Answered

Lin says, "When you add or multiply two complex numbers, you will always get an answer you can write in $a + b i$ form."

Noah says, "I don't think so. Here are some exceptions I found:"

$\begin{array}{c}

(7+2 i)+(3-2 i)=10 \\

(2+2 i)(2+2 i)=8 i

\end{array}$


ACheck Noah's arithmetic. Is it correct?


BCan Noah's answers be written in the form $a + b i$, where $a$ and $b$ are real numbers? Explain or show your reasoning.

Verified Answer :

  • Noah's arithmetic calculations are correct: \((7+2i)+(3-2i)=10\) and \((2+2i)(2+2i)=8i\).
  • Both results can be written in the form \(a+bi\) by setting \(a\) or \(b\) to zero: \(10 = 10+0i\) and \(8i = 0+8i\).
  • Since these results fit the definition, Noah's examples do not contradict Lin's claim.
  • The final answer is \(\boxed{\text{Yes, both can be written as } a+bi}\).

Explanation

  1. Verifying Noah's Arithmetic

To verify Noah's arithmetic, we perform the operations indicated. For the addition:
\[(7+2i) + (3-2i) = (7+3) + (2i-2i) = 10 + 0i = 10.\]
For the multiplication:
\[(2+2i)(2+2i) = 2(2) + 2(2i) + (2i)(2) + (2i)(2i) = 4 + 4i + 4i + 4i^2.\]
Since \(i^2 = -1\), the result is:
\[4 + 8i + 4(-1) = 4 + 8i - 4 = 0 + 8i = 8i.\]
Both calculations are correct.

  1. Analyzing the Form a + bi

A complex number is defined as any number that can be expressed in the form \(a + bi\), where \(a\) and \(b\) are real numbers and \(i\) is the imaginary unit.
For the result \(10\), we can write it as \(10 + 0i\). Here, \(a = 10\) and \(b = 0\). Since both \(10\) and \(0\) are real numbers, \(10\) is a complex number in the form \(a + bi\).
For the result \(8i\), we can write it as \(0 + 8i\). Here, \(a = 0\) and \(b = 8\). Since both \(0\) and \(8\) are real numbers, \(8i\) is a complex number in the form \(a + bi\).
Therefore, Noah's results are not exceptions to Lin's claim; they are simply special cases where one of the real components is zero. Lin's claim remains valid.

Examples

In electronics, complex numbers are used to represent impedance in AC circuits. Just as in this problem where we showed that real numbers (like 10) and purely imaginary numbers (like 8i) are just special cases of the general complex form \(a + bi\), engineers often combine resistive components (the real part) and reactive components (the imaginary part) to analyze circuits. Recognizing that any combination results in a valid complex number allows engineers to perform calculations on entire systems systematically using the same mathematical rules.

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