Draw a Length You Can Never Write Down Exactly
Learn to write any rational number as p over q and why q cannot be zero, find as many rationals between two numbers as you like, tell rational from irrational, and build root 2 and root 3 on the number line.
Can a length exist that no fraction can measure?
Draw a square of side cm and join two opposite corners. That diagonal has a definite length — you can measure it with a ruler, and Pythagoras gives it exactly:
Now try to write that length as a fraction. is close. is closer. is closer still.
No fraction is exactly right, and this is not a matter of not having found it yet. It can be proved that no fraction of whole numbers equals — and yet the diagonal is sitting there on the page, a perfectly ordinary length you just drew with a ruler.
So the number line contains points that no fraction reaches. The numbers that fractions do reach are called rational; the rest are irrational; together they are the real numbers.
This page covers the second part of the CBSE Class 9 Mathematics chapter on the world of numbers — what makes a number rational, how to find rationals between any two numbers, how to tell the two kinds apart, and how to construct an irrational length exactly on the number line.
Now try to write that length as a fraction. is close. is closer. is closer still.
No fraction is exactly right, and this is not a matter of not having found it yet. It can be proved that no fraction of whole numbers equals — and yet the diagonal is sitting there on the page, a perfectly ordinary length you just drew with a ruler.
So the number line contains points that no fraction reaches. The numbers that fractions do reach are called rational; the rest are irrational; together they are the real numbers.
This page covers the second part of the CBSE Class 9 Mathematics chapter on the world of numbers — what makes a number rational, how to find rationals between any two numbers, how to tell the two kinds apart, and how to construct an irrational length exactly on the number line.
Formula
What does it mean to write a number as p over q?
A number is rational when it can be written as a ratio of two integers with a non-zero denominator:
Worked conversions. Many numbers that do not look like fractions are rational once written properly.
- — every integer is rational, with
- — zero is rational; it is the numerator that may be zero
-
-
-
- — a square root can certainly be rational
**Why is required.** If were zero the symbol would name no number at all. Division asks a multiplication question: would be the number with , and every number times zero is zero. So the condition is not a formality — it excludes an expression that has no value.
**The same number has many forms.** . The version with and is the standard form, and is it here. Standard form matters in the next part of this chapter, where the denominator predicts the shape of the decimal expansion.
Worked example — putting a number into standard form. Write in standard form. Both signs are negative, so the value is positive; :
**Rational is a statement about what a number can be written as, not about how it is written.** carries no fraction bar and is rational; has a fraction bar and is not rational, because is not an integer. The test is always whether integers can be found for and .
Worked conversions. Many numbers that do not look like fractions are rational once written properly.
- — every integer is rational, with
- — zero is rational; it is the numerator that may be zero
-
-
-
- — a square root can certainly be rational
**Why is required.** If were zero the symbol would name no number at all. Division asks a multiplication question: would be the number with , and every number times zero is zero. So the condition is not a formality — it excludes an expression that has no value.
**The same number has many forms.** . The version with and is the standard form, and is it here. Standard form matters in the next part of this chapter, where the denominator predicts the shape of the decimal expansion.
Worked example — putting a number into standard form. Write in standard form. Both signs are negative, so the value is positive; :
**Rational is a statement about what a number can be written as, not about how it is written.** carries no fraction bar and is rational; has a fraction bar and is not rational, because is not an integer. The test is always whether integers can be found for and .
How do you find six rational numbers between 3 and 4?
Give both numbers a denominator large enough to leave the room you need, then read off the numerators.
**Worked example 1 — six numbers between and .** Six numbers need at least seven gaps, so use denominator :
Between and sit
exactly six of them. Check the first and last: and , both between and .
**Worked example 2 — five numbers between and .** Use denominator : and , giving
The mean method — one number at a time. The average of two numbers always lies between them. Between and :
Check in decimals: . Correct. Repeating the method on and gives another, and so on without end.
Worked example 3 — when the obvious denominator is too small. Find a rational between and . The common denominator gives and — **no whole number lies between and **, so that denominator has no room. Double it to :
Check: . Correct.
