How is it that we cannot measure irrational numbers.?

Your "size of the universe" argument measn you can get a close as you like to the "exact" size of an irrational number, but it also means that however close you get, you can always get a bit closer.

Euclid proved there is no fractinos that exactly represents the square root of 2, about 2500 years ago.

Suppose 2 = a^2/b^2, where a and b are integers, and we have cancelled out any common factors of a and b.

a^2 = 2b^2. Therefore a^2 is an even number. Therefore a is an even number.

Let a = 2c

4c^2 = 2b^2

2c^2 = b^2

Therefore b^2 is an even number. Therefore b is an even number.

So a and b are both even, but we said they don't have any common factors.

That is impossible because two even numbers have a common factor of 2.

So a and b don't exist.

Don't worry if you find this stuff hard. A few famous mathematicians have literally gone mad trying to understand what irrational numbers are (for example Cantor, at the end of the 19th century).

the ancient greeks though that might be true. they thought that all you might have to do to "measure" any length precisely, was to pick a large enough scale, and a (perhaps) very small unit of measurement. that way, you could exactly measure the diagonal of a square in these (very small) units.

of course, if you could actually do that, then the length of the diagonal of the square would be a multiple of our small unit (we'll call it U). so, if the length of our diagonal was D, we have:

D = kU, for some (possibly HUGE) integer k.

now if S is the side of square with length 1, we should also be able to measure it in units as well:

S = mU. again, m is a (possibly VERY big) integer. this would mean:

√2 = D/S = kU/mU = k/m*(U/U) = k/m*1 = k/m.

so let's square both sides:

2 = k²/m², or equivalently, 2m² = k²

now for every factor of 2 that k has, k² has 2 of them. so the number of factors of 2 on the right side is even. but m² also has an even number of factors of 2, so 2m² has an odd number of factors of 2. this is very disturbing, there isn't any integer which is even AND odd.

so it doesn't matter HOW big your scaling factor is, you can't find a unit U, that will work out exactly for both the edge AND the diagonal.

yeah, i know, the greeks didn't believe it, either, and when this was discovered, they were so upset, they tossed the guy who proved it into the ocean.

π is even worse. at least given a unit distance U, can draw a line of length U√2, by making a square with sides of length U, and then drawing the diagonal. but given a line of length U, you CAN'T draw a circle with area equal to U², at least not with just a straight-edge and a compass. π is not only IRRATIONAL, it is also UNCONSTRUCTABLE (these numbers are called transcendental). this really does seem irrational because nature seems to be able to make circles of any size without any problem whatsoever. believe it or not, the problem of "squaring the circle" was one that a substantial amount of money was offered for, at one time, and it has only been around 120 years or so since it was proved to be impossible (the proof is VERY complicated).

it turns out that the notion of measurement is very subtle. when we look closely at how we arrive at formulas for distance, one finds the pythagorean theorem hidden in them:

A² + B² = C² (think of A as east-west, and B as north-south).

this means C = √(A² + B²), so we are back to square roots of things (our pesky little √2 comes up when A = B = 1). in other words, to measure distances, we have to be able to measure square roots, and rational numbers just don't quite give us enough numbers to measure with...they must have "holes" in-between them. it turns out they have a LOT of holes, there are many more irrational numbers than there are rational numbers. in fact, if you think of numbers like this:

x = a.bcdefghijk..... where "." is a decimal point, it isn't hard to show that every rational number either terminates after the decimal point, or repeats in a cycle. but obviously, you can think think of a LOT more "random" patterns after the decimal point that you can these fairly simple patterns. there are so many of these "random" patterns, you can't even list them all, or even catalogue them. fortunately for us, every irrational number has a rational number nearby, or else our pocket calculators would be useless. but it is important to remember (especially for people who work with computers that make many calculations) that these rational numbers are just approximations, that there is a bit of error involved, which can get large if you don't keep an eye on it.

your question is right on the verge of opening up the way that infinite things don't act like how we expect them to, when we are used to finite things. infinity is not like a number, only bigger, it is an odd concept indeed.

The universe is infinite. There isn't a way to "size something up to the size of the universe". The fact that some numbers don't work in a rational way is why they are simply called irrational. An irrational number isn't on realistic enough terms for us to call it measurable. Pi for instance. You can't make something exactly "Pi" inches long, because Pi is infinite. You can come close by saying that it's 22/7 inches long, but you can't get the fact (the irrational fact) of Pi into the measurement.

You just can't.

The formal definition of an irrational selection is one that may not be in a position to be expressed as p/q the place p and q are integers, q no longer 0. those numbers present themselves in each and every single place. working example in case you have a wonderful attitude triangle, 2 factors of length a million, the size of the hypotenuse won't be in a position to be expressed as a decimal selection. it incredibly is approximately 2.414 however the coolest length demands an limitless form of digits. the extremely selection is the sq. root of two, and is represented as ?2. If that triangle has lengths a million and a pair of incredibly of a million and a million, then the size of the hypotenuse is ?5, yet another irrational selection. the area of a circle with radius length of a million is yet another irrational selection that's represented as ? The length of the circumference of that circle is two*?, yet another irrational selection. So the objective of those numbers is to correctly convey the length of the parts we would desire to comprehend approximately. to place ?2 on the selection line, from a pragmatic point of view it incredibly is placed at a million.414. For the coolest region, draw a wonderful attitude triangle with the hypotenuse on the selection line, the different lengths are a million, one vertex at 0. the element ?2 is the non 0 vertex on the selection line. The traces could be 0 width. when you consider that irrational numbers can't be expressed by using a finite form of decimal digits, counting them as you have indicated isn't attainable. between 0.01 and 0.02 there are an limitless form of irrational numbers. for example evaluate the selection 0.01 + ?2/one thousand. that's approximately 0.01141, that's between 0.01 and 0.02. yet another irrational selection is 0.01 + ?2/ten thousand, and 0.01 + ?2/one hundred thousand, upload infinitum. desire this helps.