[Common Math 1 Part 12] Complex Number Operations and Conjugates

In the previous post, we began with the reason complex numbers are needed at all. Now it is time to organize them as a working calculation system.

Systematically organize addition, subtraction, multiplication, and division for complex numbers, and understand why conjugates matter.

Let us fix the flow first.

• A complex number is written in the form $a+bi$.
• The calculations follow ordinary algebra, and then we use $i^2=-1$.
• Division is where things start to feel awkward because $i$ remains in the denominator.
• The key tool that resolves that awkwardness is the complex conjugate.

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1. What to Notice Before Calculating

We usually write a complex number as

$$a+bi \qquad (a, b \text{ are real numbers})$$

Here,

• $a$ is the real part
• $b$ is the coefficient of the imaginary part

Strictly speaking, the imaginary term is $bi$, and $b$ is its real coefficient. In beginner courses, however, $b$ is often informally called the imaginary part as well.

For example,

• the real part of $3+2i$ is $3$, and its imaginary coefficient is $2$
• the real part of $-1-4i$ is $-1$, and its imaginary coefficient is $-4$

This form matters because after every calculation, we usually want to rewrite the answer again in the standard form $a+bi$.

1-1. What Does It Mean for Two Complex Numbers to Be Equal?

If

$$a+bi \qquad \text{and} \qquad c+di$$

are equal, then both parts must match:

$$a=c, \qquad b=d.$$

So the real parts must be equal, and the imaginary coefficients must also be equal.

For example, suppose

$$x+2i=5+2i.$$

Then comparing parts immediately gives

$$x=5.$$

Now compare that with

$$x+2i=5-3i.$$

Then we would need

$$x=5, \qquad 2=-3,$$

and the second statement is impossible. So there is no real value of $x$ that satisfies this equation.

This viewpoint becomes useful later when we organize equations.

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2. Addition and Subtraction

Addition and subtraction are the simplest operations. We combine real parts with real parts and imaginary parts with imaginary parts.

$$(a+bi)+(c+di)=(a+c)+(b+d)i$$ $$(a+bi)-(c+di)=(a-c)+(b-d)i$$

For example,

$$(3+4i)+(2-5i)=5-i$$ $$(1-2i)-(-3+i)=4-3i$$

The rule is simply to collect the real part and the imaginary coefficient separately. The more interesting ideas usually appear in multiplication and division.

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3. Multiplication Works Like Polynomial Expansion

For multiplication, we do not need a completely new rule. We expand as usual and then use $i^2=-1$.

$$(a+bi)(c+di)=ac+adi+bci+bd i^2$$

Applying $i^2=-1$ gives

$$(a+bi)(c+di)=(ac-bd)+(ad+bc)i.$$

3-1. A Concrete Example

Let us compute

$$(2+3i)(1-4i).$$

First expand:

$$2-8i+3i-12i^2.$$

Combine like terms:

$$2-5i-12i^2.$$

Since $i^2=-1$,

$$2-5i+12=14-5i.$$

Therefore,

$$(2+3i)(1-4i)=14-5i.$$

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4. What Is a Conjugate?

For the complex number $a+bi$, the number

$$a-bi$$

is called its conjugate.

So we keep the real part the same and change only the sign of the imaginary part.

For example,

• the conjugate of $3+2i$ is $3-2i$
• the conjugate of $-1-5i$ is $-1+5i$

If we write a complex number as $z$, its conjugate is often written as $\overline{z}$.

At first this may look like a small sign change, but it becomes a decisive tool in division.

4-1. Why Is the Conjugate Useful?

Look at

$$(a+bi)(a-bi).$$

Expanding gives

$$a^2-abi+abi-b^2i^2.$$

The middle terms cancel, and since $i^2=-1$, we get

$$a^2+b^2.$$

So

$$(a+bi)(a-bi)=a^2+b^2.$$

The important point is that the result is a real number. When the denominator is complex, multiplying by its conjugate turns that denominator into a real number. That is exactly why conjugates are used in division.

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5. Division Uses the Conjugate to Clean Up the Denominator

Division of complex numbers is also possible, but we usually want the answer back in the form $a+bi$.

For example, consider

$$\frac{1+2i}{3+i}.$$

If we leave it as it is, the denominator still contains $i$. It is much cleaner to turn the denominator into a real number.

To do that, multiply the numerator and denominator by the conjugate of $3+i$, which is $3-i$:

$$\frac{1+2i}{3+i} \cdot \frac{3-i}{3-i}.$$

Then the denominator becomes

$$(3+i)(3-i)=3^2+1^2=10,$$

and the numerator is

$$(1+2i)(3-i)=3-i+6i-2i^2,$$

so first we combine like terms:

$$3+5i-2i^2,$$

and then apply $i^2=-1$:

$$3+5i-2(-1)=3+5i+2=5+5i.$$

Therefore,

$$\frac{1+2i}{3+i}=\frac{5+5i}{10}=\frac{1}{2}+\frac{1}{2}i.$$

5-1. Why Can We Multiply Top and Bottom by the Same Number?

Because

$$\frac{3-i}{3-i}=1,$$

the value does not change. Of course, this works only when the denominator is not 0.

In complex numbers, being equal to 0 means both the real part and the imaginary coefficient are 0. That is not the case for $3-i$.

5-2. The Key Idea Matters More Than the Formula

The main idea to remember is this:

Multiply by the conjugate of the denominator so the denominator becomes a real number.

That principle matters more than memorizing a ready-made formula.

Written once as a general pattern,

$$\frac{a+bi}{c+di}=\frac{(a+bi)(c-di)}{(c+di)(c-di)},$$

and since

$$(c+di)(c-di)=c^2+d^2,$$

the denominator becomes real.

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6. Common Mistakes

The rules are simple, so many errors come from small sign mistakes.

6-1. Treating $i^2=-1$ as if $i=-1$

$i^2=-1$ does not mean $i=-1$. So $-2i^2=2$, but $-2i$ does not suddenly become $2$.

6-2. Changing the Real Part in the Conjugate

The conjugate of $a+bi$ is $a-bi$. It is not $-a-bi$.

6-3. Multiplying Only the Denominator in Division

If we multiply only the denominator by the conjugate, we change the value. We must multiply both numerator and denominator by the same nonzero complex number.

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7. Key Takeaways

• After each calculation, rewrite the result in the form $a+bi$.
• In addition and subtraction, combine real parts and imaginary parts separately.
• In multiplication, expand as usual and then use $i^2=-1$.
• The conjugate of $a+bi$ is $a-bi$.
• The product $(a+bi)(a-bi)=a^2+b^2$ is real.
• That is why complex division uses the conjugate of the denominator.

In the next post, we will use this wider number system to organize the roots of quadratic equations more generally and to understand the role of the quadratic formula.

One-line conclusion:

A conjugate is not a decorative pairing. It is the key tool that turns complex division into a clean real denominator.