There are infinitely many rationals between any two rationals, however close together they are, because the mean method never runs out — and multiplying the denominator by always opens nine fresh gaps. **So the question "which rational comes next after " has no answer.** The integers have neighbours and the rationals do not, and that difference is worth noticing: it is why a list of the rationals cannot be written in increasing order the way can.
**Worked example 1 — six numbers between and .** Six numbers need at least seven gaps, so use denominator :
Between and sit
exactly six of them. Check the first and last: and , both between and .
**Worked example 2 — five numbers between and .** Use denominator : and , giving
The mean method — one number at a time. The average of two numbers always lies between them. Between and :
Check in decimals: . Correct. Repeating the method on and gives another, and so on without end.
Worked example 3 — when the obvious denominator is too small. Find a rational between and . The common denominator gives and — **no whole number lies between and **, so that denominator has no room. Double it to :
Check: . Correct.
There are infinitely many rationals between any two rationals, however close together they are, because the mean method never runs out — and multiplying the denominator by always opens nine fresh gaps. **So the question "which rational comes next after " has no answer.** The integers have neighbours and the rationals do not, and that difference is worth noticing: it is why a list of the rationals cannot be written in increasing order the way can.
How do you tell a rational number from an irrational one?
A rational number can be written as a ratio of integers; an irrational number cannot — and the test in practice is usually the square root or the decimal expansion.
Worked classification.
- — rational by definition
- — rational. Perfect squares have rational roots
- , , , — irrational. The root of a non-perfect-square positive integer is irrational
- — rational, so "has a root sign" is not a test
- — irrational
- with a growing run of zeros — irrational, because the pattern never settles into a repeating block
**Why is irrational.** Suppose it were rational, say in standard form, so and share no common factor. Squaring:
So is even, which forces to be even — an odd number squared is odd. Write :
Now is even, so is even too. But then and are both even, sharing a factor of , contradicting standard form. The assumption cannot hold, so no such fraction exists.
The identical argument works for and .
**Why is not .** with the block repeating for ever, while never repeats. They agree to two decimal places and differ after that. ** is a convenient approximation used to make arithmetic easy**, and calling it the value of is a genuine error, not a shortcut.
The decimal test. A terminating or repeating expansion means rational; a non-terminating, non-repeating expansion means irrational. That test is the subject of the next part of this chapter, and it is the practical version of everything above.
One boundary case worth knowing. Irrational numbers can combine to give rational ones: , and . So "irrational" is not a property that survives arithmetic — which is exactly why each expression has to be classified on its own rather than by inspecting its parts.
Worked classification.
- — rational by definition
- — rational. Perfect squares have rational roots
- , , , — irrational. The root of a non-perfect-square positive integer is irrational
- — rational, so "has a root sign" is not a test
- — irrational
- with a growing run of zeros — irrational, because the pattern never settles into a repeating block
**Why is irrational.** Suppose it were rational, say in standard form, so and share no common factor. Squaring:
So is even, which forces to be even — an odd number squared is odd. Write :
Now is even, so is even too. But then and are both even, sharing a factor of , contradicting standard form. The assumption cannot hold, so no such fraction exists.
The identical argument works for and .
**Why is not .** with the block repeating for ever, while never repeats. They agree to two decimal places and differ after that. ** is a convenient approximation used to make arithmetic easy**, and calling it the value of is a genuine error, not a shortcut.
The decimal test. A terminating or repeating expansion means rational; a non-terminating, non-repeating expansion means irrational. That test is the subject of the next part of this chapter, and it is the practical version of everything above.
One boundary case worth knowing. Irrational numbers can combine to give rational ones: , and . So "irrational" is not a property that survives arithmetic — which is exactly why each expression has to be classified on its own rather than by inspecting its parts.
How do you construct root 2 and root 3 on the number line?
Build a right triangle whose hypotenuse is the length you want, then swing that hypotenuse onto the line with a compass.
The tool is the Pythagoras theorem: a right triangle with legs and has hypotenuse . Choosing the legs chooses the root.
**Worked construction 1 — .** On the number line mark at and at . At draw a perpendicular of length , reaching . Then
With centre and radius , draw an arc cutting the line at . ** is exactly **, at about — between and , as a rough check requires.
