Problem 45 - College Algebra Essentials

Question

In Exercises \(45-50,\) perform the indicated operation(s) and write the result in standard form. $$ (2-3 i)(1-i)-(3-i)(3+i) $$

Step-by-Step Solution

Answer

The result is: -9 + i

Step 1 Title

Multiple the first set of complex numbers: \((2 - 3i) (1 - i)\)

Step 2 Title

Complex multiplication has the property \((a + bi)(c + di) = ac - bd + (ad + bc)i\). Using this property, we get \(2(1) + 3(1) + (2(-1) - 3(1))i = 2 -3 - 2i + 3i = -1 + i\)

Step 3 Title

Perform second multiplication operation: \((3-i)(3+i)\)

Step 4 Title

Also apply the property again: \((3-i)(3+i) = 3*3 - 1*1 + ((3*1)-(1*3))i = 9 -1 + 0i = 8 + 0i = 8\)

Step 5 Title

Subtract the result of second multiplication from the result of first

Step 6 Title

Perform the subtraction operation on both the real and the imaginary parts separately: \((-1 + i) - (8) = -1 - 8 + i = -9 + i\)

Step 7 Title

The result is in standard form

Key Concepts

Complex Number Multiplication

Understanding complex number multiplication begins with recognizing that a complex number is formed of two parts: a real part and an imaginary part, usually expressed as \(a + bi\). When multiplying two complex numbers, like \((2 - 3i)(1 - i)\), we apply the distributive property just like with regular algebra, but with an additional twist due to the imaginary unit \(i\), which has the property that \(i^2 = -1\).

Let's break down a multiplication example step by step:

• Multiply the real parts: \(2 \times 1 = 2\)
• Multiply the imaginary parts, remembering that \(i^2 = -1\): \(-3i \times -i = 3i^2 = 3(-1) = -3\)
• Multiply the cross terms: \(2 \times (-i) = -2i\) and \(-3i \times 1 = -3i\)
• Add up all the terms, combining like terms, specifically the imaginary ones: \(2 - 3 - 2i - 3i = -1 - 5i\).

The result of the product here is in standard form, but we'll get back to that concept in the next section.

Standard Form of Complex Numbers

The standard form of a complex number, referred to as \(a + bi\), neatly separates the real part \(a\) from the imaginary part \(bi\). Visualize it as a coordinate on a plane, with \(a\) representing the x-coordinate (horizontal) and \(bi\) the y-coordinate (vertical). It’s this form that allows complex numbers to be easily understood, plotted, and manipulated in algebraic operations.

When we convert the product of two complex numbers to standard form, as in the multiplication \((-1 + i) - (8)\), we ensure any like terms are combined. The simpler form, \(-9 + i\), demonstrates standard form where \(-9\) is the real part and \(i\) is the imaginary part. No matter how complex the operation, the goal is to simplify to a form that shows these two distinct components.

Subtracting Complex Numbers

Subtracting complex numbers is as straightforward as subtracting their real and imaginary parts separately. Consider the example \((-1 + i) - (8)\), which we derived from the previous complex multiplication. To subtract, we simply subtract the real part of the second number from the real part of the first and do the same with the imaginary parts.

In subtractive operation:

• Subtract the real numbers: \(-1 - 8 = -9\)
• Since the second number has no imaginary part, we don’t need to perform an imaginary subtraction, and the imaginary part of the first complex number remains unchanged: \(+ i\).

Thus, the subtraction gives us \(-9 + i\) in standard form. This illustrates how to handle subtraction even when one of the complex numbers does not have an imaginary part.