Problem 9 - A Gentle Introduction to the Art of Mathematics

Question

Perform the following computations with complex numbers (a) \((4+3 i)-(3+2 i)\) (b) \((1+i)+(1-i)\) (c) \((1+i) \cdot(1-i)\) (d) \((2-3 i) \cdot(3-2 i)\)

Step-by-Step Solution

Answer

(a) 1+i, (b) 2, (c) 2, (d) -6-13i

Step 1 - Subtract Complex Numbers in (a)

To subtract complex numbers (4+3i) - (3+2i), subtract the real parts and the imaginary parts separately. Real part: 4 - 3 = 1 Imaginary part: 3i - 2i = i Thus, (4+3i) - (3+2i) = 1 + i

Step 2 - Add Complex Numbers in (b)

To add complex numbers (1+i) + (1-i), add the real parts and the imaginary parts separately. Real part: 1 + 1 = 2 Imaginary part: i - i = 0 Thus, (1+i) + (1-i) = 2 + 0 = 2

Step 3 - Multiply Complex Numbers in (c)

To multiply complex numbers (1+i) by (1-i), use the distributive property (FOIL method): (1+i)(1-i) = 1*1 + 1*(-i) + i*1 + i*(-i) = 1 - i + i - i^2. Recall that i^2 = -1, so: 1 - i + i - (-1) = 1 + 1 = 2 Thus, (1+i)(1-i) = 2

Step 4 - Multiply Complex Numbers in (d)

To multiply complex numbers (2-3i) by (3-2i), use the distributive property (FOIL method): (2-3i)(3-2i) = 2*3 + 2*(-2i) + (-3i)*3 + (-3i)*(-2i). Calculate each term: 2*3 = 6 2*(-2i) = -4i (-3i)*3 = -9i (-3i)*(-2i) = 6i^2. Recall that i^2 = -1, so 6i^2 = 6*(-1) = -6. Combine all terms: 6 - 4i - 9i - 6 = -6 - 13i. Thus, (2-3i)(3-2i) = -6 - 13i

Key Concepts

Addition of Complex Numbers

Addition of complex numbers can be broken down into simple steps. Take each part of your complex number and add like terms. Here are the necessary steps:
1. Real parts: Add the real numbers together.
2. Imaginary parts: Add the imaginary numbers together.
For example, if you want to add \( (1+i) + (1-i) \), you do it like this:
Real part: 1 + 1 = 2
Imaginary part: i + (-i) = 0
So, \( (1+i) + (1-i) = 2 \).
If you remember to group and add real numbers separately from the imaginary numbers, you will always get the right answer!

Subtraction of Complex Numbers

Subtraction of complex numbers is just like addition, but you subtract the numbers instead. Here's how you do it:
1. Real parts: Subtract the real numbers from each other.
2. Imaginary parts: Subtract the imaginary numbers from each other.
For example, in the expression \( (4+3i) - (3+2i) \), you do this:
Real part: 4 - 3 = 1
Imaginary part: 3i - 2i = i
So, \( (4+3i) - (3+2i) = 1 + i \).
It's important to remember to subtract the components separately. This will simplify your work and ensure you're always accurate.

Multiplication of Complex Numbers

Multiplying complex numbers might look tricky, but it becomes easy with practice. You can use the distributive property, also known as the FOIL method (First, Outside, Inside, Last). Here's a step-by-step guide:
For example, if you want to multiply \( (1+i) \bullet (1-i) \), you do the following:
1. First: Multiply the first terms: 1 * 1 = 1.
2. Outside: Multiply the outside terms: 1 * -i = -i.
3. Inside: Multiply the inside terms: i * 1 = i.
4. Last: Multiply the last terms: i * -i = -i^2. Remember, \ i^2 = -1\, so -i^2 = 1.
Now combine all these: \ 1 - i + i - (-1) = 1 - i + i + 1 = 1 + 1 = 2 \.
For another example, if you want to multiply \( (2-3i) \bullet (3-2i) \), use the same method:
1. First: 2 * 3 = 6.
2. Outside: 2 * -2i = -4i.
3. Inside: -3i * 3 = -9i.
4. Last: -3i * -2i = 6i^2. And since i^2 = -1, 6i^2 = -6.
So, \ 6 - 4i - 9i - 6 = -6 - 13i \.
Breaking it down into these manageable steps makes multiplication of complex numbers straightforward.