ریاضی
What Is a Rational Number?
A rational number is one you can write as a fraction of two integers. Put 0.75, 0.3 recurring, −6 and 7 through that test and all four pass. Here is how.
Rational number
A rational number is any number that can be written as a fraction a/b where a and b are integers and b is not zero.
- طلبہ کہاں پڑھتے ہیں
- Grade 8 mathematics, and again at GCSE and IGCSE whenever a non-calculator paper gives a list of numbers and asks which of them are rational.
مختصر جواب
A rational number is any number that can be written as a fraction of two integers, such as 3/4, −6/1 or 1/3. Every integer is rational, and so is every decimal that either stops or repeats. Numbers whose decimals run on for ever without repeating, such as π and √2, are not.
ایک مثال
0.75 = 3/4 0.333… = 1/3 −6 = −6/1 7 = 7/1
Four numbers that look like four different species, all rational, and each takes one line to prove. Producing the fraction is the whole of the answer: once a/b exists with both parts whole, the classification is settled and no further reasoning is needed.
One test, and it is a yes or no
Can this number be written as a fraction with a whole number on top and a whole number underneath? That is the entire definition. Not does it look like a fraction, and not is it a nice number — can such a fraction be produced.
Students tend to treat the question as a judgement about appearance, which is why 0.75 gets marked as not a fraction and −6 gets marked as not rational because it is negative. Both readings ignore the test. The word rational comes from ratio, not from reasonable, and once that is said out loud the negatives and the decimals stop being surprising.
0.75, 0.3 recurring, −6 and 7, in one pass
Work down a list like this and the pattern that emerges is more useful than any single answer: almost everything a student meets before GCSE is rational.
Each line ends with the fraction itself, because producing the fraction is the proof. Asserting that a number is rational demonstrates nothing; writing down an a and a b that work demonstrates everything.
- 0.75 = 75/100 = 3/4 — rational. Any decimal that stops can be written over a power of ten.
- 0.333… = 1/3 — rational. Any decimal that repeats can be turned into a fraction, and there is a standard method for doing it.
- −6 = −6/1 — rational. Nothing in the definition says a and b must be positive.
- 7 = 7/1 — rational. Every integer passes, by taking the denominator to be 1.
You cannot settle this from a calculator display
Type √2 into a calculator and you get something like 1.414213562. It stops, so it looks like a terminating decimal, so it looks rational. It is none of those things. The display has run out of room and rounded, and the same thing happens to π.
This is why the question is asked on non-calculator papers, and why the reasoning has to be done from the structure of the number rather than from its decimal. √2 is irrational because no fraction of whole numbers equals it — a fact that was proved, not observed on a screen.
عام سوالات
Is 0 a rational number?
Yes. 0 can be written as 0/1, which is a fraction of two integers with a non-zero denominator, so it passes the test. The restriction in the definition is on the bottom of the fraction, never on the top: zero on top is fine, zero underneath is not allowed.
Is every fraction a rational number?
No, and this catches people. π/2 is written as a fraction but its top is not an integer, so it fails the test and is irrational. What makes a number rational is that a fraction of two whole numbers exists for it, not that it happens to be written with a line through the middle.
Are recurring decimals rational?
Yes, all of them. 0.333… is exactly 1/3, and 0.142857 recurring is exactly 1/7. A decimal that repeats for ever still names an ordinary fraction; it is only decimals that never repeat and never stop that fall outside the rationals.
How do I decide quickly on a non-calculator paper?
Check for the two things that fail: a root that is not exact, and π. Everything else in a typical list — integers, negatives, fractions, terminating decimals, recurring decimals — is rational. √9 is 3, so it passes; √10 does not. That single check handles most questions of this type.
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