1. The Problem: Square Roots of Negative Numbers
Let's start with something familiar. If someone asks you, "What number, multiplied by itself, gives you 9?" you can quickly answer: 3, because $3 \times 3 = 9$. You might also think of −3, because $(-3) \times (-3) = 9$ as well. Both are valid answers.
Now consider a trickier question: "What number, multiplied by itself, gives you −9?"
Think about it carefully. Could it be 3? No, because $3 \times 3 = +9$, not −9. Could it be −3? No, because $(-3) \times (-3) = +9$, not −9 either. A negative times a negative is always positive. This is a fundamental rule of arithmetic that we cannot break.
KEY INSIGHT
No real number, when multiplied by itself, can produce a negative result. This means that within the real number system, the square root of a negative number simply does not exist.
For centuries, mathematicians ran into this wall. Equations like $x^2 = -1$ had no solution in the real numbers. But rather than accepting this as a dead end, mathematicians asked a creative question: "What if we invented a number that could do this?"
2. Introducing the Imaginary Unit: i
In the 1500s and 1600s, mathematicians began working with a new concept. They defined a brand new number, called i (for "imaginary"), with one simple but powerful definition:
DEFINITION
The imaginary unit i is defined as:
Or equivalently:
This means i is the number that, when squared, gives −1.
The word "imaginary" can be a bit misleading. It makes it sound like these numbers are fake or made up in a way that real numbers aren't. But in mathematics, i is just as legitimate as any other number. It follows consistent rules, it solves real problems in engineering and physics, and it extends our number system in a useful and beautiful way.
Think of it this way: at one point in history, negative numbers were also considered "fake." People asked, "How can you have less than nothing?" But eventually, negative numbers became an essential tool. Imaginary numbers followed the same path — from skepticism to acceptance to everyday use.
3. How to Simplify Square Roots of Negative Numbers
Now that we have i, we have a systematic method for handling any square root of a negative number. The process works in three steps.
The Three-Step Method
Step 1: Separate the negative sign
Rewrite the expression $\sqrt{-n}$ as $\sqrt{-1 \cdot n}$. You are simply factoring out the −1 from inside the square root.
Step 2: Apply the product rule
Split it into two separate square roots: $\sqrt{-1} \cdot \sqrt{n}$. The product rule tells us that $\sqrt{a \cdot b} = \sqrt{a} \cdot \sqrt{b}$ when at least one factor is non-negative.
Step 3: Replace and simplify
Replace $\sqrt{-1}$ with i. Then simplify $\sqrt{n}$ if possible. Your final answer will be in the form $i\sqrt{n}$ or a whole number times i.
Worked Examples
| Expression | Step 1: Factor | Step 2: Split | Step 3: Replace & Simplify |
|---|---|---|---|
| $\sqrt{-9}$ | $\sqrt{-1 \cdot 9}$ | $\sqrt{-1} \cdot \sqrt{9}$ | $i \cdot 3 = 3i$ |
| $\sqrt{-25}$ | $\sqrt{-1 \cdot 25}$ | $\sqrt{-1} \cdot \sqrt{25}$ | $i \cdot 5 = 5i$ |
| $\sqrt{-16}$ | $\sqrt{-1 \cdot 16}$ | $\sqrt{-1} \cdot \sqrt{16}$ | $i \cdot 4 = 4i$ |
| $\sqrt{-7}$ | $\sqrt{-1 \cdot 7}$ | $\sqrt{-1} \cdot \sqrt{7}$ | $i\sqrt{7}$ |
| $\sqrt{-50}$ | $\sqrt{-1 \cdot 50}$ | $\sqrt{-1} \cdot \sqrt{50}$ | $i \cdot 5\sqrt{2} = 5i\sqrt{2}$ |
Notice the pattern: when the number under the square root is a perfect square (like 9, 25, or 16), you get a clean whole number times i. When it's not a perfect square (like 7 or 50), you leave the square root in your answer, with i out front.
