الرياضيات
What Is an Irrational Number?
π, √2 and √3 are irrational; √9 is 3 and is not. The root sign is not the test, and knowing what the real test is takes about a minute to learn.
Irrational number
An irrational number is a real number that cannot be written as a fraction of two integers, so its decimal runs on for ever without ever repeating.
- أين يقابله الطلاب
- Grade 9 and 10, and again on GCSE Higher and IGCSE Extended papers, where surd manipulation sits in the additional content for higher-attaining pupils.
الإجابة باختصار
An irrational number cannot be written as a fraction of two integers, which means its decimal never stops and never settles into a repeating block. π, √2, √3 and e are the ones you are expected to recognise. A root sign is not the test: √9 is 3, and 3 is rational.
مثال
√2 = 1.41421356… √9 = 3 √2 × √2 = 2
The middle line is the one that matters. √9 carries a root sign and is a perfectly ordinary integer, which shows that the symbol tells you nothing on its own. The third line goes further: two irrational numbers multiplied together have given a whole number, so even containing an irrational is no guide.
The short list you actually have to recognise
At school level the irrational numbers that appear are few enough to name. Everything else in a classification question will be rational, so learning this list is most of the work.
Three of the four are roots, and that is not a coincidence. The square root of any whole number that is not a perfect square is irrational, which is where almost every exam example comes from.
- π = 3.14159… — the ratio of a circle's circumference to its diameter, and the one that appears in nearly every list
- √2 = 1.41421… — the diagonal of a unit square, and historically the first number known to be irrational
- √3 = 1.73205… — and in general the root of any whole number that is not a perfect square
- e = 2.71828… — met at A level rather than GCSE, but included in lists often enough to be worth knowing
Why the root sign is not the test
The habit students fall into is to scan a list for √ and tick whatever has one. It fails immediately: √9 is 3, √16 is 4, √(1/4) is 1/2, and every one of those is rational. What decides the matter is whether the number underneath is a perfect square.
It fails from the other direction too. π ÷ π is 1. √2 × √2 is 2. Irrational numbers can combine into rational ones, so a number is not irrational merely because an irrational number appears somewhere inside it. The only reliable question is the same one as before: does a fraction of two whole numbers equal this number?
Surd: the school word for leaving it exact
A surd is a root left in root form because writing it as a decimal would mean rounding it. 3√2 is exact. 4.24 is a rounded stand-in that is very slightly wrong, and every subsequent line of working inherits that error.
This is what an instruction like give your answer in surd form is really asking for: the number itself, not an approximation of it. The same reasoning explains why an area is written as 12π rather than 37.7, and why a Pythagoras answer of √52 is often preferred to 7.21.
أسئلة شائعة
Is π irrational?
Yes, and it was proved to be so by Johann Heinrich Lambert in 1761. 22/7 and 3.14 are approximations used for convenience; neither equals π. This matters in practice, because a question asking for an exact answer wants the π left in place rather than replaced by a decimal.
Is √9 irrational?
No. √9 is 3, which is an integer and therefore rational. The root sign is a way of writing a number, not a category of number. The test is whether what sits underneath is a perfect square: √9, √16 and √225 are all rational, and √2, √3 and √10 are not.
Can an irrational number be negative?
Yes. −√2 and −π are irrational, because being expressible as a fraction of two integers has nothing to do with sign. If a number is irrational then so is its negative, since attaching a minus sign could never turn an impossible fraction into a possible one.
Is a recurring decimal irrational?
No, and this is the most common confusion about the word. 0.333… goes on for ever but repeats, and it is exactly 1/3. An irrational decimal goes on for ever and never repeats, which is why no fraction can capture it. Forever is not the test; the absence of a repeating block is.
المصادر
- National curriculum in England: mathematics programmes of study — Department for Education
آخر تحديث
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