[Common Math 1 Part 11] Why Quadratic Equations Lead to Complex Numbers

In the previous post, we saw that factorization is the key tool for reading the roots of an equation. But some quadratic equations can no longer be factored completely within the real numbers. In this post, we examine that stopping point and introduce complex numbers as the number-system extension that lets us move beyond it.

Understand why complex numbers are needed through quadratic equations that cannot be solved in the real numbers, and learn their basic form and simple calculations.

Let us fix the flow first.

• We compare quadratic equations that still work in the real numbers with ones that stop there.
• We introduce the imaginary unit $i$ and complex numbers to move past that stopping point.
• We learn the basic calculations and see how the roots of quadratic equations become wider.
• Then, at the end, we connect this to why complex numbers form a completed stage for polynomial roots, so we do not have to keep adding larger number systems.

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1. The Moment the Real Numbers Stop Working

First, consider an equation that works well inside the real numbers:

$$x^2-1=0$$

Factoring gives

$$(x-1)(x+1)=0$$

so the solutions are $x=1,-1$.

Now compare that with

$$x^2+1=0.$$

Rewriting gives

$$x^2=-1.$$

Here is the problem: the square of a real number is always at least 0.

• $2^2=4$
• $(-3)^2=9$
• $0^2=0$

No real number can square to $-1$. So this equation has no solution in the real numbers.

The important point is not that mathematics has failed. It is that the number system we are using is still too small.

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2. The Imaginary Unit $i$ and Complex Numbers

Since no real number satisfies $x^2=-1$, we introduce a new symbol.

$$i^2=-1$$

A number $i$ with this property is called the imaginary unit.

So the starting point is not the notation $\sqrt{-1}$. It is the definition:

define a new number $i$ so that $i^2=-1$

Once we accept that definition, we can view $i$ as a square root of $-1$.

A number of the form

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

is called a complex number.

2-1. What Does a Complex Number Look Like?

• $3+2i$ is a complex number.
• $-4+i$ is also a complex number.
• $5$ can be written as $5+0i$, so it is also a complex number.
• $-2i$ can be written as $0-2i$, so it is also a complex number.

So the real numbers are contained inside the complex numbers. Complex numbers do not replace the real numbers; they extend them.

2-2. Why Do We Need Them?

The equation

$$x^2+1=0$$

becomes, in the complex numbers,

$$x^2=-1=i^2,$$

so its solutions are

$$x=\pm i.$$

A problem that stopped in the real numbers starts working again once we enlarge the number system.

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3. Complex Numbers Still Belong to the General Pattern of Extension

This pattern did not begin with complex numbers.

• $x+3=1$ cannot be solved inside the natural numbers, so we need the integer $-2$.
• $2x=1$ cannot be solved inside the integers, so we need the rational number $\frac{1}{2}$.
• $x^2=2$ cannot be expressed exactly inside the rational numbers, so we need the real number $\sqrt{2}$.

So the number system has expanded whenever the current one was too small to contain the solution of an equation. Complex numbers belong to that same pattern.

Complex numbers also belong to this pattern. At this stage, however, the most important point is simpler: complex numbers are the extension that lets us keep solving equations when the real numbers stop. Why this extension also has a special kind of completeness will be easier to see after we first work with some actual complex roots.

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4. How Do We Calculate with Complex Numbers?

The key idea is not to memorize many separate rules. We expand and simplify as usual, and only at the end use $i^2=-1$.

4-1. Addition and Subtraction

Add or subtract the real parts together, and the $i$-parts together.

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

4-2. Multiplication

Multiply as with polynomials, then use $i^2=-1$.

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

Grouping like terms gives

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

Now use $i^2=-1$:

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

So

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

4-3. Powers of $i$ Form a Cycle

Higher powers of $i$ appear often, and they follow a repeating pattern:

$$i^0=1, \qquad i^1=i, \qquad i^2=-1, \qquad i^3=-i, \qquad i^4=1$$

because

$$i^4=(i^2)^2=(-1)^2=1.$$

So after $i^4$, the same pattern starts again. For example,

$$i^6=i^4 \cdot i^2=1 \cdot (-1)=-1.$$

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5. Quadratic Equations Gain a Wider Set of Roots

In Common Math 1, the most direct meaning of complex numbers is that they let us carry quadratic equations farther than the real numbers can.

5-1. The Simplest Example

$$x^2+1=0$$

gives

$$x^2=-1,$$

so the roots are

$$x=\pm i.$$

Inside the real numbers, this equation had no solution. Inside the complex numbers, it has two distinct roots.

5-2. A More Typical Example

Consider

$$x^2-4x+5=0.$$

To complete the square, note that

$$(x-2)^2=x^2-4x+4.$$

So rewrite the equation as

$$x^2-4x+4+1=0,$$

that is,

$$(x-2)^2+1=0.$$

Then

$$(x-2)^2=-1,$$

so

$$x-2=\pm i,$$

and therefore

$$x=2\pm i.$$

So complex numbers are not a temporary trick for one strange equation. They are an extended language of roots that lets us keep going when square roots of negative numbers appear.

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6. "No Real Root" Is Not the Same as "No Root"

This is one of the most important distinctions for beginners.

• No real root means there is no solution within the real numbers.
• No root can sound like there is no solution in the chosen number system at all.

We study complex numbers in order to see this difference clearly.

For example,

$$x^2+1=0$$

has

• no solution in the real numbers, but
• the solutions $x=\pm i$ in the complex numbers.

So whenever we talk about solving an equation, we should also ask:

In which number system are we solving it?

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7. Why Are Complex Numbers a Completed System for Polynomial Roots?

So far, we have focused on why complex numbers are needed. Now we can take one more step and ask why they are often described as a completed stage for polynomial equations.

A standard result that supports this idea is the Fundamental Theorem of Algebra.

Every polynomial of degree $n\ge 1$ with complex coefficients has at least one complex root, and in fact has exactly $n$ roots in the complex numbers when multiplicity is counted.

Equivalently, every polynomial with complex coefficients factors completely into linear factors over the complex numbers.

This is important for a clear reason.

• Once polynomial coefficients are allowed to be complex, their roots can also be contained inside the complex numbers.
• In other words, we do not have to introduce yet another larger number system just to hold those roots.
• That is why, from the viewpoint of polynomial coefficients and roots, complex numbers can be viewed as a kind of completion of number-system expansion.

In Common Math 1, we do not prove this theorem rigorously. But it is still an important reason for studying complex numbers. Complex numbers are not just a temporary device for solving $x^2+1=0$; they are the number system in which polynomial roots no longer force us to keep extending outward.

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8. Practice

Use the checkpoint below to review why complex numbers expand the idea of roots.

9. Key Takeaways

• There is no real number that satisfies $x^2=-1$.
• At that stopping point, we introduce the imaginary unit $i$ with $i^2=-1$.
• A number of the form $a+bi$ is called a complex number, and the real numbers are contained inside the complex numbers.
• Once we introduce complex numbers, quadratic equations that fail in the real numbers can be solved all the way through.
• From the viewpoint of polynomial coefficients and roots, complex numbers form a completed stage, because polynomial roots no longer force us to add still larger number systems.

In the next post, we will organize complex-number operations more systematically and introduce an important tool: conjugates.

In one sentence:

Complex numbers are the number-system extension that lets us continue past quadratic equations that stop in the real numbers, and they are special because they hold both polynomial coefficients and roots in one completed setting.