Problem 45 - Algebra and Trigonometry Real Mathematics, Real People

Question

Perform the operation and write the result in standard form. $$(4+5 i)^{2}-(4-5 i)^{2}$$

Step-by-Step Solution

Answer

The result of the operation is \( 80i \)

Step 1: Square the first complex number

Perform the squaring operation on the complex number \( (4+5i) \): \\( (4+5i)^{2} = (4 + 5i)*(4 + 5i) \= 16 + 20i + 20i + 25i^{2} \= 16 + 40i - 25 \ (since \ i^{2} = -1) \= -9 + 40i \)

Step 2: Square the second complex number

Perform the squaring operation on the complex number \( (4-5i) \): \\( (4 - 5i)^{2} = (4 - 5i)*(4 - 5i) \= 16 - 20i - 20i + 25i^{2} \= 16 - 40i - 25 \ (since \ i^{2} = -1) \= -9 - 40i \)

Step 3: Subtract the two results

Subtract the result obtained in Step 2 from that obtained in Step 1: \\( (-9 + 40i) - (-9 - 40i) \= -9 + 40i + 9 + 40i \= 80i \)

Key Concepts

Squaring Complex Numbers

When we square a complex number, we multiply it by itself. Interestingly, this isn't much different from squaring real numbers, yet it introduces the twist of dealing with the imaginary unit, 'i'. For example, squaring the complex number \((4+5i)\) involves a process similar to FOIL in algebra, where we multiply each term by every other term one by one. Here's the breakdown:

• Multiply the real parts: \(4 \times 4 = 16\).
• Multiply the real part of the first number by the imaginary part of the second, and the imaginary part of the first by the real part of the second: \(4 \times 5i + 5i \times 4 = 20i + 20i\).
• Multiply the imaginary parts, remembering that \(i^2 = -1\): \(5i \times 5i = 25i^2 = 25(-1) = -25\).

We then add up these results: \(16 + 40i - 25\), simplifying down to \(-9 + 40i\) as our squared result.

Subtracting Complex Numbers

Subtracting complex numbers works in a similar vein to subtracting real numbers — with the main difference being that we perform the operation separately on the real and imaginary parts. For our example, we need to subtract the complex number \((4-5i)^2\) from \((4+5i)^2\). We computed the squares previously, resulting in \(-9 + 40i\) and \(-9 - 40i\). To subtract, we align the like terms (real with real, imaginary with imaginary) and simply subtract:

• Subtract the real parts: \(-9 - (-9) = 0\).
• Subtract the imaginary parts: \(40i - (-40i) = 80i\).

Notice, the real parts cancel out, and we are left with \(80i\), a purely imaginary number. All operations strictly follow basic algebraic rules for addition and subtraction.

Imaginary Unit i

The imaginary unit 'i' is a fundamental part of complex numbers and is defined as the square root of -1. This might seem abstract since no real number squared gives a negative result. However, in the realm of complex numbers, 'i' is the cornerstone that allows us to expand our number system to include solutions to equations like \(x^2 + 1 = 0\). An important property to remember is that \(i^2 = -1\), which helps simplify expressions involving complex numbers. For instance, when we square \((4+5i)\), we eventually reach the term \(25i^2\) which we simplify by substituting \(i^2 = -1\), turning the term into \(-25\).

Algebraic Operations with Complex Numbers

Algebraic operations with complex numbers, such as addition, subtraction, multiplication, and division, follow the rules of algebra for real numbers, with special attention to the properties of the imaginary unit 'i'. When performing these operations, separate the real and imaginary components and treat 'i' as an algebraic variable, mindful of the fact that \(i^2 = -1\).
For multiplication, use the distributive property to expand the product, then combine like terms. If performing division, we may need to 'rationalize' the denominator by multiplying the numerator and the denominator by the conjugate of the denominator. Through these algebraic manipulations, we maintain the calculations within the framework of standard arithmetic, allowing complex numbers to be applied in a range of mathematical scenarios.