Class 9 Maths Number Systems Solutions

Is article mein hum Class 9 ke Number Systems chapter ke solutions dekhenge, bilkul NCERT aur CBSE pattern ke hisaab se. Agar aapko concepts clear karne mein dikkat aa rahi hai, toh ye section aapke liye hi hai. Bas dhyan se padhte jaiye, sab kuch step by step samjhaya gaya hai. Aur haan, thoda sa patience rakhiye—kyunki numbers kabhi kabhi aise hi ullu banate hain, par solution mein sab kuch neatly sorted milta hai, chahe wo rational ho ya irrational, chhota ho ya bada.

Kya Seekhenge

• Okay, so first up, we’ve got Natural Numbers, or Prakritik Sankhya as they’re called in Hindi. Then we move on to Whole Numbers. That’s the core of this part.
• Integers—yaani Purnank—aur Rational Numbers, dono ko hum ek saath samjhenge. Pehle dekhenge integers kya hote hain, phir rational numbers ki taraf badhenge, aur beech mein yeh bhi clear karenge ki inka aapas mein kya rishta hai.
• So here’s the thing—irrational numbers and real numbers. They sound heavy, right? But really, they’re just the building blocks of everything you’ll count, measure, or even imagine. We’re not just memorizing definitions here. We’re getting to the why. Why do some numbers never end? Why do others just keep going without any pattern? That’s your irrational crowd. And then you’ve got the real numbers—the whole gang, rational and irrational together, filling up the number line completely. No gaps. No empty spots. Just one continuous story from the smallest negative to the biggest positive you can dream up. So when we dig into this, it’s not about ticking boxes. It’s about seeing the logic that holds the number universe together. And honestly? Once you see it, you can’t unsee it.

Links for Chapter-wise Download NCERT Solution for Class 9 गणित in hindi Language

Here we have provided NCERT Solution for Class 9 गणित in hindi Language, Just select the chapters below to get solution of the same:

बहुपद

निर्देशांक ज्यामिति

दो चरों वाले रैखिक समीकरण

यूक्लिड की ज्यामिति का परिचय

रेखाएँ और कोण

त्रिभुज

चतुर्भुज

समांतर चतुर्भुजों और त्रिभुजों के क्षेत्रफल

वृत्त

रचनाएँ

हीरोन का सूत्र

पृष्ठीय क्षेत्रफल और आयतन

सांख्यिकी

प्रायिकता

Class 9 Maths Number Systems: Detailed Solutions

Class 9 Maths Number Systems mein aapka swagat hai. Yaani, yahan se hi maths ki asli shuruaat hoti hai — number systems base hain, poori building isi par khadi hai. Is chapter mein hum number types ko tod-te phod-te samjhenge, aur haan, examples ke bina toh baat hi adhoori hai. Toh chaliye, different numbers aur unki properties ko aasaan bhasha mein, step by step, dekhte hain.

1. Natural Numbers (Prakritik Sankhya)

Look, natural numbers are just your plain counting numbers. 1, 2, 3, and so on. That's it. You use them every day without even thinking about it—counting apples, counting steps, counting the minutes until the weekend. They're the positive integers, pure and simple. Take 5, for instance — is it a natural number? Yeah, absolutely — it's right there in the count. No fuss, no zero, no negatives. Just the good old starting block of math.

2. Whole Numbers (Poorn Sankhya)

Whole numbers are basically the natural numbers with zero thrown into the mix. So you've got 0, 1, 2, 3, and it just keeps going. That little zero might seem like no big deal, but trust me, it matters a lot. You can't just skip it.

3. Integers (Purnank)

Integers are basically whole numbers, but they also bring their negative side along. Think of it like this: -3, -2, -1, 0, 1, 2, 3. No fractions, no decimals sneaking in—just clean, whole numbers with their negative twins.

