- Noah's arithmetic is correct: \((7 + 2i) + (3 - 2i) = 10\) and \((2 + 2i)(2 + 2i) = 8i\).
- The result 10 can be written as \(10 + 0i\), where \(a = 10\) and \(b = 0\).
- The result \(8i\) can be written as \(0 + 8i\), where \(a = 0\) and \(b = 8\).
- Since both results fit the form \(a + bi\), they are not exceptions, and Lin's claim is verified.
- The final conclusion is that the calculations are correct and both results fit the requested form: \(\boxed{\text{Yes, they can be written as } a+bi}\).
Explanation
- Problem Analysis
We are examining the claim that adding or multiplying any two complex numbers results in an answer of the form \(a + bi\), where \(a\) and \(b\) are real numbers. We are evaluating Noah's arithmetic for two operations:
- Addition: \((7 + 2i) + (3 - 2i)\)
- Multiplication: \((2 + 2i)(2 + 2i)\)
Noah claims that the results of these operations (10 and \(8i\)) are exceptions to the form \(a + bi\). Our goal is to verify his calculations and determine if these results can indeed be written in the specified form.
- Verifying Noah's Arithmetic
First, we check the addition: \((7 + 2i) + (3 - 2i)\). To add complex numbers, we combine the real parts and the imaginary parts separately:
\[(7 + 3) + (2i - 2i) = 10 + 0i = 10.\]
Noah's arithmetic for the addition is correct.
Next, we check the multiplication: \((2 + 2i)(2 + 2i)\). We use the distributive property (often called FOIL):
\[(2 \times 2) + (2 \times 2i) + (2i \times 2) + (2i \times 2i)\]
\[= 4 + 4i + 4i + 4i^2\]
Since \(i^2 = -1\), the term \(4i^2\) becomes \(4(-1) = -4\). Substituting this back:
\[= 4 + 8i - 4 = 8i.\]
Noah's arithmetic for the multiplication is also correct.
- Analyzing the Results
A complex number is defined as \(a + bi\), where \(a\) and \(b\) are real numbers. We must see if 10 and \(8i\) can be expressed this way.
For 10:
We can write this as \(10 + 0i\). Here, \(a = 10\) and \(b = 0\). Since both 10 and 0 are real numbers, 10 is indeed in the form \(a + bi\).
For \(8i\):
We can write this as \(0 + 8i\). Here, \(a = 0\) and \(b = 8\). Since both 0 and 8 are real numbers, \(8i\) is indeed in the form \(a + bi\).
Because both results can be written in the form \(a + bi\), they are not exceptions to the rule. Lin's statement holds true even for these examples.
Examples
Understanding complex numbers is essential in fields like electrical engineering and physics. For instance, when analyzing alternating current (AC) circuits, voltages and currents are represented as complex numbers, where the real part might represent active power and the imaginary part reactive power. By mastering the form \(a + bi\), engineers can easily add or multiply these values to predict how circuits will behave, ensuring that power distribution systems function safely and efficiently.