Number System
संख्या पद्धति — स्थानीय मान, विभाज्यता, अभाज्य और परिमेय
Numbers, place value and the two ways of grouping digits
The families
Indian and international grouping
| System | Commas | 50801592 written as |
|---|---|---|
| Indian | after the first three digits from the right, then after every two | 5,08,01,592 |
| International | after every three | 50,801,592 |
Roman numerals — I 1, V 5, X 10, L 50, C 100, D 500, M 1000
The trap: in the Indian system the first comma comes after three digits and every later one after two.
संख्या पद्धति — स्थानीय मान, विभाज्यता, अभाज्य और परिमेय
Divisibility — the tests a paper sets
| Divisible by | Test |
|---|---|
| 2 | the last digit is even |
| 3 | the sum of the digits is divisible by 3 |
| 4 | the number formed by the last two digits is divisible by 4 |
| 5 | the last digit is 0 or 5 |
| 6 | divisible by both 2 and 3 |
| 8 | the number formed by the last three digits is divisible by 8 |
| 9 | the sum of the digits is divisible by 9 |
| 10 | the last digit is 0 |
| 11 | the sums of the digits in the odd and the even places differ by 0 or a multiple of 11 |
The test for 11, worked — 3,08,605: digits in the odd places from the right, 5 + 6 + 0 = 11; in the even places, 0 + 8 + 3 = 11; the difference is 0, so 11 divides it.
The product rule — a number divisible by two co-prime numbers is divisible by their product: divisible by 3 and by 4 means divisible by 12.
The trap: the rule needs co-primes. A number divisible by 4 and by 6 need not be divisible by 24 — 12 is divisible by both — because 4 and 6 share the factor 2.
संख्या पद्धति — स्थानीय मान, विभाज्यता, अभाज्य और परिमेय
Primes, factors, HCF and LCM
HCF and LCM
Reading a word problem — the greatest length that measures, the largest tile that fits, the most equal groups: HCF. When bells ring together again, when runners meet at the start: LCM.
The trap: HCF × LCM = product holds for two numbers only — it fails for three.
संख्या पद्धति — स्थानीय मान, विभाज्यता, अभाज्य और परिमेय
Rational numbers — the properties that sort the families
A rational number is p/q with integers p, q and q ≠ 0. Every integer, every fraction and every terminating or recurring decimal is rational; π and √2 are not.
| Closed under | Whole numbers | Integers | Rationals |
|---|---|---|---|
| addition | yes | yes | yes |
| subtraction | no | yes | yes |
| multiplication | yes | yes | yes |
| division | no | no | no — division by zero |
The trap: rationals fail closure under division only because of zero — for non-zero divisors the quotient is always rational.
संख्या पद्धति — स्थानीय मान, विभाज्यता, अभाज्य और परिमेय
Points to Remember