Irrational Numbers

Welcome class! Today, we are going to discuss irrational numbers. To start off, can someone remind me what rational numbers are?

Rational numbers can be expressed as a fraction p/q, where p and q are integers.

That's correct! Now, what do you think irrational numbers are?

Are they numbers that can’t be written as fractions?

Exactly! Irrational numbers cannot be expressed in the form p/q. Instead, they have decimal expansions that are non-terminating and non-repeating.

Can you give an example of an irrational number?

Sure! A classic example is $\sqrt{2}$. Just like in our previous sessions, remember that every time we encounter a number we can't write as a fraction, it falls under the realm of irrationals.

How do we locate these on the number line?

Great question! We can locate $\sqrt{2}$ by using geometric methods. Imagine drawing a square with side lengths of 1; the diagonal represents $\sqrt{2}$. This helps visualize where irrational numbers fit on the number line!

So, key takeaway: irrational numbers are real numbers but cannot be expressed as simple fractions. Remember this as we continue our lesson.

Let’s delve into the history a bit. Who can tell me who were among the first mathematicians to discover irrational numbers?

Was it Pythagoras and his followers?

Yes! The Pythagoreans discovered that the square root of 2 couldn’t be expressed as a ratio of integers, leading them to conclude that such numbers are irrational. This was quite controversial for them.

Did any specific philosopher face consequences for this discovery?

Indeed, myths suggest Hippasus of Metapontum met a violent fate for revealing that $\sqrt{2}$ was irrational. This indicates the profound impact irrational numbers had on ancient mathematics.

Where does π fit into this discussion?

Good question! Like $\sqrt{2}$, π has been proven irrational, and its implications stretch far into mathematics, particularly geometry. We still use approximations for practical calculations, but its essence cannot be reduced to a fraction.

Remember, the historical journey of numbers enriches our understanding of their place in mathematics.

Let’s look at some practical examples now. Who can give me an example of a number we discussed that is considered irrational?

I think $\sqrt{3}$ is another example, like $\sqrt{3}$.

Well done! $\sqrt{3}$ also cannot be expressed as a simple fraction. Now, let’s see how we can find irrational numbers in between two rational numbers.

How would we do that?

"For instance, if we want an irrational number between 1 and 2, we can use the number 1.5, but to get an irrational, we could consider something like 1.414213...