Problem 72

Question

Find the sum, difference, product, and quotient of $1+3 i$ and $1-3 i$

Step-by-Step Solution

Verified

Answer

Sum: 2, Difference: 6i, Product: 10, Quotient: 1+0.6i.

1Step 1: Find the Sum

To find the sum of the complex numbers $1 + 3i$ and $1 - 3i$, we add the real parts and the imaginary parts separately: real parts: $1 + 1 = 2$ and imaginary parts: $3i - 3i = 0$. Hence, the sum is $2 + 0i = 2$.

2Step 2: Find the Difference

To find the difference, subtract $1 - 3i$ from $1 + 3i$. Subtract the real and imaginary parts separately: $1 - 1 = 0$ for the real parts and $3i - (-3i) = 3i + 3i = 6i$ for the imaginary parts. So, the difference is $0 + 6i = 6i$.

3Step 3: Find the Product

To find the product, multiply the complex numbers $1 + 3i$ and $1 - 3i$. Use the formula $(a + bi)(c + di) = ac + adi + bci + bdi^2$: $1\cdot1 + 1(-3i) + 3i(1) + 3i(-3i)$. Simplify to get $1 - 3i + 3i - 9i^2$; since $i^2 = -1$, this becomes $1 - 9(-1) = 1 + 9 = 10$. The product is $10$.

4Step 4: Find the Quotient

To find the quotient of $1 + 3i$ by $1 - 3i$, multiply numerator and denominator by the conjugate of the denominator: \[\frac{1+3i}{1-3i} \times \frac{1+3i}{1+3i}\]. Simplify the denominator using the formula $a^2 + b^2$ which gives $1^2 + (3)^2 = 10$. The numerator becomes $(1+3i)^2 = 1 + 6i - 9(-1) = 10 + 6i$. So, the quotient is $\frac{10 + 6i}{10} = 1 + 0.6i$.

Key Concepts

Complex Number OperationsAddition of Complex NumbersSubtraction of Complex NumbersMultiplication of Complex NumbersDivision of Complex Numbers

Complex Number Operations

Complex numbers are numbers that consist of a real and an imaginary part. Their operations involve addition, subtraction, multiplication, and division, which follow specific rules. Understanding these operations is essential for working with complex numbers in mathematics. Each operation requires combining like terms or using properties unique to complex numbers.

Real part: Represented by a real number.
Imaginary part: Represented by an imaginary number, using the symbol 'i' where $i^2 = -1$.
Complex numbers are often written in the form $a + bi$.

By using these operations, complex numbers can be manipulated and used in various mathematical contexts.

Addition of Complex Numbers

To add complex numbers, you simply add their real parts and their imaginary parts separately. This is a straightforward process, much like regular addition. Take two complex numbers $a + bi$ and $c + di$, you add:

Real parts: $a + c$
Imaginary parts: $b + d$

For example, to add $1 + 3i$ and $1 - 3i$, we find:

Real part: $1 + 1 = 2$
Imaginary part: $3i - 3i = 0$

This results in the sum of $2 + 0i$, simplifying to just $2$. Simple, right? It's like organizing apples and oranges; real numbers and imaginary numbers are added separately.

Subtraction of Complex Numbers

Subtracting complex numbers is just as simple as addition. You subtract the real parts and imaginary parts separately. The concept is as follows:

Real parts: $a - c$
Imaginary parts: $b - d$

So if you have $1 + 3i$ and $1 - 3i$, subtract like this:

Real part: $1 - 1 = 0$
Imaginary part: $3i - (-3i) = 3i + 3i = 6i$

Thus, the difference is $0 + 6i = 6i$. It’s similar to balancing a budget; you handle the real and imaginary parts individually to find the difference.

Multiplication of Complex Numbers

When you multiply complex numbers, you use distributive property and the fact that $i^2 = -1$. If you have two complex numbers $(a + bi)$ and $(c + di)$, use the formula:\[(a + bi)(c + di) = ac + adi + bci + bdi^2\]Let's see how it works with $1 + 3i$ and $1 - 3i$:

Apply distribution: $1\cdot1 + 1(-3i) + 3i(1) + 3i(-3i)$
Simplify: $1 - 3i + 3i - 9i^2$
Since $i^2 = -1$: $1 + 9 = 10$

The product is $10$. It’s like expanding a factored expression; distributive steps take place, then consolidation.

Division of Complex Numbers

Dividing complex numbers involves multiplying by the conjugate of the denominator. This process eliminates the imaginary part from the denominator. For $\frac{a + bi}{c + di}$, multiply by $\frac{c - di}{c - di}$: