Irrational Numbers Proofs

12 Questions

Why is the square root of 2 considered an irrational number?

Its decimal representation never ends or repeats in a pattern.

What assumption leads to the proof of irrationality for the square root of 2?

(\sqrt{2}=x+\frac{y}{z})

How does the proof regarding the square root of 2 use parity to establish its irrationality?

By ensuring both sides of an equation have the same parity.

Which number is defined as the limit of ((1+\frac{1}{n})^n) as (n) approaches infinity?

(e)

What kind of numbers cannot be expressed as the ratio of two integers?

Irrational numbers

In the context of irrational numbers, what does it mean for a decimal representation to be non-repeating?

It has a sequence that never ends.

Based on the proof provided, what conclusion can we draw about the number 'e'?

e is irrational

If we assume that π can be expressed as a ratio of two integers, what conclusion does the proof lead us to?

π is irrational

What method was used to prove the irrationality of 'e' in the text?

Assuming e can be expressed as a ratio of two integers and deriving a contradiction

Why does assuming that π is rational lead to a contradiction in the given proof?

It results in an irrational number equaling 2

What role do irrational numbers play in mathematics, based on the text?

They provide insights into the nature of numbers

Why did the proof show that π must be irrational?

Because π squared equals 2

Study Notes

Irrational Numbers

Irrational numbers are real numbers that cannot be expressed as the ratio of two integers. They are neither rational nor whole numbers; their decimal representations never end, repeat in a pattern, or terminate. Some well-known examples of irreational numbers include (\sqrt{2}), (\pi), (e) and their multiples. Here are some proofs of irrationality related to these numbers:

Sqrt(2) is Irrational

One of the most famous proofs of irrationality is the one concerning the square root of 2. It is based on the assumption that (\sqrt{2}=x+\frac{y}{z}), where (x), (y), and (z) are integers and (z>0). This leads to the conclusion that (\sqrt{2}=\frac{2x+y}{z}). Since both sides of this equation must have the same parity (either odd or even), we can conclude that either (\frac{y}{z}) or (\frac{2x+y}{z}) must have an odd denominator.

However, both the numerators and denominators are integers, so they can't both have an odd denominator. This contradiction shows that our assumption was wrong, and therefore (\sqrt{2}) cannot be expressed as the ratio of two integers. Thus, (\sqrt{2}) is irrational.

The Irrationality of e

Another proof of irrationality involves the number (e), which is the base of the natural logarithm. It is defined as the limit of ((1+\frac{1}{n})^n) as (n) approaches infinity. To show that (e) is irrational, we can use the fact that (\sqrt{2}) is irrational.

Assume that (e) is rational, which means it can be expressed as the ratio of two integers, say (e=p/q). Then, we can write:

[ \begin{aligned} e^2 &= \left(p/q\right)^2 \ &= 1/(q^2) \end{aligned} ]

Since (e) is irrational, we know that (\sqrt{2}) is irrational, and hence cannot be expressed as the ratio of two integers. This contradiction shows that our assumption was wrong, and therefore (e) is also irrational.

The Irrationality of Pi

The irrationality of (\pi) can be proven using the fact that (\pi) is the circumference of a unit circle. To show that (\pi) is irrational, we make the assumption that (\pi=p/q), where (p) and (q) are integers.

Squaring both sides, we get:

[ \begin{aligned} \pi^2 &= p^2/q^2 \ &= 2 \end{aligned} ]

This is a contradiction, as it implies that (\pi) is a rational number (since it is a square root of 2, which is a rational number). Thus, our assumption that (\pi) is rational must be false, and therefore (\pi) is irrational.

In conclusion, irrational numbers play an important role in mathematics, and the proofs of their irrationality provide insights into the nature of these numbers.