hw 6.3.1 – complex numbers
if $z_1 = 5 + 9i$ and $z_2 = 8 - 3i$, find:
1.
a. $z_1 + z_2$
b. $z_1 - z_2$
2. $z_1 \cdot z_2$
3. $(z_2)^2$
find the modulus of the complex numbers.
4. $|13 + 2i|$
5. $|-4 + i|$
6. $|8 - 9i|$
perform the division and write the result in $a + bi$ form.
7. $\frac{-5 + i}{1 - 5i}$
8. $\frac{6 - 2i}{2 - 3i}$
9. $\frac{-10 + 11i}{12 - 7i}$
write the coordinates of the points on the complex plane in the form $x + yi$.
10. a
11. b
12. c
13. d
14. is the point a or the point d closer to the origin? justify your answer.
15. what is $i^3$ equal to?
16. what is $i^{63}$ equal to?

Question

hw 6.3.1 – complex numbers
if $z_1 = 5 + 9i$ and $z_2 = 8 - 3i$, find:
1.
  a. $z_1 + z_2$
  b. $z_1 - z_2$
2. $z_1 \cdot z_2$
3. $(z_2)^2$
find the modulus of the complex numbers.
4. $|13 + 2i|$
5. $|-4 + i|$
6. $|8 - 9i|$
perform the division and write the result in $a + bi$ form.
7. $\frac{-5 + i}{1 - 5i}$
8. $\frac{6 - 2i}{2 - 3i}$
9. $\frac{-10 + 11i}{12 - 7i}$
write the coordinates of the points on the complex plane in the form $x + yi$.
10. a
11. b
12. c
13. d
14. is the point a or the point d closer to the origin? justify your answer.
15. what is $i^3$ equal to?
16. what is $i^{63}$ equal to?

Accepted Answer

### 1a. $ z_1 + z_2 $
# Explanation:
## Step1: Identify $ z_1 $ and $ z_2 $
$ z_1 = 5 + 9i $, $ z_2 = 8 - 3i $
## Step2: Add real and imaginary parts
$ (5 + 8) + (9i - 3i) = 13 + 6i $
# Answer: $ 13 + 6i $

### 1b. $ z_1 - z_2 $
# Explanation:
## Step1: Identify $ z_1 $ and $ z_2 $
$ z_1 = 5 + 9i $, $ z_2 = 8 - 3i $
## Step2: Subtract real and imaginary parts
$ (5 - 8) + (9i + 3i) = -3 + 12i $
# Answer: $ -3 + 12i $

### 2. $ z_1 \cdot z_2 $
# Explanation:
## Step1: Use distributive property (FOIL)
$ (5 + 9i)(8 - 3i) = 5 \cdot 8 + 5 \cdot (-3i) + 9i \cdot 8 + 9i \cdot (-3i) $
## Step2: Simplify each term
$ 40 - 15i + 72i - 27i^2 $, and $ i^2 = -1 $, so $ -27(-1) = 27 $
## Step3: Combine like terms
$ (40 + 27) + (-15i + 72i) = 67 + 57i $
# Answer: $ 67 + 57i $

### 3. $ (z_2)^2 $
# Explanation:
## Step1: Write $ z_2^2 $ as $ z_2 \cdot z_2 $
$ (8 - 3i)(8 - 3i) $
## Step2: Use distributive property (FOIL)
$ 8 \cdot 8 + 8 \cdot (-3i) - 3i \cdot 8 + (-3i) \cdot (-3i) $
## Step3: Simplify each term
$ 64 - 24i - 24i + 9i^2 $, $ i^2 = -1 $, so $ 9(-1) = -9 $
## Step4: Combine like terms
$ (64 - 9) + (-24i - 24i) = 55 - 48i $
# Answer: $ 55 - 48i $

### 4. $ |13 + 2i| $
# Explanation:
## Step1: Recall modulus formula $ |a + bi| = \sqrt{a^2 + b^2} $
$ a = 13 $, $ b = 2 $
## Step2: Calculate $ \sqrt{13^2 + 2^2} = \sqrt{169 + 4} = \sqrt{173} $
# Answer: $ \sqrt{173} $

### 5. $ |-4 + i| $
# Explanation:
## Step1: Recall modulus formula $ |a + bi| = \sqrt{a^2 + b^2} $
$ a = -4 $, $ b = 1 $
## Step2: Calculate $ \sqrt{(-4)^2 + 1^2} = \sqrt{16 + 1} = \sqrt{17} $
# Answer: $ \sqrt{17} $

### 6. $ |8 - 9i| $
# Explanation:
## Step1: Recall modulus formula $ |a + bi| = \sqrt{a^2 + b^2} $
$ a = 8 $, $ b = -9 $
## Step2: Calculate $ \sqrt{8^2 + (-9)^2} = \sqrt{64 + 81} = \sqrt{145} $
# Answer: $ \sqrt{145} $

