# Irrational Numbers and Real Life I need 6 answers

Once again my professor asked us to ask 6 people the following question and see how they answer it so if you could respond and give an answer, I would really appreciate it. And if possible can you also tell me a little bit about your mathematics background? We are supposed to write up what people say along with a little bit of their background.

What is an example of a real life application that uses irrational numbers?

Char. Limit
Gold Member
Let's say you want to make a card in the shape of a circle. Let's say you give this card a width. Now let's say you want to put a border on the card. How long will the border need to be? It's going to involve pi, an irrational number.

The possibilities are endless.

EDIT: Since I just saw the "background" part, I'm a 17-year-old high school student taking AP Calculus in school and independently studying DEs and linear algebra.

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A quite simple example was described by Socrates to a slave boy, i.e. simpleton, where he examines squares of 1 unit sides. How long is the diagonal?

The purpose of this explanation was to show how anyone can understand these things!

Mentallic
Homework Helper
If you are to deposit a dollar at 100% interest rate per annum, then after a year the interest added to your original deposit becomes $2. If you however compound the interest each half year, so the interest added is 50% each time, then you'll end up having$2.25. Quarterly, the interest will be 25% each and you'll have about $2.44. Monthly, about$2.61.

If you keep compounding in shorter and shorter time intervals, eventually you'll tend towards an irrational sum of money after the first year, nearly $2.72, or$e.

I'm just out of high school, about to begin my major in mathematics and physics at university and I'm 18.

mathman
One difficulty in trying to interpret mathematical concepts in the real word is the fact that physical measurements are never exact, so that the question of rational or irrational is irrelevant.

I think the OP's question was about applications of the fact that certain real numbers are irrational, not the applications of real numbers which happen to be irrational. Yes, irrational numbers occur all the time, but how often do we ever use the fact that they are irrational? Would our lives be affected in any way if we still believed, as the ancient Greeks once did, that all real numbers were rational? No, I don't think so. Whether a real number can be written as a terminating decimal or not probably doesn't matter, and whether a real number can be written as a periodic decimal or not definitely doesn't matter.

However, the statement that, for example, the square root of two is irrational can be written as a statement of number theory, and some results in number theory do have certain applications in the field of cryptography. So maybe one day people will use their knowledge of numbers being irrational to secure your email account. But don't hold your breath.

I am a freshman in college double majoring in physics and math, and I have taken Calculus, Vector Calculus, Linear Algebra, and Real Analysis, and am currently taking Differential Equations.

From what I understand the Pythagoreans (the cult of number started by Pythagorus) held close the great secret of the irrationals. To them it was sacred but also frightening news about the imperfection of reality. So when one of them finally told someone and the secret was out that member of the cult was sentenced to execution. I remember something about the unfortunate mathematician cultist being thrown from a boat.

So I don't know what your professor is getting at, but be careful.

Oh, and I am a graduate student in mathematics.

Lugita: Whether a real number can be written as a terminating decimal or not probably doesn't matter, and whether a real number can be written as a periodic decimal or not definitely doesn't matter.

That's crazy! Some people like fractions, like 1/3.

Mark44
Mentor
Here's a real-world example that uses irrational numbers. A friend of mine who was a carpenter needed to install a round ceiling light fixture and wanted to know how large a square hole could he cut in the ceiling so that the hole didn't extend beyond the rim of the fixture. (It's easier to make cuts with straight edges then curved or circular cuts.)

If the diameter of the light fixture is 6", a square that is $\sqrt{18}$" (or about 4.24") on each side will just touch the circle, so anything smaller will be covered by the light fixture.

I taught mathematics in a very small high school in Washington state, US, for two years, and in a Seattle-area community college for 18 years.

mathman
Here's a real-world example that uses irrational numbers. A friend of mine who was a carpenter needed to install a round ceiling light fixture and wanted to know how large a square hole could he cut in the ceiling so that the hole didn't extend beyond the rim of the fixture. (It's easier to make cuts with straight edges then curved or circular cuts.)

If the diameter of the light fixture is 6", a square that is $\sqrt{18}$" (or about 4.24") on each side will just touch the circle, so anything smaller will be covered by the light fixture.

I taught mathematics in a very small high school in Washington state, US, for two years, and in a Seattle-area community college for 18 years.
You have just illustrated my point. The carpenter cannot measure √18", but he can measure 4.24" to some degree of approximation. Irrational or rational makes no sense in this context.

Char. Limit
Gold Member
Irrational and rational numbers make perfect sense. The only problem is that whenever a human measures something, there will always be an error of
measurement. However, an irrational number, defined as "cannot be written in a fraction" is, despite being as immeasurable as a rational number, a different concept. Just look at half-lives of first order reactions.

Half-life= ln(2)/k

ln(2) is much different from .693.

Let's say you want to make a card in the shape of a circle. Let's say you give this card a width. Now let's say you want to put a border on the card. How long will the border need to be? It's going to involve pi, an irrational number.

The possibilities are endless.

EDIT: Since I just saw the "background" part, I'm a 17-year-old high school student taking AP Calculus in school and independently studying DEs and linear algebra.

3.14 will do.

Irrational numbers are useful within mathematics only, but for that exact reason they are useful in the real world. They allow us to develop theories with useful concepts like derivatives, integrals, the various results of analytical geometry, the rules trigonometry etc. These theories are of course very useful in solving real world problems.

Lugita: Whether a real number can be written as a terminating decimal or not probably doesn't matter, and whether a real number can be written as a periodic decimal or not definitely doesn't matter.

That's crazy! Some people like fractions, like 1/3.
Then those people will approximate every real number as a fraction with a small denominator, whether that number happens to be a fraction with a very large denominator, or a nonrepeating decimal.

I can't really decide how to answer this question. The question has way too much room for interpretation. I'm not sure if they are ubiquitous, or if they are not useful at all. Depends on what you mean by use, and what you mean by real life application.

Well I guess that's my answer,
I am a senior physics major.

Mark44
Mentor
You have just illustrated my point. The carpenter cannot measure √18", but he can measure 4.24" to some degree of approximation. Irrational or rational makes no sense in this context.
But I think you might be missing mine. An exact measurement of √18" isn't important. My carpenter friend didn't remember how to do the calculation, which requires irrational numbers. Once I did the calculation for him, he knew that all he had to do was measure 4.24" or smaller and his cut wouldn't show under the light fixture.

jmlink:What is an example of a real life application that uses irrational numbers?

What is now being argued about is what, exactly, is the meaning of this question, "What is a real life application?"

Apparently, I take it, the answer is not expected to be, "None." However, it is a fact of the carpenter that all numbers we use are approximations. So, I assume the question indicates that knowledge of a number's status is understood, and the answer must bear on that. Thus, I answered that one possible answer is the diagonal of a square.

I might add that whole numbers often relate to Physics as well. As one author puts it "The revelance of integral quanties in Physics goes back at least 25 centuries and relates to Pythagoras in terms of a stretched sting and the theory of musical harmony.