4. Rational Numbers (Parimey Sankhya)

Rational numbers are simply those numbers you can write as p/q, where p and q are integers, and q is definitely not zero. So yeah, 1/2 works, so does -4/5, and even 7 counts because 7 is just 7/1 in disguise.

5. Irrational Numbers (Aparimey Sankhya)

Irrational numbers are the ones that refuse to fit neatly into that p/q box—you simply can't write them as a simple fraction. Try it with √2. You'll just keep chasing a decimal that never settles down. Same with π. That decimal expansion goes on forever and ever, and here's the kicker: it never even bothers to repeat itself. No pattern, no rhythm, just an endless string of digits that keeps you guessing. So yeah, they're a bit wild like that. But that's exactly what makes them interesting.

6. Real Numbers (Vastvik Sankhya)

Real numbers—yaani vastvik sankhya—rational aur irrational dono ka mel hain. Inko number line par dikhaya ja sakta hai, bilkul seedha sa.

Solved Examples with Steps

Example 1: Rational Numbers between Two Numbers

Alright, let’s try this one: find two rational numbers that sit between 1 and 2. Seems easy enough, right — honestly, it kind of is. Rational numbers are just fractions—any number you can write as a/b where a and b are integers. So you’re not looking for anything fancy here. One obvious pick is 3/2, which is exactly 1.5. That’s right in the middle. Then you could grab 4/3, which is about 1.33, or maybe 5/3, which lands around 1.67. Both work fine. In fact, there are infinite choices. The trick is just remembering that between any two numbers, you can always squeeze in a rational one—actually, tons of them. So for 1 and 2, the solution’s almost too simple: pick a couple of fractions and you’re done.

Solution: Pehle toh average hi nikal lo. (1+2)/2 = 1.5, aur yeh seedha ek rational number hai, koi shak nahi. Ab isko aage badhao—1 aur 1.5 ka average lo: (1+1.5)/2 = 1.25. Dekho, bas yahi hai. Toh ab tumhare paas 1.25 aur 1.5 hain, dono rational, aur dono 1 aur 2 ke beech fit ho rahe hain.

Example 2: Irrational Number Proof

Here’s the rewritten paragraph, keeping the heading context and all facts intact. --- Problem: Prove that √3 is irrational. This one’s a classic. You start by assuming the opposite—that √3 can be written as a fraction, say a/b, where a and b are integers with no common factors. That’s a standard trick. It works because if the fraction could be simplified, you’d just do it. So you square both sides: 3 = a²/b², which gives a² = 3b². Now here’s where things get interesting. That equation tells you a² is divisible by 3. And since 3 is prime, that means a itself must be divisible by 3 too. So you write a = 3k, plug it back in, and get 9k² = 3b², which simplifies to b² = 3k². Same logic applies—b has to be divisible by 3 as well. But wait, that’s a contradiction. You said a and b had no common factors, yet here they both are, sharing a factor of 3. So the whole premise falls apart, and you’re forced to conclude that √3 simply can’t be a fraction. It’s irrational — end of story.

Alright, so here’s the deal. We start by assuming √3 is rational. That means we can write it as a fraction, say √3 = a/b, where a and b are integers with no common factor—coprime, in math speak—and b isn’t zero. So far so good. Now, square both sides. You get 3 = a²/b², which rearranges to a² = 3b². And here’s where things get interesting. Since a² is 3 times something, that tells us a itself has to be divisible by 3. Makes sense, right? Because if a weren’t divisible by 3, its square wouldn’t be either. So let’s say a = 3c for some integer c. Plug that back in, and you get (3c)² = 3b², which simplifies to 9c² = 3b², and then b² = 3c². And what does that imply? Yeah, you guessed it—b must also be divisible by 3. But hold on. We said a and b were coprime, no shared factors. If both are divisible by 3, that’s a straight-up contradiction. So our original assumption must be wrong. √3 can’t be rational. It’s irrational. Done.

