# Can a Real number be both rational and irrational ?

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### 13 Answers

- RaymondLv 74 months ago
No. At least, not in the same base.

A "rational" number is one that can be expressed as a ratio of two integers. For example 2/3, 19/7, or even 4 (which is the same as 4/1, 8/2, 12/3, etc.)

If a number CAN be expressed as a rational number (even though it is shown differently), then it is rational.

The number 2.857142857142857142857142..... (forever) is a rational number ( = 20/7)

Any number that CANNOT be expressed as a ratio is "irrational" (meaning, not rational).

If you start mixing bases, then the partition of rational and irrational stops making sense. If you create a situation where a number could be both, then it is neither (it means that the partition of rational vs irrational cannot exist in that situation).

- KrishnamurthyLv 74 months ago
A number is rational if we can write it as a fraction where the top number of the fraction and bottom number are both whole numbers. ... Alternatively, an irrational number is any number that is not rational. It is a number that cannot be written as a ratio of two integers (or cannot be expressed as a fraction).

- ZirpLv 74 months ago
revised answer because I mixed up irrational with imaginary. (hence the 4 thumbs down).

No, because irrational vs rational is a way to categorise, just like odd vs even is. Numbers are called irrational whenever they have a linear value but cannot be expressed as one integer divided by another integer.

the number e is irrational, as is pi and many roots and logaritms

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- JOHNLv 74 months ago
A real number cannot be both rational and irrational because a rational real number results from one integer divided by another, whereas an irrational real number, though it exists somewhere on the number line, is never the result of one integer divided by another. And that such numbers do exist, consider a right triangle with sides enclosing the right angle each of length 1. What then is the length of the hypotenuse? The fact is that there are no integers m and n such that m/n exactly measures the hypotenuse. We can however find rational numbers which approximate it closely, in fact as closely as we wish. But the exact length is an irrational number.

- PuzzlingLv 74 months ago
No they are mutually exclusive.

A rational number is one that *can* be represented as the ratio of two integers.

An irrational number is one that *can't* be represented as the ratio of two integers.

- ToddLv 74 months ago
A real number is anything not imaginary. So, anything including "I" is not real. Everything else is. Now, in many computer languages, you have to make a distinction between integers and floating point numbers on purpose, but they're both real. Irrational just means the number can't be formed by a fraction of integers. If you mean both at the same time than no, but both are real.

- MorningfoxLv 74 months ago
No, it can't be both. The defintions do not overlap. "IR"-rational = not rational.

No, Todd, you are wrong

--Anything including "i" is not real.

//

e^(πi) =

e to the power "pi times i" //

is real, is rational, is an integer, is a unit that is its own inverse...