Is 0 101100101010 an irrational number Justify your answer?

Is 0.101100101010 an Irrational Number? Let’s Get Real.

Rational or irrational? These terms might sound like something straight out of a math textbook (and, well, they are), but understanding them is actually pretty crucial for making sense of the number world. So, let’s dive in and figure out if that number, 0.101100101010, is one of those “irrational” types.

Rational vs. Irrational: The Lowdown

Okay, so what’s the deal with rational numbers? Simply put, a rational number is any number you can write as a fraction – think p/q, where p and q are just regular ol’ integers (and q can’t be zero, because, you know, math). The cool thing about rational numbers is that when you turn them into decimals, they either stop eventually (like 0.75, which is just 3/4) or they start repeating the same pattern over and over (like 0.20454545…, where “45” goes on forever). So, integers, fractions, decimals that end, decimals that repeat – all part of the rational club.

Now, irrational numbers? They’re the rebels. These are real numbers that you just can’t write as a simple fraction. And their decimal expansions? Forget about it! They go on forever, with no repeating pattern in sight. Pi (π) is probably the most famous example, but the square root of 2 (√2) is another classic. They just keep going and going…

So, What About 0.101100101010?

Alright, let’s get back to our number: 0.101100101010. The big question is, does it stop, or does it repeat? Take a good look.

Here’s the thing: it stops. It ends nice and neatly after that last ‘0’. That’s a pretty big clue.

The Verdict: Why It’s Rational

Because 0.101100101010 ends, it’s actually a rational number. We can even prove it by turning it into a fraction. Just count the number of digits after the decimal – there are 12. That means we can write it as:

See? Both the top number (101100101010) and the bottom number (1000000000000) are integers. So, we’ve successfully written 0.101100101010 as a fraction, which means it has to be rational.

The Bottom Line

So, there you have it: 0.101100101010 is definitely not irrational. It’s a good ol’ rational number because it ends, and we can write it as a fraction. Understanding this difference is key to unlocking more mathematical secrets down the road. It’s all connected!