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Number systems is the first idea-heavy topic of Class 9, and it feels strange because most of it is about proving things you already believe. Why is 0.333... a fraction? Why can't root 2 be written as p/q? This guide walks through each idea with worked examples, the mistakes that cost marks, and a short practice routine.
What is in the Class 9 number systems chapter now?
NCERT has a new Class 9 textbook for 2026-27, Ganita Manjari. Number systems is its Chapter 3, The World of Numbers, which moves from counting and zero through integers and rational numbers to irrational numbers, Madhava's series for pi and decimal patterns. The CBSE Class IX curriculum for 2026-27 gives the Number System unit 7 of the 80 theory marks and lists what you are assessed on: representing rational numbers on the number line, density of rational numbers, decimal representation, the proof that root 2 and root 3 are irrational, and the square root spiral. Check your school's copy of the syllabus each year, because CBSE revises it.
How do you find rational numbers between two fractions?
A rational number is one that can be written as p/q, where p and q are integers and q is not zero. Between any two rational numbers there is always another one, which is what "density" means. The quick proof: the average of two rationals is rational and lies between them.
Worked example: find three rational numbers between 1/3 and 1/2.
Use a common denominator large enough to leave room: 1/3 = 8/24 and 1/2 = 12/24.
Read off the fractions in between: 9/24, 10/24 and 11/24.
Simplify: 3/8, 5/12 and 11/24.
As decimals these are 0.375, about 0.417 and about 0.458, all between 0.333... and 0.5. If you need more numbers, use a larger common denominator.
How do you turn a repeating decimal into a fraction?
Every rational number has a decimal that either stops (7/40 = 0.175) or repeats (1/7 = 0.142857142857...). The reverse also holds: a repeating decimal is a fraction. The trick is to multiply so that the repeating parts line up, then subtract.
Worked example 1: x = 0.272727...
100x = 27.272727...
100x minus x gives 99x = 27, so x = 27/99 = 3/11.
Worked example 2: x = 0.4777..., where only the 7 repeats.
10x = 4.777... and 100x = 47.777...
100x minus 10x gives 90x = 43, so x = 43/90.
The common mistake in example 2 is subtracting x from 100x. That gives 99x = 47.3, which still has a decimal and does not finish cleanly. Line up the repeating parts first: both numbers you subtract must have the same block after the decimal point.
Why is root 2 irrational, and how do you write the proof?
An irrational number cannot be written as p/q. The standard proof is by contradiction: assume the opposite and show it leads to something impossible.
Suppose root 2 = p/q, where p and q are integers with no common factor (the fraction is in lowest terms).
Square both sides: 2 = p²/q², so p² = 2q². This means p² is even.
If p were odd, p² would be odd. So p is even. Write p = 2k.
Then 4k² = 2q², so q² = 2k². This means q² is even, so q is even.
Now p and q are both even, so they share the factor 2. This contradicts step 1. So root 2 is not rational.
Examiners look for two things: the "lowest terms" assumption in step 1, and a reason for "p² even means p even" in step 3. Students often skip both. The proof for root 3 has the same shape, using "multiple of 3" in place of "even".
What mistakes do students make with irrational numbers?
Treating 22/7 as pi. 22/7 is a rational approximation. Pi itself is irrational, and the CBSE curriculum mentions its irrationality and Madhava's infinite series.
Assuming irrational plus irrational is irrational. Root 2 plus (minus root 2) is 0, and root 2 times root 2 is 2. Both are rational. What is always true: a non-zero rational number times an irrational number, or a rational plus an irrational, is irrational.
Calling every square root irrational. Root 49 is 7. Root n is irrational only when n is not a perfect square (for whole numbers n).
Drawing the square root spiral without checking. Each new triangle has legs root n and 1, so the hypotenuse is the square root of (n + 1). Start from legs 1 and 1 to get root 2, then root 3, and so on. If your spiral drifts, check that each new unit side is perpendicular to the last hypotenuse.
The real numbers are all the rationals and irrationals together, and every real number has a point on the number line. That idea returns in Class 10; our guide to real numbers in Class 10 shows where it goes next.
How should you practise this chapter?
Recalling a method from memory helps it stick better than rereading it. A large review of study techniques by Dunlosky and colleagues (2013) rated practice testing and spaced-out (distributed) practice as the two most useful techniques it examined. For this chapter that means:
Day 1: write the root 2 proof from memory on a blank page, then compare it with the five steps above. Mark any step you missed.
Day 3: convert five repeating decimals (mix pure ones like 0.1515... with mixed ones like 0.8333...) and check each by dividing.
Day 7: prove root 3 is irrational without looking, find four rationals between two fractions, and draw the spiral up to root 5.
If a step in the proof still feels like a trick you memorised, look it up in the Fermi Concept Library, which covers every concept in physics, chemistry, biology, mathematics and computing, so you can read why "p² even" forces "p even" in another way.
Tonight, write the root 2 proof on a blank page without notes and check it against the five steps.