**Worked construction 2 — .** Continue from . At draw a perpendicular of length , reaching . Since ,
Swing onto the line to locate .
**Worked construction 3 — in one step.** Legs of and give
So a perpendicular of length erected at the point does it immediately, without passing through and .
The square root spiral. Repeating construction 2 builds every root in turn. Starting from a unit segment, each stage erects a perpendicular of length on the previous hypotenuse:
The hypotenuses come out as and the figure coils outward as a spiral. Check one link: from , the next is , which is a rational stage sitting in the middle of the spiral.
What the construction actually proves. It gives the point exactly, not approximately. A decimal such as is a rounded value that misses the true point, while the compass arc lands on it — an irrational number has a precise location even though its decimal expansion never ends.
Measuring the constructed length with a ruler is not a check. The ruler reads about cm and would read the same for , a different point. The check is the Pythagoras calculation, which is why every construction answer should carry the line beside the diagram.
The tool is the Pythagoras theorem: a right triangle with legs and has hypotenuse . Choosing the legs chooses the root.
**Worked construction 1 — .** On the number line mark at and at . At draw a perpendicular of length , reaching . Then
With centre and radius , draw an arc cutting the line at . ** is exactly **, at about — between and , as a rough check requires.
**Worked construction 2 — .** Continue from . At draw a perpendicular of length , reaching . Since ,
Swing onto the line to locate .
**Worked construction 3 — in one step.** Legs of and give
So a perpendicular of length erected at the point does it immediately, without passing through and .
The square root spiral. Repeating construction 2 builds every root in turn. Starting from a unit segment, each stage erects a perpendicular of length on the previous hypotenuse:
The hypotenuses come out as and the figure coils outward as a spiral. Check one link: from , the next is , which is a rational stage sitting in the middle of the spiral.
What the construction actually proves. It gives the point exactly, not approximately. A decimal such as is a rounded value that misses the true point, while the compass arc lands on it — an irrational number has a precise location even though its decimal expansion never ends.
Measuring the constructed length with a ruler is not a check. The ruler reads about cm and would read the same for , a different point. The check is the Pythagoras calculation, which is why every construction answer should carry the line beside the diagram.
Exam tip
Exam tip: give the denominator enough room, and justify every classification
**For rationals between two numbers, use denominator or larger.** Six between and : write and , then list to .
If the common denominator leaves no gap, multiply it. Between and , sixths give and with nothing between — use twelfths and take .
The mean method gives one number per step: . Use it when the question wants just one.
Always check your answers lie in range by converting to decimals — one line, and it catches a wrong denominator.
Never leave a classification bare. Write *rational, since or irrational, since is not a perfect square*. The justification carries the marks.
A root sign is not a verdict: is rational, is not.
** is an approximation to , not its value** — say "taking " when you use it.
For a construction, show the Pythagoras line: , so . A measured ruler value proves nothing.
Use a compass arc centred at the origin to transfer the hypotenuse to the line, and label the point.
And in standard form, keep and cancel the HCF: .
If the common denominator leaves no gap, multiply it. Between and , sixths give and with nothing between — use twelfths and take .
The mean method gives one number per step: . Use it when the question wants just one.
Always check your answers lie in range by converting to decimals — one line, and it catches a wrong denominator.
Never leave a classification bare. Write *rational, since or irrational, since is not a perfect square*. The justification carries the marks.
A root sign is not a verdict: is rational, is not.
** is an approximation to , not its value** — say "taking " when you use it.
For a construction, show the Pythagoras line: , so . A measured ruler value proves nothing.
Use a compass arc centred at the origin to transfer the hypotenuse to the line, and label the point.
And in standard form, keep and cancel the HCF: .
Did you know
Why the rationals leave gaps they seem too crowded to leave
Between any two rationals lie infinitely many more. Pick and and there are still infinitely many rationals squeezed between them. On any drawing, the rationals look like a solid line.
And yet is a gap. So is , and , and infinitely many others. A set can be crowded everywhere and still miss most of the points it seems to cover.
The reason is that crowded and complete are different properties. Being crowded means no smallest gap — between any two members there is another. Being complete means no gaps at all, and the rationals fail that test while looking exactly as though they pass it.