### 7. $ \frac{-5 + i}{1 - 5i} $
# Explanation:
## Step1: Multiply numerator and denominator by conjugate of denominator ($ 1 + 5i $)
$ \frac{(-5 + i)(1 + 5i)}{(1 - 5i)(1 + 5i)} $
## Step2: Simplify numerator (FOIL)
$ -5 \cdot 1 + (-5) \cdot 5i + i \cdot 1 + i \cdot 5i = -5 - 25i + i + 5i^2 $, $ i^2 = -1 $, so $ -5 - 24i - 5 = -10 - 24i $
## Step3: Simplify denominator (difference of squares)
$ 1^2 - (5i)^2 = 1 - 25i^2 = 1 + 25 = 26 $
## Step4: Divide numerator by denominator
$ \frac{-10 - 24i}{26} = \frac{-5 - 12i}{13} = -\frac{5}{13} - \frac{12}{13}i $
# Answer: $ -\frac{5}{13} - \frac{12}{13}i $

### 8. $ \frac{6 - 2i}{2 - 3i} $
# Explanation:
## Step1: Multiply numerator and denominator by conjugate of denominator ($ 2 + 3i $)
$ \frac{(6 - 2i)(2 + 3i)}{(2 - 3i)(2 + 3i)} $
## Step2: Simplify numerator (FOIL)
$ 6 \cdot 2 + 6 \cdot 3i - 2i \cdot 2 - 2i \cdot 3i = 12 + 18i - 4i - 6i^2 $, $ i^2 = -1 $, so $ 12 + 14i + 6 = 18 + 14i $
## Step3: Simplify denominator (difference of squares)
$ 2^2 - (3i)^2 = 4 - 9i^2 = 4 + 9 = 13 $
## Step4: Divide numerator by denominator
$ \frac{18 + 14i}{13} = \frac{18}{13} + \frac{14}{13}i $
# Answer: $ \frac{18}{13} + \frac{14}{13}i $

### 9. $ \frac{-10 + 11i}{12 - 7i} $
# Explanation:
## Step1: Multiply numerator and denominator by conjugate of denominator ($ 12 + 7i $)
$ \frac{(-10 + 11i)(12 + 7i)}{(12 - 7i)(12 + 7i)} $
## Step2: Simplify numerator (FOIL)
$ -10 \cdot 12 + (-10) \cdot 7i + 11i \cdot 12 + 11i \cdot 7i = -120 - 70i + 132i + 77i^2 $, $ i^2 = -1 $, so $ -120 + 62i - 77 = -197 + 62i $
## Step3: Simplify denominator (difference of squares)
$ 12^2 - (7i)^2 = 144 - 49i^2 = 144 + 49 = 193 $
## Step4: Divide numerator by denominator
$ \frac{-197 + 62i}{193} = -\frac{197}{193} + \frac{62}{193}i $
# Answer: $ -\frac{197}{193} + \frac{62}{193}i $

### 10. Point A
# Explanation:
## Step1: Determine coordinates on complex plane
Real (x) and imaginary (y) axes. Point A: x = -3, y = 4 (assuming grid units)
## Step2: Write as $ x + yi $
$ -3 + 4i $
# Answer: $ -3 + 4i $

### 11. Point B
# Explanation:
## Step1: Determine coordinates on complex plane
Point B: x = 2, y = 3
## Step2: Write as $ x + yi $
$ 2 + 3i $
# Answer: $ 2 + 3i $

### 12. Point C
# Explanation:
## Step1: Determine coordinates on complex plane
Point C: x = 4, y = -2
## Step2: Write as $ x + yi $
$ 4 - 2i $
# Answer: $ 4 - 2i $

### 13. Point D
# Explanation:
## Step1: Determine coordinates on complex plane
Point D: x = -2, y = -5
## Step2: Write as $ x + yi $
$ -2 - 5i $
# Answer: $ -2 - 5i $

### 14. Closer to Origin (A or D)
# Explanation:
## Step1: Calculate modulus of A ($ -3 + 4i $)
$ |A| = \sqrt{(-3)^2 + 4^2} = \sqrt{9 + 16} = 5 $
## Step2: Calculate modulus of D ($ -2 - 5i $)
$ |D| = \sqrt{(-2)^2 + (-5)^2} = \sqrt{4 + 25} = \sqrt{29} \approx 5.39 $
## Step3: Compare moduli
$ 5 < \sqrt{29} $, so A is closer.
# Answer: Point A is closer because its modulus (5) is less than D's modulus ($ \sqrt{29} \approx 5.39 $).

### 15. $ i^3 $
# Explanation:
## Step1: Recall $ i^2 = -1 $
## Step2: $ i^3 = i^2 \cdot i = -1 \cdot i = -i $
# Answer: $ -i $

### 16. $ i^{63} $
# Explanation:
## Step1: Find remainder of 63 divided by 4 (cycle of $ i $: $ i^1 = i, i^2 = -1, i^3 = -i, i^4 = 1 $)
$ 63 \div 4 = 15 $ remainder $ 3 $
## Step2: $ i^{63} = i^{4 \cdot 15 + 3} = (i^4)^{15} \cdot i^3 = 1^{15} \cdot (-i) = -i $
# Answer: $ -i $

Sovi.AI - AI Math Tutor

QUESTION IMAGE

Question

Explanation:

1a. \( z_1 + z_2 \)

Step1: Identify \( z_1 \) and \( z_2 \)

Step2: Add real and imaginary parts

Step1: Identify \( z_1 \) and \( z_2 \)

Step2: Subtract real and imaginary parts

Step1: Use distributive property (FOIL)

Step2: Simplify each term

Step3: Combine like terms

Answer:

1b. \( z_1 - z_2 \)