Real Numbers
Lipi Shorts
Karnataka · KSEEB · Class 10 · Mathematics
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Real Numbers
Central idea
Every composite number has exactly one blueprint of prime factors, in exactly one combination, no matter how you go about finding it. That single fact of uniqueness turns out to be the one tool this whole chapter runs on -- it builds HCF and LCM directly, and it is also the hidden engine that proves numbers like the square root of 2 can never be written as a fraction.
Central idea
Sonia takes 18 minutes per lap and Ravi 12. Both start together. Each returns to the start after a whole number of laps, so the first shared return is the Lowest Common Multiple. Sonia's return times are 18, 36, 54, ... minutes; Ravi's are 12, 24, 36, ...
Sonia and Ravi, and when two loops line back up
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Factor 32760 into primes, following the book's large example. Divide by 2 three times to reach 4095; divide by 3 twice to reach 455; then 455=5×91=5×7×13. Thus 32760=2×2×2×3×3×5×7×13=2³×3²×5×7×13.
One blueprint, however you take a number apart
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Use the book's first prime-factor example, 6=2×3 and 20=2²×5. HCF takes the smallest power of each prime common to both numbers: only 2 is shared, so HCF=2. LCM takes the greatest power of every prime present: 2²×3×5=60.
HCF and LCM, read straight off the blueprint
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The book's prime-square theorem says that if a prime p divides u², then p divides u: squaring repeats prime factors but creates no new prime. To prove √2 irrational, assume √2=a/b in lowest terms, where a and b share no factor greater than 1.
The same uniqueness, turned into a proof that a number is NOT a fraction
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The book has already established that √3 is irrational. To prove 5−√3 is irrational, assume the opposite: let 5−√3=a/b for integers a and b with b nonzero. Isolating the root gives √3=5−a/b=(5b−a)/b.
Book Example 6: why 5−√3 is irrational
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The book first proves that √2 is irrational. Now suppose, for contradiction, that 3√2 is rational. Then 3√2=a/b for integers a,b with b nonzero. Dividing both sides by 3 gives √2=(a/b)/3=a/(3b).
Book Example 7: why 3√2 is irrational
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Myth vs truth
✕ Myth
HCF(a,b) x LCM(a,b) = a x b works no matter how many numbers are involved.
✓ Truth
It only works for EXACTLY 2 numbers. For 3 or more numbers (e.g. 6, 72, 120), the product of all the numbers does NOT equal HCF x LCM -- a different, more complex formula is needed for 3+ numbers.
Myths vs truth
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Word meanings
composite number
a natural number greater than 1 that is not prime, i.e. it has at least one factor other than 1 and itself
coprime
two integers that share no common factor other than 1
irrational number
a real number that cannot be expressed as p/q for any integers p and q with q not zero
proof by contradiction
a proof technique that assumes the opposite of what is to be shown, then derives a logical impossibility, concluding the original assumption must be false
Word meanings
Worked examples
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Themes & message
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One single fact -- that prime factorisation is unique -- is genuinely powerful enough to build two seemingly unrelated tools: a direct formula for HCF and LCM, and a watertight method for proving a number can never be written as a fraction.
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A clean identity that holds for 2 numbers (HCF x LCM = product) does not automatically generalise to 3 or more -- always worth checking a pattern's boundary rather than assuming it stretches indefinitely.
•
Textbooks get edited and trimmed over time; an intro paragraph can occasionally reference content that a later rationalisation quietly removed. Reading the actual section that follows, rather than assuming the intro is a perfectly accurate table of contents, is a genuinely useful habit.
Themes & message
Key facts at a glance
Fundamental Theorem of Arithmetic
every composite number factorises into primes in exactly one way, ignoring order
HCF via prime factorisation
product of the SMALLEST power of each prime common to all the numbers
LCM via prime factorisation
product of the GREATEST power of each prime appearing in any of the numbers
Key facts at a glance
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