10-க்கு அருகில் உள்ள எண்களை பெருக்கல்
Multiply Numbers Near 10
When two numbers are both close to 10, their product has a beautiful shortcut. 9 × 8, 7 × 9, 8 × 8 — all solved in your head using the base-10 method.
Let's Learn
What you will learn today
Multiply any two numbers that are both close to 10 using the Nikhilam base method.
How far from 10?
How far is 9 from 10? (1) How far is 7 from 10? (3) How far is 8 from 10? (2) These 'distances' are called deficiencies — and they are the key to today's technique.
When two numbers are both close to 10, their product has a hidden structure. 9 × 8 = 72. You can find this without multiplying directly — using only the distance each number is from 10. Once you see the pattern, you will never think about near-10 multiplication the same way again.
10-க்கு அருகில் உள்ள எண்களின் பெருக்கல் — ஒரு மறைந்த வடிவம்.
The Nikhilam base-10 method
For two numbers both close to 10:
- Find the deficiency of each number from 10
- Cross: take either number, subtract the OTHER number's deficiency
- Multiply the two deficiencies together → this is the units (ones) part
- Combine: cross-result | units-result = final answer
📐 Example: 9 × 8
Deficiency of 9 from 10: 1. Deficiency of 8 from 10: 2.
- 1Cross: 9 − 2 = 7 (or: 8 − 1 = 7 — same result either way)
- 2Units: 1 × 2 = 2
- 3Combine: 7 | 2 = 72
- 49 × 8 = 72 ✓
📐 Example: 7 × 8
Deficiency of 7: 3. Deficiency of 8: 2.
- 1Cross: 7 − 2 = 5 (or: 8 − 3 = 5)
- 2Units: 3 × 2 = 6
- 3Combine: 5 | 6 = 56
- 47 × 8 = 56 ✓
Why does this work?
Let a = 10 − p and b = 10 − q (where p, q are deficiencies). Then a × b = (10−p)(10−q) = 100 − 10p − 10q + pq = 10(10−p−q) + pq = 10(a−q) + pq. So the tens part is (a − q) and the units part is (p × q). That is exactly what the cross and units calculations give.
When units product exceeds 9
If the deficiency product is ≥ 10 (e.g., 7 × 6: deficiencies 3 and 4, product 12), carry the tens digit into the cross result: cross=3, units=12 → 3+1=4 tens, 2 units = 42. Verify: 7×6=42 ✓
Common error: using the wrong deficiency in the cross
The cross uses one number minus the OTHER number's deficiency. Not one number minus its own deficiency. 9 × 7: cross = 9 − 3 (3 is 7's deficiency) = 6. Not 9 − 1 (which would be 8, wrong).
Try: 8 × 8
Both are 8. Deficiency of each from 10 is 2. Apply the method.
Challenge Round
Challenge: 6 × 7
Both are close to 10. Apply the method. Watch the units product carefully.
Base-10 Nikhilam: find deficiencies from 10, cross-subtract for tens part, multiply deficiencies for units part. If units ≥ 10, carry.
↪ Next: Base-100 — the same method scaled up to numbers near 100.
Key Points
முக்கிய குறிப்புகள்
- ✓Deficiency = how far the number is below 10
- ✓Cross: either number minus the OTHER's deficiency → tens part
- ✓Units: multiply the two deficiencies → ones part
- ✓If units ≥ 10, carry to the tens part
Glossary
சொல் அகராதி
Deficiency
குறைபாடு
Base
அடிப்படை
Cross-subtraction
குறுக்கு கழித்தல்
Nikhilam
நிகிலம்
Practice Activities
Quizவினாடி வினா
Answer each question to check your understanding.
What is 9 × 6 using the base-10 method?
Fill in the Blanksஇடைவெளி நிரப்புக
Type the missing word and press Check or Enter.
Type the missing word and click Check.