How to Prove √3 * √5 is Irrational: A Simple Guide

Understanding Irrational Numbers

Alright, before we dive into proving that √3 * √5 is irrational, let’s take a moment to refresh what we mean by irrational numbers. You've probably encountered terms like "rational" and "irrational" in your math classes, but let’s break them down quickly.

• Rational numbers are numbers that can be expressed as a fraction of two integers. So, if you can write a number as p/q, where both p and q are integers, it’s rational.

• Irrational numbers, on the other hand, cannot be written as a fraction of integers. These are numbers like √2, π, and e—numbers that go on forever without repeating in a regular pattern.

So, now you might be thinking, "Okay, I get it, but how do we prove that √3 * √5 is irrational?" Let's break it down.

The Basics: √3 and √5 are Irrational

You need to start by recognizing that √3 and √5 are both irrational. I know, it might seem obvious, but let’s quickly go over why they are irrational.

1. Why is √3 irrational?

If √3 were rational, we could write it as p/q, where p and q are integers with no common factors (in other words, they are coprime). But here’s the thing: √3 doesn’t simplify nicely into a fraction. If you assume √3 = p/q, and square both sides, you get:

3 = p²/q², which gives us p² = 3q².

Now, p² must be divisible by 3, which means p must also be divisible by 3 (since if a square is divisible by a number, the original number must be divisible by that same number). Let’s say p = 3k for some integer k. Substituting that into the equation, we get:

(3k)² = 3q², or 9k² = 3q², which simplifies to 3k² = q².

Now, q² must also be divisible by 3, and so q must be divisible by 3. But if both p and q are divisible by 3, they’re not coprime, which contradicts our assumption. So, √3 is irrational.

2. Why is √5 irrational?

The proof for √5 is very similar. You assume √5 = p/q (with p and q being coprime integers), square both sides to get:

5 = p²/q², or p² = 5q².

For p² to be divisible by 5, p must also be divisible by 5. Let’s say p = 5k for some integer k. Substituting into the equation:

(5k)² = 5q², or 25k² = 5q², which simplifies to 5k² = q².

So, q² must be divisible by 5, and therefore, q must be divisible by 5. But just like in the case of √3, if both p and q are divisible by 5, they are not coprime, which is a contradiction. Hence, √5 is also irrational.

Proving √3 * √5 is Irrational

Now that we know √3 and √5 are both irrational, we’re ready to prove that √3 * √5 is irrational too. Let’s assume the opposite, for the sake of contradiction.

Step 1: Assume √3 * √5 is Rational

Let’s assume, for a moment, that √3 * √5 is rational. This means we could write it as p/q, where p and q are integers with no common factors. So, we assume:

√3 * √5 = p/q.

Squaring both sides, we get:

3 * 5 = p²/q², or 15 = p²/q², which simplifies to:

p² = 15q².

Step 2: Examine the Result

Now, look at this equation: p² = 15q². This implies that p² is divisible by 15, and so p must also be divisible by 3 and 5 (since 15 = 3 * 5). Let’s say p = 15k for some integer k.

Substituting into the equation:

(15k)² = 15q², or 225k² = 15q², which simplifies to:

15k² = q².

Now, q² must be divisible by 15, and thus q must be divisible by both 3 and 5. So, both p and q are divisible by 3 and 5, which means they share common factors—contradicting our original assumption that p and q were coprime.

Conclusion

Since we’ve reached a contradiction, our assumption that √3 * √5 is rational must be false. Therefore, √3 * √5 is irrational.

Why Does This Matter?

You might be wondering, "Why is this important? What’s the big deal?" Well, proving that certain combinations of irrational numbers remain irrational helps us better understand the structure of numbers. It’s also a great exercise in logical thinking, and it teaches us how assumptions can lead to contradictions, which is fundamental in mathematics.

Plus, who doesn’t love proving something that seems simple at first, but turns out to be more complex than expected? I’ve always found these little "Aha!" moments in math to be pretty satisfying.

In any case, next time someone asks you if √3 * √5 is irrational, you’ll know exactly how to prove it!

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