Problem 70 - College Algebra Essentials

Question

Determine whether each statement makes sense or does not make sense, and explain your reasoning. When I add or subtract complex numbers, I am basically combining like terms.

Step-by-Step Solution

Answer

Yes, the statement makes sense. During addition or subtraction of complex numbers, like terms are combined which means the real parts are added or subtracted with each other and the imaginary parts are also added or subtracted separately, similar to adding or subtracting real numbers.

Step 1: Understand the structure of a complex number

A complex number is of the form \(a + bi\), where \(a\) and \(b\) are real numbers, and \(i\) is the imaginary unit with the property that \(i^2 = -1\). \(a\) is called the real part and \(bi\) is the imaginary part of the complex number.

Step 2: How adding/subtracting real numbers works

When adding or subtracting real numbers, like terms are combined. That means, if we are given \(3x + 4x\), we can combine the like terms ('x' in this case) to simplify the expression to \(7x\).

Step 3: Compare to adding/subtracting complex numbers

Now, considering the addition of two complex numbers: if \(z1 = a + bi\) and \(z2 = c + di\), \(z1 + z2\) will be \((a + c) + (b + d)i\), combining the real parts and the imaginary parts separately. Similarly for subtraction, the like terms (real part with real part, imaginary part with imaginary part) are combined, proving the initial statement correct.