# Irrational Numbers: Infinite Numerical Values Explained

• icystrike
In summary, irrational numbers are numbers that cannot be written as fractions and have an infinite number of digits after the decimal point. However, even some rational numbers can have an infinite number of digits after the decimal point, such as 1/7 or 1/3. This depends on the base of the number system used. And while every irrational number has an infinite non-repeating decimal expansion, rational numbers can have either a finite or infinite repeating decimal expansion. The representation of numbers as decimals is based on conventions and does not necessarily reflect the true nature of the numbers themselves.
icystrike
Irrational Numbers are contained by infinite numerical values?

huh?!i didn't get what u mean:D

meaning if we would to write a irrational number out , we need a infinite number of digits?

icystrike said:
meaning if we would to write a irrational number out , we need a infinite number of digits?

Do you care? If we write out 1/7 we would require an infinite number of digits.

Depends on the base of the number system used. 1/7 is 0.17, 1/3 is 0.13, both require infinite number of digits if they are to be written base 10.

Edit: do you hate it when you make an idiot out of yourself just because you think in your first language when you should in English? I do. Irrational as it sounds, I was all the time thinking about rational numbers.

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An irrational number cannot be written as a fraction. (A fraction is a "ratio," so it is considered to be "rational.") Therefore, it has an infinite number of digits after the decimal point. (If there was a finite number of digits after the decimal point, it could be written as a fraction and would therefore NOT be irrational).

On the other hand, just because there are an infinite number of digits following the decimal point, doesn't mean that the the value is irrational. (0.111111111... can be written as 1/9, so it is rational, whereas $$\pi, \: e, \: and \: \sqrt{2}$$ are all examples of irrational numbers).

Borek said:
Depends on the base of the number system used. 1/7 is 0.17, 1/3 is 0.13, both require infinite number of digits if they are to be written base 10.
1/7 can also be written as 0.06666666...7, and 1/3 can be written as 0.02222222...3, requiring an infinite number of digits in these bases.

Every irrational number has an infinite non-repeating decimal expansion. Every rational number has either a finite decimal expansion, or an infinite repeating decimal expansion. I.e. for every rational number with an infinite decimal expansion, there is a repetion in the expansion, e.g. the above example 1/9=0.1111..., or 5/12=0.416666...

icystrike said:
meaning if we would to write a irrational number out , we need a infinite number of digits?
I can write $\sqrt{2}$ with two symbols: 2 and $\sqrt{\ }$.

The decimal expansion of $\sqrt{2}$ has infinitely many digits, though. And so does the decimal expansion of every irrational number, most rational numbers, and even every integer. (don't forget about the infinitely many zeros!)

Hurkyl said:
don't forget about the infinitely many zeros!

That's what I was thinking. We can write "2" without all the zeroes (2.000000...) because by convention we leave them off. What if by convention we left off .4142135623731...? Then things would be different (we'd write sqrt(2) as "1"). The point is, representations of numbers tell you about conventions, not so much about the numbers themselves.

maybe you'd be interested in http://www.dpmms.cam.ac.uk/~wtg10/decimals.html" article on the topic of thinking about numbers as infinite decimals

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hey qntty ! thanks for your help. its greatly appreciated!

Several years ago, a poster asked how to prove that a rational number could be written as a fraction! My first reaction was that that is the definition of "rational number" and could not be proven.

Then I realized that he had been taught "a rational number can be written as a terminating or repeating decimal" as the definition of rational number and now wanted to prove that a number satisfies that definition if and only if it can be written as a fraction, the opposite to the way it is normally done.

## 1. What are irrational numbers?

Irrational numbers are numbers that cannot be expressed as a simple fraction or ratio of two integers. They are infinite decimal values that do not repeat or terminate.

## 2. How are irrational numbers different from rational numbers?

Rational numbers can be expressed as a simple fraction or ratio, while irrational numbers cannot. Rational numbers also have a finite number of decimal places, while irrational numbers have an infinite number of decimal places.

## 3. What is an example of an irrational number?

One of the most famous examples of an irrational number is π (pi). Other examples include √2 (the square root of 2), √3 (the square root of 3), and e (Euler's number).

## 4. How are irrational numbers used in real life?

Irrational numbers are used in many real-life applications, such as in geometry, physics, and engineering. For example, the value of π is used in calculating the circumference and area of a circle.

## 5. Can irrational numbers be approximated?

Yes, irrational numbers can be approximated by rounding off the decimal values. However, this will result in a less precise value, as the decimal places of irrational numbers are infinite.

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