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Irrational Numbers
An irrational number is a number that can't be expressed by a fraction having integers in both its numerator and denominator. While their existence was once kept secret from the public for philosophical reasons, they are now well accepted, yet still surprisingly hard to prove on an individual basis. Please post all questions about irrational numbers, including the famous examples of π, e, and √2, into this category.
3,962 Questions
Are all non-terminating decimals irrational?
0.3333... non-terminating = 1/3 (rational)
0.1666... non-terminating = 1/6 (rational)
0.0666... non-terminating = 1/15 (rational)
0.1111... non-terminating = 1/9 (rational)
0.3636... non-terminating = 4/11 (rational)
0.428571428... non-terminating = 3/7 (rational)
I guess not .
How do you prove that the square root of 7 is an irrational number?
The proof is by the method of reductio ad absurdum.
We start by assuming that sqrt(7) is rational.
That means that it can be expressed in the form p/q where p and q are co-prime integers.
Thus sqrt(7) = p/q.
This can be simplified to 7*q^2 = p^2
Now 7 divides the left hand side (LHS) so it must divide the right hand side (RHS).
That is, 7 must divide p^2 and since 7 is a prime, 7 must divide p.
That is p = 7*r for some integer r.
Then substituting for p gives,
7*q^2 = (7*r)^2 = 49*r^2
Dividing both sides by 7 gives q^2 = 7*r^2.
But now 7 divides the RHS so it must divide the LHS.
That is, 7 must divide q^2 and since 7 is a prime, 7 must divide q.
But then we have 7 dividing p as well as q which contradict the requirement that p and q are co-prime.
The contradiction implies that sqrt(7) cannot be rational.
Are there numbers that are rational and irrational?
Irrational numbers are have decimal points that never stop. Like pi. They cannot be expressed as a fraction of two integers.
Rational numbers, no matter how ridiculous and long, can be expressed in a fraction of two integers. If they are decimals, they will stop at some point. like 0.47
They can't be both. That's a paradox. Unless your math teacher says so because he's probably smarter than me. :)
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For rational numbers, think of the word ratio. A rational number can be expressed as the ratio (or fraction) of 2 integers. Irrational means not rational.
Any whole number is RATIONAL, since you can write it as a fraction with a denominator 1. In this case, 22/1.
How can you deal with irrational numbers by expressing them in decimals?
You can deal with them approximately.
What does irrational answers mean?
In mathematics it means numbers which cannot be represented by ratios of two integers.
How are decimal that are rational number described?
A "rational number" is one that can be written as a fraction, with integers in the numerator and the denominator. When written as a decimal number, you'll get either a terminating decimal (e.g., 8 / 5 = 1.6), or a periodic decimal - one that repeats eventually, over and over again (e.g., 1 / 7 = 0.142857 142857 142857 ...).
Is -77.95 a rational or irrational number?
It is rational. -77.95 can be expressed as the ratio of two integers, so -7795/100 or -1559/20 are both equivalent fractions to -77.95
In general, any decimal which truncates after a finite number of digits (such as two digits in this case) or repeats (but with a pattern, such as 0.33333.... or 0.121212...) can be expressed as a fraction, so it's rational.
If the digits do not have a repeating pattern, then the number is irrational.
Can you reduce an irrational number?
An irrational fraction can be reduced.
For example, sqrt(20)/6 = sqrt(5)/3
What can you formulate from the real numbers?
This question is kind of vague.
Many of your everyday situations use the rational numbers (fractions and integers), which are a subset of real numbers.
If you do anything with angles (like surveying, or navigation) then irrational (which are still real) come into play.
There are situations when it's necessary to consider complex (a combination of real and imaginary) numbers. When dealing with waves or periodic motion, imaginary numbers become an important thing to consider.
Is sum of two irrationals is irrational give example?
It can be but need not be.
sqrt(2) + sqrt(2) = 2sqrt(2)
Is 2.2020020002 an irrational number?
This is an irrational number because of it's decimal point... Rational numbers are whole numbers that are divisible by 2 and do not leave a remainder or form a decimal fraction.
Is 0.3233233323333 an irrational number?
It is not an irrational number, and can be represented by the fraction:
4 669 723 / 14 442 889
