Two Questions About Irrational Numbers

Factual Questions
1

As I said on another message board, I have a hard time understanding how irrational numbers never repeat. Nonterminating, repeating numbers I can understand. ⅓ is .3333333333333333333333… I understand that well. You go to the quadrillionth place and it’s still three.

But what about pi? How can it never repeat? That somehow seems impossible or at least hard to wrap your mind around to me. Any kernel of wisdom someone has to offer would be helpful.

And lastly, what do we need all those non repeating numbers for?

All I know personally, from high school now, was significant figures (—correct name?). The more precise your instruments, the more accuracy is involved, right? Also, while in grade school now, I recall reading some place that the bigger the circle, the more values of pi apply. Is that correct?

But what about trigonometric values? The Sine of 30° is ½. Okay. But the Sine of 45° is .707106781… What are all those extra numbers for? I hope you can see my confusion.

Thank you in advance for your help and kindly replies :slight_smile:

2

Try something simpler…

Consider the number 0.12345678910111213…

How could it possibly repeat?

3

As for why we need all those digits: we don’t need all of them for any practical purposes. But that doesn’t change the truth that they exist.

4

They’re for the converting degrees to the ‘more natural’ radians.

5

Great example!

Another approach: Think of the number line. Between the numbers 3 and 4 there has to be an infinite number of decimal numbers, they have to cover all possible sequence of decimals, how could it not be that some of them never repeat?

Take the number you can imagine that is closest to never repeating. Now change a single digit in that number. Now it no longer repeats.

6

We don’t. We just think it is really neat that we can calculate them. That is, we do need more digits than you’d work with in school back in the days of sine tables, but we do know a lot more digits than we could every possibly need even if we do a practical project with absolutely outrageous precision.

7

You will note, many numbers can be calculated that are irrational. Any given rational number is computable (just write it down…), but the limit of a sequence of rational numbers is not necessarily computable.

8

What will happen when @Jim_B encounters imaginary numbers? :ghost:

9

Note that, just as the set of irrational numbers is far larger than the set of rational numbers, so too is the set of uncomputable numbers, or even the set of undefinable numbers. Ultimately, the entire set of all definable numbers is still no larger than the set of integers.

10

@kayaker I know that you’re joking. But you actually bring up a good point.

Because imaginary numbers are based on the square root of -1, they don’t exist.

Yet in many ways they are the cornerstone of math and science. For example, I think aeronautical science is based on imaginary numbers.

How can bona fide science and math be based on something that doesn’t exist :confused: ? :slight_smile:

11

You can do engineering math without imaginary numbers - it’s just really inconvenient, just like it would be inconvenient to do accounting without negative numbers. In some sense no numbers really exist - but they make a lot of practical things more easy to do, so we use them.

12

As others have said, from a physical stand point they aren’t “for” anything. Beyond around 60 digits of pi you have enough precision to estimate the radius of the visable universe to planc length. You can’t even theoretically get more precise than that.

However we talk about them because they are convenient to talk about mathematically even if they require us to move a bit beyond the grade school arithmetic notion of number that is the rationals. For example we want to talk about a number which when squared equals 2. If we are stuck in rational numbers this number doesn’t exist (if you want a proof of this, ask and I can give one).

We can get really really damn close, for example (1.414214^2)=1.999999237796, but if we use (1.414215)^2=2.000004066225 we’ve gone too far. In fact we can get as close as you want and then find rational numbers that are closer. We know exactly where it fits between the rationals on the number line. For any given rational number we can tell you when you square it its larger or smaller than 2. We are so close to the sqrt(2) wecan almost taste it, but we can’t actually get there with the rationals. So what we do is mark that spot on the number with a pushpin labeled Sqrt(2) and call it a day. The decimal expansion is just a roadmap that will get you closer and closer to this elusive location on the number line even if it won’t ever let you reach your destination.

13

Indeed it’s not hard to prove that the square root of any positive integer n is either another integer (i.e. n is a perfect square) or is irrational.

