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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
How do you determine whether a rational number has a terminating decimal expansion?
When the rational number is expressed as a ratio in its simplest form, if the only factors of the denominator are 2 and 5, then the number has a terminating representation. Otherwise it has an infinitely recurring sequence.
An irrational number is a real number that cannot be expressed as a ratio of two integers.
In decimal terms, it is a real number that has either a terminating decimal or an infinitely recurring decimal. This would apply whatever the integer base, such as binary, octal, hexadecimal.
Is -127 a rational or irrational number?
It's rational ... it's the ratio of -127 to ' 1 '.
Here's a tip that'll save you lot of future wonder and uncertainty:
Any number that you can completely write down on paper is rational.
73 is a whole number (an integer).
As all integers are rational, 73 is rational.
yes. it has a definite end to it; that is, it does not continue infinitely.
Can an irrational number be a repeating decimal?
No. Any fraction can be written as a repeating decimal - and vice versa. For example, take the repeating decimal (which I call "x"):
x = 0.4123123123...
Now multiply that by 1000:
1000x = 412.3123123123...
Subtract the first quation from the second, and you get:
999x = 411.9
Solving for x:
9990x = 4119
x = 4119/9990
This is a fraction of whole numbers, therefore, a rational number.
-5.16 is rational or irrational?
It is rational because it can be expressed as -516/100.
An irrational number cannot be expressed as such. For example , PI is irrational.
How is the sum of a rational and irrational number irrational?
This can easily be proved by contradiction. Without loss of generality, I will take specific numbers as an example. The proof can easily be extended to any rational + irrational number.
Assumption: 1 plus the square root of 2 is rational. (It is a well-known fact that the square root of 2 is irrational. No need to prove it here; you can use any other irrational number will do.)
This rational sum can be written as p / q, where "p" and "q" are whole numbers (this is basically the definition of a "rational number").
Then, the square root of 2, which is equal to the sum minus 1, is:
p / q - 1
= p / q - q / q
= (p - q) / q
Since the difference of two whole numbers is a whole number, this makes the square root of 2 rational, which doesn't make sense.
What is an irrational number between 5 and 7?
4*sqrt(2)
Rational multiples of irrational numbers are irrational.
sqrt(2) is about 1.414, and 5/4 = 1.25 < 1.414... < 1.75 = 7/4 so
4*sqrt(2) is between 5 and 7, and is irrational.
Is 2 to the 3 power irrational number?
No because 2 to the power of 3 is 8 which is a rational number
