Why do irrational numbers exist

Active Oldest Votes. Add a comment. David Joel Joel Can't believe I missed that. Integer: A real number which can be expressed without a fractional component. Irrational number: A number which cannot be expressed as a ratio of two integers. Peter Woolfitt Peter Woolfitt Yashbhatt 8 8 bronze badges. AmbretteOrrisey AmbretteOrrisey 3 3 silver badges 12 12 bronze badges. Sign up or log in Sign up using Google. Sign up using Facebook. Sign up using Email and Password.

Post as a guest Name. Email Required, but never shown. Upcoming Events. And ultimately, those circle-lookalikes you've seen probably were all made of atoms anyway, which means they could not have been a circle, as they have volume, while the circle is an infinitely thin line. One odd thing about the question is the apparent belief that the decimal notation of a number is somehow fundamental to its meaning, and so an unending decimal expansion is a sign that the number itself is somehow unending or imprecise.

I'm not exactly sure what this means, but I suspect it's not true. Anyway, this fixation on decimal expansions is unnecessary.

Here's how I choose to imagine the "true nature" of a real number: the essence of a real number is that it measures some continuous quantity — for example, each real number is the length of some idealized line segment.

In this way, the decimal expansion isn't the "true nature" of a number, but simply as the view of a number you get when you look at it through the lens of powers of ten. This becomes more obvious when you realise that you can write the same number in different bases, view it through many different lenses, and indeed in some cases you get very different looking results, e.

Now ask again what it means for some real number, the length of an idealized line segment, to be a rational number. Now the question becomes: why should it be rational? Why should that relationship ever hold exactly? Philosophically, you could consider an "irrational quantity" as the limit of a sequences of "rational quantities", so e. There is a legend about Kronecker saying integers were god-made, and the rest was a creation of mankind , but we leave that to the history of mathematics for the moment.

These relations therefore accommodate a variety of unit systems. The celerity of light, if constant, does not care about being measured in meter per second, or foot per lunar month. What matters, if laws are accurate enough, is to have sufficient precision. So, with "quadruple precision" you can almost work with "decimals", with respect to your floating point system.

An interesting question. It relates mathematical, i. Let us assume for the purpose of this discussion that there is a reality, and that it exists outside of our mind. Then the relation of the well-understood mental concept of irrational numbers and that reality depends on the properties of that reality which, in all reality, are probably not completely known.

I give three examples. If the underlying structure of space and matter were completely continuous i. The same goes for the International Prototype Kilogram and all other artifacts and natural phenomena. If reality were like Minecraft, and all physical properties were "blocky" albeit at an atomic scale , all properties would be rational : Everything is a multiple of block lengths, weights, times etc. Our reality is a bit like Minecraft, except that things get fuzzy if you look too close.

This means that no property of an object we want to measure can be measured exactly; on a fundamental level there is always a, well, uncertainty. This is not a deficiency in our way of measuring -- those errors would come on top --, but a property of our universe. Because exactness is limited one needs not be afraid of running out of decimal places.

Any interval produced by the amount of uncertainty has many rational numbers in them; so in a way you can probably get away without using irrational numbers in describing reality. But you could also get away without ever using any rational numbers, mind you.

Take a line segment of length 1. With compass and straight edge it is possible to construct a line segment starting from its endpoint that is perpendicular to it and also of length 1. Draw that line segment.

Irrational number pi is the ratio of circumference of a circle to its diameter or circumference of a circle of unit diameter. Hence many things can be comprehended better by irrational numbers. So, they do exist in some form in nature, though the a common person may not find it easy to comprehend. The fact is these numbers make understanding of many thing easy. In fact, even complex numbers, though were very difficult to comprehend even by mathematicians till 17th century, make easy to understand electromagnetic phenomena and flow of current through electronic circuits using resistances, inductance and capacitors.

Hence, though common person may find many things in mathematics as incomprehensible or difficult to understand, they do exist in some form and serve the purpose of understanding of nature. Why do irrational numbers exist? This is a BETA experience. You may opt-out by clicking here. More From Forbes. Nov 11, , pm EST. Nov 11, , am EST.