Problem 70 - College Algebra

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

The statement makes sense since adding or subtracting complex numbers does involve combining like terms, where 'like terms' refers to the real part being added to the real part, and the imaginary part being added to the imaginary part.

Step 1: Understand Complex Numbers

A complex number has two parts: a real part and an imaginary part, it takes the form \(a + bi\) where \(a\) and \(b\) are real numbers and \(i\) is the imaginary unit with the property that \(i^2 = -1\). When we refer to 'like terms', it usually means terms that have the same variable and exponent, which can be combined.

Step 2: Analyze the statement in the context of complex numbers

The statement claims that adding or subtracting complex numbers is essentially combining like terms. In the context of complex numbers, 'like terms' can be considered the real parts and the imaginary parts. The addition or subtraction of two complex numbers does indeed involve combining 'like terms' i.e., the real parts with the real parts and the imaginary parts with the imaginary parts.

Step 3: Conclude and provide reasoning

Based on the nature of complex numbers and the action of addition and subtraction in the context of these numbers, the statement does make sense. When you add or subtract complex numbers, you are essentially combining like terms - the real components with the real components, and the imaginary components with the imaginary components, much akin to how we combine like terms in algebraic expressions.