My answer to a question in Quora, slightly edited:
There are many good answers in this thread; I am trying to give a an answer that is simpler than most of them, but has deep historic roots.
Irrational numbers had been introduced into mathematics because it was discovered, quite a long time ago, that measurement of magnitudes cannot be reduced to counting of units of measurements. As it was explained in this thread many times, it was a great discovery of Pythagoreans that there is no unit of length which can be used for simultaneous absolutely precise measurement of the side and diagonal of the square.
Magnitudes and numbers are two very different species and need to be treated with equal respect. A simplest example: temperature is a magnitude, but, strictly speaking, is not a number. Why? because we expect from numbers that they can be compared by size, added, and multiplied. We can can compare two temperatures (“water in jar A is warmer than water in jar B”), but addition is already a problem: if we pour water from both jars into jar C, resulting warmth of water depends on the amount of water in each of jars. And what about multiplication of two temperatures? Can you suggest a real life situation when it is needed and used?
Finally, there is one particular magnitude for which even units of measurement do not exist: strength of smell (or odour).
It is one of the misconceptions about mathematics: it is frequently claimed that all mathematical magnitudes can be measured by real numbers. There is a simple and striking counterexample: there are rigorous, in the most modern sense, axiomatisations of Euclidean geometry, where angles cannot be measured by distances. Remarkably, Euclid never claimed that angles and lengths were the same magnitude.
So, acceptance of irrational numbers is a form of mental hygiene: it is equivalent to acceptance of the sad fact of reality that there is no one universal unit (with subunits centi- milli-, micro- , nano-) of measurement for everything.
The most prolific and popular responder to mathematics questions on Quora, Alon Amit
There’s no physical significance to the product of two temperatures, but there’s a lot of significance to the product of temperature and mass, for example. This provides ample motivation for attaching numerical value to temperature.
What follows is my reply to him:
Yes, of course. But measurement of time is remarkably imprecise in comparison with measurement of time and distance. And specific heat depends on temperature. And actually no phenomenon of thermodynamic nature could be measured with absolute precision.
Unit of time, second, is now defined via counting:
Since 1967, the second has been defined as exactly 9,192,631,770 times the period of the radiation corresponding to the transition between the two hyperfine levels of the ground state of the caesium-133 atom.
Wikipedia further says:
A set of atomic clocks throughout the world keeps time by consensus: the clocks “vote” on the correct time, and all voting clocks are steered to agree with the consensus, which is called International Atomic Time (TAI).
As we can see, there is no escape from the number/magnitude and counting/measurement dichotomies, Historically, mathematicians of Ancient Greece had to handle geometric proportions and discovered that they cannot be expressed by rational numbers. In modern terminology (and in a very crude description) Eudoxes dealt with real numbers as equivalent classes of proportions. His approach was brilliantly revived very recently, by Schanuel who suggested a new construction of the field of real numbers as the factor ring of the ring of “almost additive” maps on the additive group of integers (that is, maps from integers to integers such that the difference
\(f(m + n) – f(m) -f(n)\)
takes only finitely many values), by the ideal of maps which take only finitely many values. The reverse interpretation is obvious: if \(t\) is a real number, we can associate with it a map from integers to integers defined as
\(f(n) = [tn]\),
where square brackets denote, as usual, the integer part of real number, and give a counting approximation to measurement.