Is 3.62566 an Irrational Number?

In summary: However, you can count the digits of any single irrational number, it's just that there are infinite digits. It's like trying to count all of the numbers between 1 and 2. You can count 1.1, 1.2, 1.3... 1.99999999999... on and on, but you'll never get to 2. However, you can still count each individual number in that range. It's just that there are an infinite number of numbers in that range, so you'll be counting for an eternity. A countably infinite set is a set whose members can be put into one-to-one correspondence with
  • #1
adjacent
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An irrational number is any real number which cannot be expressed as the ratio of two real numbers.

Then is 3.62566 is also an irrational number?
I thought all irrational numbers are uncountable.

I am not sure that the above is an irrational number :confused:
 
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  • #2
Not quite right. An irrational number is any real number which cannot be expressed as the ratio of two integers. 3.62566 = 362566 / 100000 and so it is rational. Any decimal which terminates is therefore rational.
 
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  • #3
phyzguy said:
Not quite right. An irrational number is any real number which cannot be expressed as the ratio of two integers.
Oh, thank you for that.
phyzguy said:
3.62566 = 362566 / 100000 and so it is rational. Any decimal which terminates is therefore rational.
:shy:
I think I was relying on my stupid calculator too much
 
  • #4
But also, any decimal that has a repeating pattern is rational too.

[itex]0.333... = 0.\overline{3} = 1/3[/itex] (repeats every digit)

[itex]0.142857142857... = 0.\overline{142857} = 1/7[/itex] (repeats every 6 digits)

Generally, if we have a number between 0 and 1 which repeats after n digits, then it's of the form

[tex]0.\overline{a_1a_2a_3...a_n}[/tex]

So if we let

[tex]x=0.\overline{a_1a_2a_3...a_n}[/tex]

then

[tex]10^nx = a_1a_2a_3...a_n.\overline{a_1a_2a_3...a_n} = a_1a_2a_3...a_n + 0.\overline{a_1a_2a_3...a_n}[/tex]

and so

[tex]10^nx-x=a_1a_2a_3...a_n[/tex]

Solving for x,

[tex]x(10^n-1)=a_1a_2a_3...a_n[/tex]

[tex]x=\frac{a_1a_2a_3...a_n}{10^n-1}[/tex]

which is a ratio of 2 integers, hence x is a rational number.

You can use this procedure to easily find, for example, what 0.5454545... is in fractions.

Let [tex]x=0.\overline{54}[/tex]

[tex]100x=54.\overline{54}[/tex]

[tex]100x-x = 54.\overline{54} - 0.\overline{54} = 54[/tex]

[tex]x=\frac{54}{99}[/tex]
 
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  • #5
adjacent said:
I thought all irrational numbers are uncountable.

There are uncountably many irrational numbers. An irrational number itself cannot be uncountable, I have no idea what that would mean.
 
  • #6
micromass said:
An irrational number itself cannot be uncountable

Count ##\pi##. I am sure you can't
 
  • #7
adjacent said:
Count ##\pi##. I am sure you can't

Yes, I can: "Pi!"
That wasn't so difficult.

If you meant the digits of pi, these are still not uncountable, even if there is an infinite number of them.
 
  • #8
willem2 said:
Yes, I can: "Pi!"
That wasn't so difficult.
So funny. But I asked you to count it :p and ##\pi !## returns a math error
willem2 said:
If you meant the digits of pi, these are still not uncountable, even if there is an infinite number of them.

You will have to do that forever then.
 
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  • #9
adjacent said:
So funny. But I asked you to count it :p

Can you count 2.5 to show us what you mean? :confused:

adjacent said:
and ##\pi !## returns a math error

Because your calculator's factorial function is only defined for non-negative integers. Wolfram alpha gives 7.188... (an irrational).


adjacent said:
You will have to do that forever then.

There are a countably infinite amount of digits in any irrational or rational that doesn't terminate. The number of irrational numbers however is uncountably infinite.
 
