Master Complex Numbers: Simple A+bi Form Guide

Welcome to the exciting world of complex numbers! Today, we're going to tackle some common expressions and simplify them into the standard $a+bi$ form. Don't worry, it's not as daunting as it sounds. We'll break down each step to make it clear and easy to follow. Think of $a+bi$ as a special way to write numbers where 'a' is the real part and 'b' is the imaginary part, with 'i' representing the square root of -1. Ready to dive in and become a complex number whiz?

Understanding the Basics of Complex Numbers

Before we start simplifying, let's quickly recap what makes complex numbers tick. A complex number is generally expressed in the form $a+bi$, where 'a' represents the real part and 'b' represents the imaginary part. The symbol 'i' is the imaginary unit, defined as the square root of -1 (i.e., $i = \sqrt{-1}$). This means that $i^2 = -1$. This fundamental property is key to simplifying many complex number expressions. When we work with complex numbers, we often perform operations like addition, subtraction, and multiplication, much like we do with regular algebraic expressions. The main difference is how we handle the 'i' terms. Remember that combining like terms is crucial – real parts are combined with real parts, and imaginary parts are combined with imaginary parts. When you see $i^2$, always replace it with -1. This simple substitution is the magic wand that transforms complex expressions into the clean $a+bi$ format we're aiming for. We'll be using these rules throughout our examples to ensure accuracy and clarity.

Simplifying Complex Number Expressions

Now, let's get down to business and simplify those expressions one by one. We'll focus on the techniques needed to get each one into the coveted $a+bi$ form.

a. $-3+6i - (-5-3i) - 8i$

To simplify this expression, we first need to distribute the negative sign to the terms inside the second parenthesis. This changes the signs of $-5$ and $-3i$ to $+5$ and $+3i$ respectively. So, the expression becomes: $-3+6i + 5+3i - 8i$.

Next, we group the real terms and the imaginary terms together. The real terms are $-3$ and $+5$. The imaginary terms are $+6i$, $+3i$, and $-8i$.

Now, combine the real terms: $-3 + 5 = 2$.

Combine the imaginary terms: $6i + 3i - 8i = 9i - 8i = 1i$, which can be written simply as $i$.

Putting it all together in $a+bi$ form, we get $2+i$.

b. $4i(-2-8i)$

For this expression, we'll use the distributive property. Multiply $4i$ by each term inside the parenthesis: $4i \times -2$ and $4i \times -8i$.

First, $4i \times -2 = -8i$.

Next, $4i \times -8i = -32i^2$.

Now, remember that $i^2 = -1$. Substitute $-1$ for $i^2$ in the second term: $-32(-1) = +32$.

So, the expression simplifies to $-8i + 32$. To put this in the standard $a+bi$ form, we write the real part first: $32-8i$.

c. $5i \cdot i \cdot -2i$

This involves multiplying three imaginary terms. Let's do it step-by-step. First, multiply the numbers: $5 \times 1 \times -2 = -10$.

Next, multiply the 'i' terms: $i \times i \times i = i^3$.

We know that $i^2 = -1$. So, $i^3$ can be written as $i^2 \times i = -1 \times i = -i$.

Combining the number and the 'i' term, we get $-10 \times (-i) = +10i$.

Since there is no real part (the 'a' term is 0), the expression in $a+bi$ form is $0+10i$ or simply $10i$.

d. $(1-7i)^2$

This expression means we need to multiply $(1-7i)$ by itself. We can use the FOIL method (First, Outer, Inner, Last) or the square of a binomial formula $(x-y)^2 = x^2 - 2xy + y^2$.

Using FOIL:

• First: $1 \times 1 = 1$
• Outer: $1 \times -7i = -7i$
• Inner: $-7i \times 1 = -7i$
• Last: $-7i \times -7i = +49i^2$

Combine these terms: $1 - 7i - 7i + 49i^2$.

Combine the like terms (the imaginary terms): $1 - 14i + 49i^2$.

Now, substitute $i^2 = -1$: $1 - 14i + 49(-1)$.

This simplifies to $1 - 14i - 49$.

Finally, combine the real terms: $1 - 49 = -48$.

So, the simplified expression in $a+bi$ form is $-48-14i$.

e. $-3i \cdot 6i - 3(-7+6i)$

This expression involves multiplication and subtraction. Let's tackle the multiplication part first: $-3i \cdot 6i$.

Multiply the numbers: $-3 \times 6 = -18$.

Multiply the 'i' terms: $i \times i = i^2$.

So, $-3i \cdot 6i = -18i^2$. Since $i^2 = -1$, this becomes $-18(-1) = +18$.

Now, let's distribute the $-3$ to the terms in the second parenthesis: $-3(-7+6i)$.

$-3 \times -7 = +21$.

$-3 \times 6i = -18i$.

So, $-3(-7+6i) = 21 - 18i$.

Now, combine the results from both parts: $18 + (21 - 18i)$.

Remove the parenthesis: $18 + 21 - 18i$.

Combine the real terms: $18 + 21 = 39$.

The imaginary term is $-18i$.

Putting it all together in $a+bi$ form, we get $39-18i$.

f. $(6+2i)^2$

Similar to part (d), we need to multiply $(6+2i)$ by itself. We can use the FOIL method or the square of a binomial formula $(x+y)^2 = x^2 + 2xy + y^2$.

Using FOIL:

• First: $6 \times 6 = 36$
• Outer: $6 \times 2i = +12i$
• Inner: $2i \times 6 = +12i$
• Last: $2i \times 2i = +4i^2$

Combine these terms: $36 + 12i + 12i + 4i^2$.

Combine the like terms (the imaginary terms): $36 + 24i + 4i^2$.

Now, substitute $i^2 = -1$: $36 + 24i + 4(-1)$.

This simplifies to $36 + 24i - 4$.

Finally, combine the real terms: $36 - 4 = 32$.

So, the simplified expression in $a+bi$ form is $32+24i$.

Conclusion: Your Journey into Complex Numbers

Congratulations! You've successfully navigated through simplifying various complex number expressions and converted them into the standard $a+bi$ form. We've seen how essential it is to remember that $i^2 = -1$ and how to properly combine real and imaginary terms. Working with complex numbers is a fundamental skill in many areas of mathematics, physics, and engineering, and practicing these simplification techniques will build a strong foundation. Remember, the key is careful attention to detail, especially when distributing negative signs and substituting for $i^2$. Don't be discouraged if you make a mistake; it's all part of the learning process! Keep practicing, and you'll find these operations become second nature.

For further exploration and to deepen your understanding of complex numbers, you can visit resources like Khan Academy for excellent tutorials and practice exercises, or delve into advanced topics on the Wolfram MathWorld website.