Example 3: Decimal Expansion

Here’s the rewrite: Here’s the deal: we’re staring at 0.666… and the task is to prove it’s rational. Now, that repeating 6 might look sneaky, but it’s actually a dead giveaway. Because when a decimal repeats forever, you can always trap it in a fraction. So let’s do that — call the number x. So x equals 0.666…, right? Then multiply both sides by 10—that shifts the decimal one spot, giving you 10x = 6.666…. Here’s the neat part: subtract the original x from that. So 10x minus x is 9x, and 6.666… minus 0.666…? The repeating parts wipe each other out, leaving you with a clean 6. So 9x = 6, which means x = 6/9. Simplify that bad boy, and you get 2/3. Boom—a fraction, plain and simple. That’s rational, no question.

Take x = 0.666… and then multiply both sides by 10. That gives you 10x = 6.666… Now subtract the original from this — so 10x minus x leaves you with 9x. On the other side, 6.666… minus 0.666… just cancels out the repeating part, giving you a clean 6. So 9x = 6, which means x = 6/9, and that simplifies down to 2/3. And there you have it — 0.666… turns out to be a rational number, because it’s exactly equal to 2/3.

Important Points to Remember

• Every integer is a rational number—no exceptions.
• Square root of a prime number? Always irrational — no exceptions, no special cases. You can test it with 2, 3, 5, 7—every single one lands you in irrational territory. The proof is classic, the kind you'd see in any number theory class, and it hinges on the fact that primes don't factor into smaller integers. So if you ever spot a prime under that radical sign, just know you're dealing with a number that goes on forever without repeating. Simple rule, but it holds every time.
• Real numbers can actually be shown on a number line. Simple as that—you just plot them, and they fit right in. Every single real number, whether it’s a whole number, a fraction, or something with endless decimals, has its own exact spot on that line. It’s not a rough estimate or a maybe; it’s precise. So when you’re picturing them, don’t think of abstract symbols floating in space—think of a straight line stretching left and right. Each number sitting right where it belongs. That’s the trick to remembering it.
• Rational numbers ki decimal expansion hamesha do tarah ki hoti hai—ya toh terminating hoti hai, ya phir repeating. Matlab, chahe aap kisi bhi rational number ko dekho, uske decimal points ke baad ya toh finite digits milenge, jo ke terminate ho jayenge, ya phir koi pattern baar-baar repeat hoga. Simple hai, bas yehi yaad rakhna.

Practice Problems for Self-Study

Here’s your rewrite. Sure, here are some practice problems to work through on your own. Start with this one: find three rational numbers that sit somewhere between -1 and 0.

2. Okay, here’s a classic — prove that 5 − √2 is irrational. You’ve probably seen a proof like this before. The trick is to assume the opposite. Say 5 − √2 equals some rational number, p/q, where p and q are integers with no common factors and q isn’t zero. Rearrange it a bit — you get √2 = 5 − p/q. And 5 minus a rational is still rational. So that would mean √2 is rational. But we already know √2 is irrational—that’s a well-known fact. Contradiction. Done. So 5 − √2 can’t be rational. It has to be irrational. Try writing it out step by step; it’s a good warm-up for trickier proofs.

Here’s number three: go ahead and represent √5 on the number line. Sounds simple enough, right? But it’s one of those problems that really tests whether you actually get the construction method, not just the theory. So grab a ruler and compass, take your time, and see if you can place it accurately. If you get stuck, that’s fine—just trace your steps back and check where things went sideways. This one’s a classic, and once you nail it, you’ll feel a lot more confident with irrational numbers.

Problems khud solve karo, toh understanding apne aap badhti hai—koi shortcut nahi hai. Teachers se poochhne mein kyun jhijhak? Agar koi question atka ho, toh wohi log hain jo sabse sahi rasta dikha sakte hain, aur unse baat karne mein thodi si deri bhi future mein bada confusion paida kar sakti hai. Bas practice karte raho, har roz thoda sa, aur jab lage ki kahin phas gaye ho, toh turant madad maango—wait mat karo.