  • #10
Mentallic said:
Can you count 2.5 to show us what you mean? :confused:
Two point five
Mentallic said:
Because your calculator's factorial function is only defined for non-negative integers. Wolfram alpha gives 7.188... (an irrational).
That's why my calculator is stupid. (Sometimes I forget to set to degrees and get the wrong answer in tests)

Mentallic said:
There are a countably infinite amount
That is so strange. There is a joke:
A mathematician and an engineer are sitting at a table drinking when a very beautiful woman walks in and sits down at the bar.

The mathematician sighs. "I'd like to talk to her, but first I have to cover half the distance between where we are and where she is, then half of the distance that remains, then half of that distance, and so on. The series is infinite. There'll always be some finite distance between us."

The engineer gets up and starts walking. "Ah, well, I figure I can get close enough for all practical purposes."
Hahah, if you can count a set of things infinite in number it means you do that for eternity?
 
  • #11
adjacent said:
Hahah, if you can count a set of things infinite in number it means you do that for eternity?

A countably infinite set is a set whose members can be put into one-to-one correspondence with the set of integers. That's clearly the case for the digits of ∏; just map the n'th digit to the integer n. On the other hand, the set of real numbers is not countably infinite; it can be proved that no such mapping can be constructed.

If you want an amusing diversion... You might try googling for "Hilbert's hotel"
 
  • #12
adjacent said:
Two point five

Ok then count wasn't the word you were looking for. Reciting the digits is what you want.

adjacent said:
That's why my calculator is stupid. (Sometimes I forget to set to degrees and get the wrong answer in tests)

It's not stupid, it just wasn't programmed to solve anything and everything that Mathematicians have ever discovered. The calculator wouldn't have enough memory for that anyway. If you need to find the factorial of a non-integer, then you're likely at a level where a calculator just isn't going to be enough to do everything for you. That's what computers and math programs like MATLAB were made for. Do you need to work with larger numbers than 10100? Then use a computer :smile:

Well then that's not the calculator's fault, is it :tongue:
Unless you're doing a test in physics or some other science that works solely in degrees, keep to radians and then can convert when needed (which should be rarely). Also, you want to work towards having an intuition about the magnitude of the numbers you would be expecting. If you are calculating something in radians but your calculator is unfortunately set to degrees, then you should be able to spot the fault when you get a value that is 2 orders of magnitude less than expected.
adjacent said:
Hahah, if you can count a set of things infinite in number it means you do that for eternity?
Countably infinite means that you would technically be counting for an eternity, yes. The difference however is that you can't find a way to count all of the irrational numbers (if you could count for eternity).
 
  • #13
adjacent said:
Count ##\pi##. I am sure you can't
There is one and only one number that qualifies as ##\pi##. So the answer to your question is one.

The integers are a subset of the rationals, which in turn are a subset of the algebraic numbers, which in turn are a subset of the computable numbers. Even though the computable numbers are a superset of the integers, both sets have the same cardinality. They are countable sets.

Note that ##\pi## and ##e## are computable numbers.

The reals are not countable. Most ("almost all" is the technical term) or the reals cannot be computed.
 
  • #14
adjacent said:
Count ##\pi##. I am sure you can't

Count ##1/3##, you can't do that either. It also has an infinite number of digits.
 
  • #15
Mentallic said:
Then use a computer :smile:

And spend some time understanding the limitations of the algorithms your computer is using to perform arithmetic... There have to be some limitations somewhere, because the computer can only manipulate a finite number of finite-length bit strings.

An entire generation has been raised in blissful ignorance of what life was like before we discovered fire, toolmaking, and IEEE floating-point arithmetic. (Not that IEEE arithmetic is perfect - far from it - but it is good enough that we can generally survive despite our complacency).
 
  • #16
There is some terminology confusion on adjacent's part. Just because a set is infinite doesn't mean it is uncountable. That I am a human and have a short lifespan is irrelevant to mathematical truth. Being countable or uncountable has nothing to do with what humans can do in their lifetimes, and everything to do with an existence of a one-to-one correspondence with the naturals. Think about it, if being an infinite set meant you were uncountable, then the naturals would be uncountable, but that is the very set upon which counting is based!


As for the π issue, unfortunately, people associate numbers so much with how they are written, rather than how they are defined, that the concept of a nonterminating, nonrepeating decimal number like π somehow becomes this mysterious entity that we can "never know exactly" in their minds.