14

What? Of course they exist. i is literally just a left turn, and saying “i^2 = -1” just means that if you make two left turns in a row, you’re reversing your direction.

I don’t know why everyone always insists on making out imaginary numbers to be some sort of weird, difficult, esoteric thing. They’re perfectly ordinary and mundane.

15

Well you have to admit that getting from real to complex numbers is a bit more of a stretch conceptually than getting from rationals to reals. There is no way to even get something close to resembling i if you are settled in the reals, and you need a whole 'nother dimension to express them. Even if your settled on the idea that “ i^2=−1 ” that you have to discuss what the difference is between i and -i. I sometimes prefer to ignore the whole sqrt(-1) thing and just think of complex numbers more as ordered pairs of reals or special 2x2 matrices.

PS to Chronos: how the heck do you get those nice math fonts. I thought I could learn your secret by quoting your post but they just show up as normal characters. :slightly_frowning_face:

16

Admittedly, it’s not obvious why pi should be irrational. It wasn’t until aroun 1760 that anyone was able to prove that pi is irrational, despite knowing about pi and probably suspecting that it’s irrational for many hundreds of years.

By contrast, the ancient Greeks knew (and could prove) that the square root of 2 is irrational.

17

It’s just because someone at some time called them “imaginary” numbers. And so the confusion started.

18

I have a different theory about what the ancient Greeks knew, or thought they knew, about irrational numbers.

Prior to Pythagoras, they thought of all fractions as being ratios of integers – that is, what we today call the rational numbers. They had no concept that any others existed. It made sense: They understood that, between any two rational numbers, you could find infinitely many more. Drawn on a number line, this appeared to fill the entire line. (That is, the property of rationals that we now call density.)

Pythagoras understood, from the rules about sides of right triangles, that he could construct a triangle whose hypotenuse was a length that must be square root of two. He could draw this length on a number line, showing that there must be an exact point on the line for this number.

But he also discovered that no rational number could represent this length.

Think for a moment how that must have looked back in the day: Rational numbers were all they knew. But Pythagoras found a length on the number line that was not any such number! This discovery upset everything they thought they knew about numbers, down to the foundations!

I don’t think Pythagoras came up with the idea of irrational numbers. Instead, I think they interpreted this to mean that there are magnitudes that are not numbers! That is, he discovered the property that we now call incompleteness of the (rational) numbers. It is said that he sacrificed a hundred cattle to the gods because of this.

This meant that they could no longer rely on the number system (meaning the rationals) to measure anything. But they could rely on a geometric figure, the number line, to measure anything. They assumed that the number line was complete, in a way that the number system (as they knew it) was not.

I’ve hypothesized that this is why the ancient Greeks came to focus their mathematical studies on geometry (see: Euclid), with the result that anything like algebra was not developed until centuries later.

19

Maybe. On the other hand, I think very few people actually “get” the real numbers. They aren’t as obviously weird as complex numbers, so the differences get less attention. But I suspect that when most people think of pi or whatever, they are really thinking of a finite approximation to it, which is just a rational. They aren’t really grokking the infinite nature.

It’s inline \LaTeX notation. Surround the formula with dollar signs, like $\LaTex$ or $i^2 = -1$.

20

More and more, I appreciate what Gauss thought about complex numbers.

He did not even want to call them “imaginary” because of the connotation people got from that.

Complex numbers are true numbers (not going to use ‘real’ because math ‘real’ has a different meaning from ‘real as in actual’).

And beyond the Champernowne’s number example above, there are other numbers like:

0.1010010001000010000010000001… (add one more zero each time)

It’s simple to construct numbers that do not ever repeat. The ‘why’ of it take up with Og or the universe.

Oh, and sin(45 deg) is sqrt(2)/2, which may make more sense. The square root of 2 itself does not repeat, so that pushes the decimal question up to that level. The Pythagoreans were with you about not liking this fact.