4. Why We Need the ± (Plus or Minus) Symbol
Before we apply imaginary numbers to equations, we need to understand a critical concept about square roots in algebra.
Consider the equation $x^2 = 25$. What value(s) of x make this true? If you think about it, two numbers work:
- ✔ $x = 5$ works, because $5 \times 5 = 25$
- ✔ $x = -5$ also works, because $(-5) \times (-5) = 25$
This happens because squaring "erases" the sign of a number. Whether you start with a positive or negative number, squaring it always produces a positive result. So when you reverse the process (taking a square root to solve for x), you must account for both possibilities.
RULE
Whenever you solve an equation by taking the square root of both sides, you must include ± (read "plus or minus") to capture both solutions.
This is shorthand for writing two separate equations: $x = +\sqrt{k}$ AND $x = -\sqrt{k}$
Important: The ± symbol is not optional — forgetting it means you're only finding half of the solutions. This is one of the most common mistakes students make when solving quadratic equations.
5. Solving Quadratic Equations with Non-Real Solutions
Now we can combine everything we've learned. Let's solve several equations step by step, building from simpler to more complex.
Example 1: $x^2 + 7 = 0$
Step 1: Isolate $x^2$
Subtract 7 from both sides:
Step 2: Take the square root
Remember to include ±:
Step 3: Simplify using i
Solutions:
(two non-real solutions)
Example 2: $x^2 + 36 = 0$
Step 1:
Step 2:
Step 3:
Solutions:
(two non-real solutions)
Example 3: $2x^2 + 18 = 0$
Step 1: Subtract 18
Step 2: Divide by 2
Step 3: Take square root and simplify
Solutions:
(two non-real solutions)
6. Quick Trick: How to Predict the Type of Solutions
You don't always need to fully solve an equation to know what kind of solutions it has. Here's a quick shortcut for equations in the form $x^2 = k$ (or equivalently $x^2 + c = 0$, where $k = -c$):
| If $x^2 = ...$ | Square root of... | Number of solutions | Type |
|---|---|---|---|
| Positive number (e.g., 9) | A positive number | Two | Two REAL solutions |
| Zero | Zero | One | One REAL solution |
| Negative number (e.g., −7) | A negative number | Two | Two NON-REAL solutions |
For more complex quadratics: For equations in the form $ax^2 + bx + c = 0$, mathematicians use something called the discriminant (the expression $b^2 - 4ac$ from the quadratic formula) to determine the nature of solutions.
But for simple equations like $x^2 + 7 = 0$, the shortcut above is all you need: since $x^2$ equals a negative number (−7), you immediately know the solutions are two non-real (imaginary) numbers.
Chapter Summary
The imaginary unit i was created to extend the real number system so that every square root has a value, even square roots of negative numbers. It is defined by the property $i = \sqrt{-1}$, or equivalently $i^2 = -1$.
To simplify the square root of any negative number, you separate out the −1, replace $\sqrt{-1}$ with i, and then simplify the remaining positive square root as you normally would.
The ± symbol is essential whenever you solve by taking a square root, because both the positive and negative versions of a number produce the same result when squared.
When a quadratic equation requires you to take the square root of a negative number, it has two non-real solutions (one positive imaginary and one negative imaginary).
Practice Problems
Simplify each expression:
- Simplify $\sqrt{-49}$
- Simplify $\sqrt{-100}$
- Simplify $\sqrt{-11}$
- Simplify $\sqrt{-72}$
Solve each equation and describe the solutions:
- $x^2 + 16 = 0$
- $x^2 + 5 = 0$
- $3x^2 + 48 = 0$
- $x^2 - 9 = 0$ (compare this one to the others!)
Answer Key
Simplification:
1) $7i$ 2) $10i$ 3) $i\sqrt{11}$ 4) $6i\sqrt{2}$
Equations:
5) $x = \pm 4i$ (two non-real)
6) $x = \pm i\sqrt{5}$ (two non-real)
7) $x = \pm 4i$ (two non-real)
8) $x = \pm 3$ (two real solutions — different because $x^2 = +9$!)
Related Topics
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