Irrationals, rationals, etc are not defined by what they look like or how they can be written. This is merely a result of the mathematical definition. Numbers are not symbols on paper. Symbols on paper are a means of communicating the idea of a number. I actually had to convince someone that 4/2 was an integer recently.

An irrational number is not a number that does not terminate or repeat. Numbers do not terminate or repeat, such a statement is nonsense. An irrational number is a number that is not equal to any ratio of integers. As a result, when writing this number in decimal form, that notation will not allow for repetition or termination, but that is not a definition.
 
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  • #17
Ok everyone, thanks for helping me. I think I get it now :smile:
 
  • #18
Nugatory said:
And spend some time understanding the limitations of the algorithms your computer is using to perform arithmetic... There have to be some limitations somewhere, because the computer can only manipulate a finite number of finite-length bit strings.

An entire generation has been raised in blissful ignorance of what life was like before we discovered fire, toolmaking, and IEEE floating-point arithmetic. (Not that IEEE arithmetic is perfect - far from it - but it is good enough that we can generally survive despite our complacency).

Maybe they should start with coding in C without using imported packages, run into a few overflow and precision issues and a lack of functions, then they'll begin to appreciate calculators again.
 
  • #19
adjacent said:
So funny. But I asked you to count it :p and ##\pi !## returns a math error


You will have to do that forever then.

In mathematics, there are different "sizes" of infinity. There are infinitely many integers: 0, 1, 2, ... It never stops, so it's infinite. There are also infinitely many real numbers. But there is a sense in which there are more real numbers than there are integers. The sense is this:

Two sets [itex]A[/itex] and [itex]B[/itex] are said to be "the same size" (technically, the same cardinality) if you can set up a correspondence between the two sets, so that every element of [itex]A[/itex] is matched with exactly one element of [itex]B[/itex], and vice-verse (technically, a one-to-one mapping). For example, the sets

[itex]A = \{ cat, dog, pig \}[/itex]
[itex]B = \{ red, yellow, blue\}[/itex]

are the same size because they can be put into correspondence many different ways, but here's one: [itex]cat \leftrightarrow red,\ dog \leftrightarrow yellow,\ pig \leftrightarrow blue[/itex]

Infinite sets can be put into a one-to-one correspondence, also. For example, the set [itex]A = [/itex] the positive integers and the set [itex]B = [/itex] all integers:

[itex]1 \leftrightarrow 0[/itex]
[itex]2 \leftrightarrow -1[/itex]
[itex]3 \leftrightarrow +1[/itex]
[itex]4 \leftrightarrow -2[/itex]
[itex]5 \leftrightarrow +2[/itex]
etc.

You can also set up a one-to-one correspondence between the integers and the rationals. That's a little harder to describe, but it can be done.

Any set that can be put into a one-to-one correspondence with the positive integers is called a "countable" set.

Some sets are not countable. The easiest example is the set of reals. There is no way to set up a one-to-one correspondence between the positive integers and the reals.
 
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  • #20
I see, thank you so much :smile:
 

1. Is 3.62566 an Irrational Number?

No, 3.62566 is not an irrational number. It is a rational number because it can be expressed as a fraction, 3 62566/100000, where both the numerator and denominator are integers.

2. How can you determine if a number is irrational?

To determine if a number is irrational, it must first be written in decimal form. If the decimal is non-terminating (goes on forever) and non-repeating, then the number is irrational. If the decimal terminates or has a repeating pattern, then the number is rational.

3. Can a decimal number be both rational and irrational?

No, a decimal number cannot be both rational and irrational. A number is either rational or irrational, but not both. This is because a rational number can be expressed as a fraction, while an irrational number cannot.

4. What are some examples of irrational numbers?

Some examples of irrational numbers include pi (3.1415926...), the square root of 2 (√2 = 1.4142135...), and Euler's number (e = 2.7182818...).

5. How are irrational numbers used in the real world?

Irrational numbers are used in various fields of science and mathematics, such as physics, engineering, and statistics. They are used to accurately represent measurements and calculations that cannot be expressed as whole numbers or fractions. For example, pi is used in geometry to calculate the circumference and area of a circle.

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