Here is the failure in a form you can hold. Consider the fractions whose squares are less than : , , , , and so on, each closer to the diagonal of the unit square than the last. Among the rationals this list approaches a boundary that is not a rational number. The sequence has somewhere to go and nothing to arrive at.
Filling those gaps is what the real numbers do, and the number line is the picture of the finished job — every point a number, every number a point, with no holes anywhere. The square root spiral is a way of walking to some of the filled-in points with nothing but a ruler and a compass.
What makes the whole thing sit oddly is that the gaps were there all along, in a set that looked seamless. The diagonal of a cm square is an easy thing to draw and an impossible thing to write as a fraction, and both of those facts are about the same short line on the page.
And yet is a gap. So is , and , and infinitely many others. A set can be crowded everywhere and still miss most of the points it seems to cover.
The reason is that crowded and complete are different properties. Being crowded means no smallest gap — between any two members there is another. Being complete means no gaps at all, and the rationals fail that test while looking exactly as though they pass it.
Here is the failure in a form you can hold. Consider the fractions whose squares are less than : , , , , and so on, each closer to the diagonal of the unit square than the last. Among the rationals this list approaches a boundary that is not a rational number. The sequence has somewhere to go and nothing to arrive at.
Filling those gaps is what the real numbers do, and the number line is the picture of the finished job — every point a number, every number a point, with no holes anywhere. The square root spiral is a way of walking to some of the filled-in points with nothing but a ruler and a compass.
What makes the whole thing sit oddly is that the gaps were there all along, in a set that looked seamless. The diagonal of a cm square is an easy thing to draw and an impossible thing to write as a fraction, and both of those facts are about the same short line on the page.
Exam relevance
How do rational and irrational numbers feed into JEE Main?
Because the real number system defined here is the setting for every later chapter, and the specific skill of handling irrational expressions is used constantly.
This is the foundation for Class 11 Mathematics Sets, where , , and become standard notation, and for Complex Numbers and Quadratic Equations, examined in JEE Main. The hierarchy established here — naturals inside integers inside rationals inside reals — extends there to include the complex numbers, and the reason for each extension is the same one seen on this page: an equation with no solution in the smaller system. needs the integers, needs the rationals, needs the irrationals, and needs the complex numbers.
Irrational roots in quadratic equations are where this most directly reappears. When the discriminant of a quadratic is positive but not a perfect square, the roots are irrational and come in the conjugate pair — and the fact that and add to a rational number is exactly why the sum of such roots is rational. That observation, made on this page as a boundary case, becomes a standard tool for Class 11 Quadratic Equations.
The proof by contradiction is itself the content. Class 11 Mathematical Reasoning treats this as a named method, and the argument is the example every treatment uses. Being able to reproduce it — assume standard form, derive that both and are even, contradict — is worth more than the conclusion.
Rationalising a denominator grows out of the classification skill here and becomes routine in Class 11 Limits and Derivatives, where expressions such as are handled by multiplying by the conjugate. A student who is comfortable that already has the move.
Where the constructions lead. The square root spiral is also a geometry exercise, and the Pythagoras step inside it is the same step used throughout Class 11 Straight Lines for distances. Olympiad and NTSE style questions do sometimes ask directly for a construction or for a proof that a given number is irrational.
What the questions look like. For board work, expect classify and justify, **insert rationals between two numbers, express in form, and a construction on the number line — each self-contained. For JEE Main, the topic surfaces as surd manipulation, conjugate pairs, and the nature of the roots of a quadratic, never as a definition question.
How board and competitive emphasis differ. A board paper rewards the justification sentence and the labelled construction; a competitive paper assumes both and tests whether irrational expressions can be simplified quickly.
The single trap that costs the most marks.** Treating as equal to , and then asserting that is rational. It is a legitimate approximation for calculation and an illegitimate equality in a classification question — and questions are deliberately built to separate students who know the difference.
This is the foundation for Class 11 Mathematics Sets, where , , and become standard notation, and for Complex Numbers and Quadratic Equations, examined in JEE Main. The hierarchy established here — naturals inside integers inside rationals inside reals — extends there to include the complex numbers, and the reason for each extension is the same one seen on this page: an equation with no solution in the smaller system. needs the integers, needs the rationals, needs the irrationals, and needs the complex numbers.
