum of its real and imaginary parts. Typically, a complex number is denoted as \(a + bi\), where \(a\) is the real part, and \(b\) is the imaginary part.
- The real part is simply any real number such as 1, -3, or 0.
- The imaginary part is always a real number multiplied by the imaginary unit \(i\), which we will clarify further in this article.

When looking at our exercise, we were tasked with writing the result in standard form. Thus, after performing any necessary operations, make sure to simplify and arrange your result as \(a + bi\).
This approach makes it clear which component is real and which is imaginary, essential for many mathematical operations. Understanding the standard form is a fundamental skill that will assist students in identifying, comparing, and performing operations on complex numbers.

Complex Conjugate

The **complex conjugate** of a complex number is found by changing the sign of its imaginary part. If you have a complex number expressed as \(a + bi\), its complex conjugate is \(a - bi\). Complex conjugates are particularly useful because they help in simplifying division of complex numbers and are often used in various mathematical computations.

For instance, in the exercise, although we didn't explicitly use the concept of complex conjugate, understanding it helps in simplifying expressions. The subtraction part of the problem could involve recognizing patterns where complex conjugates were involved in simplifying calculations. By recognizing that the square of a complex number automatically produces terms that can be complex conjugates, simplifying becomes more intuitive.
Thus, appreciating the role of complex conjugates not only aids in specific calculations, but also enhances overall understanding of complex number operations.

Imaginary Unit

One essential idea in complex numbers is the **imaginary unit**, which is designated as \(i\). It's defined by the property \(i^2 = -1\). This concept is crucial as it helps in the formation and manipulation of complex numbers.

Understanding \(i\) allows us to formulate expressions such as imaginary numbers \(bi\), where \(b\) is a real number. For example, \(5i\) can be computed by leveraging \(i^2 = -1\) during calculations.
In the exercise, we used \(i^2 = -1\) during the expansion of \((2-5i)^2\) and \((2+5i)^2\).
By substituting \(i^2\) with \(-1\), we transitioned from a seemingly complex expression to a simplified form. This direct application of the imaginary unit showcases its importance, without which, operations involving complex numbers could become cumbersome.
Understanding the imaginary unit is pivotal for decoding the imaginary part of any complex number and for performing complex arithmetic efficiently.