Irrational roots in quadratic equations are where this most directly reappears. When the discriminant of a quadratic is positive but not a perfect square, the roots are irrational and come in the conjugate pair — and the fact that and add to a rational number is exactly why the sum of such roots is rational. That observation, made on this page as a boundary case, becomes a standard tool for Class 11 Quadratic Equations.
The proof by contradiction is itself the content. Class 11 Mathematical Reasoning treats this as a named method, and the argument is the example every treatment uses. Being able to reproduce it — assume standard form, derive that both and are even, contradict — is worth more than the conclusion.
Rationalising a denominator grows out of the classification skill here and becomes routine in Class 11 Limits and Derivatives, where expressions such as are handled by multiplying by the conjugate. A student who is comfortable that already has the move.
Where the constructions lead. The square root spiral is also a geometry exercise, and the Pythagoras step inside it is the same step used throughout Class 11 Straight Lines for distances. Olympiad and NTSE style questions do sometimes ask directly for a construction or for a proof that a given number is irrational.
What the questions look like. For board work, expect classify and justify, **insert rationals between two numbers, express in form, and a construction on the number line — each self-contained. For JEE Main, the topic surfaces as surd manipulation, conjugate pairs, and the nature of the roots of a quadratic, never as a definition question.
How board and competitive emphasis differ. A board paper rewards the justification sentence and the labelled construction; a competitive paper assumes both and tests whether irrational expressions can be simplified quickly.
The single trap that costs the most marks.** Treating as equal to , and then asserting that is rational. It is a legitimate approximation for calculation and an illegitimate equality in a classification question — and questions are deliberately built to separate students who know the difference.
Key takeaways
Rational numbers, irrational numbers and constructions: quick revision
- A number is rational when it can be written as with integers and ****.
- is required because names no number — no satisfies for .
- Every integer is rational: . Zero is rational — it is the numerator that may be zero.
- , , , .
- Standard form has and : .
- **For rationals between two numbers**, use denominator or more. Six between and : to , from and .
- Five between and with sixths: .
- Mean method: , and .
- If the common denominator leaves no gap, multiply it: sixths fail between and , twelfths give .
- Infinitely many rationals lie between any two rationals, so no rational has a "next" one.
- Irrational: , , , , and . Rational: , , .
- ** is irrational**: in standard form gives , so is even; gives , so is even too — contradicting standard form.
- **** — it repeats the block , while never repeats. It is an approximation.
- Irrationals can combine to rationals: and .
- **To construct **: perpendicular of at the point , since ; swing the hypotenuse onto the line.
- **To construct **: perpendicular of at , since . For directly, legs and .
- The square root spiral repeats this: , and appears as a rational stage.
- A construction locates the point exactly; a ruler reading of cm does not distinguish from .
Draw a cm square, measure its diagonal, then square your measurement and see how far from you land — the gap is the difference between measuring and proving.
- is required because names no number — no satisfies for .
- Every integer is rational: . Zero is rational — it is the numerator that may be zero.
- , , , .
- Standard form has and : .
- **For rationals between two numbers**, use denominator or more. Six between and : to , from and .
- Five between and with sixths: .
- Mean method: , and .
- If the common denominator leaves no gap, multiply it: sixths fail between and , twelfths give .
- Infinitely many rationals lie between any two rationals, so no rational has a "next" one.
- Irrational: , , , , and . Rational: , , .
- ** is irrational**: in standard form gives , so is even; gives , so is even too — contradicting standard form.
- **** — it repeats the block , while never repeats. It is an approximation.
- Irrationals can combine to rationals: and .
- **To construct **: perpendicular of at the point , since ; swing the hypotenuse onto the line.
- **To construct **: perpendicular of at , since . For directly, legs and .
- The square root spiral repeats this: , and appears as a rational stage.
- A construction locates the point exactly; a ruler reading of cm does not distinguish from .
Draw a cm square, measure its diagonal, then square your measurement and see how far from you land — the gap is the difference between measuring and proving.