0, 1, 2, 3, ..... 8, 9 are called digits.
"0" is insignificant digit and others are significant digits.
A group/set of digits is called a Numeral i.e. Number.
Place value or local value of a digit in a numeral:
In the numeral 98765432 we have.
Place value of 2 is 2 units = 2.
Place value of 3 is 3 tens = 30.
Place value of 4 is 4 hundreds = 400 and so on.
Face value: The face value of a digit in a numeral is the value of the digit it self at whatever place it may be. [i.e irrespective of its place]
Natural Numbers: The numbers which are used for counting are called natural numbers, and denoted by N. N = {1,2,3,....∞}.
1 is the least natural number and the greatest natural number does not exist.
Even Numbers: The integers which are exactly divisible by 2 are called even numbers. Ex: ..., -8, -6, -4, -2, 2, 4, 6, 8, 10, 12, ....∞ i.e starting from 2 every alternate number is even.
Odd Number: The integers which are not divisible by 2 are odd numbers. Ex: 1, 3, 5, 7, 9, 11 ... ∞.
Whole Numbers: Set of All Natural numbers including ‘0’ are called whole numbers. Denoted by ‘W’. W = {0, 1, 2, 3, ......∞}.
0 is the least whole number and the greatest does not exist.
Integers: Set of all whole numbers including negative numbers are called Integers, and are denoted by ‘I’ or ‘Z’. Z = { ….-3 , -2, -1, 0, 1,2,3,….∞}
Least and greatest integers does not exist.
Rational Number: A number which can be expressed in the form of p/q where q ≠0 is called a rational number and denoted by Q. Ex. 1/2, 1/4, 1/3, 6/5, .....∞
Irrational Numbers: A number which can’t be expressed in the form of p/q is called irrational number and denoted by ’Q’ Ex: $\displaystyle \sqrt{2}, \sqrt{3}, \ \sqrt{15}$, π.
Real Numbers: Numbers including both rational and irrational numbers are called real numbers and denoted by ‘R’.
Prime Number: A natural number other than 1, which has factors only 1 and itself. Ex: 2,3,5,7,11,13 ......
2 is the only even prime and all the other primes are odd.
Up to 100 there are 25 prime numbers, up to 1000 there are 168 prime numbers.
Composite Numbers: Natural numbers other than prime are called composite numbers. Ex: 4,6,8,9 ......
1 is neither prime nor composite.
A composite number has at least 3 factors.
Twin primes: Twin primes which are differ by 2 are called twin primes. Ex : 3,5; / 5,7 / 11,13 / 17,19 / 29,31 / 41,43 etc.
Co- prime: Two numbers having no common factors other than 1 are called co-primes or relatively prime to each other., or HCF of that two numbers is 1. Ex: 2,3; 4,9 etc.
Perfect Number: If the sum of all the factors of a number is twice the number itself is called a perfect number. Ex: 6, 28, 496 etc.
A number is divisible by:
1. By 2 => If the last digit is either 0 or even.
2. By 3 => If the sum of the digits is divisible by 3
3. By 4 => If the number so formed by the last two digits is divisible by 4.
4. By 5 => If the last digit is either 0 or 5.
5. By 6 => If it is divisible by both 2 and 3.
6. By 7 => double the units place digit and subtract it from the remaining part of the number if the difference is 0 or the difference is divisible by 7 then that number is also divisible by 7.
7. By 8 => If the number so formed by the last three digits is divisible by 8.
8. By 9 => If sum of the digits is divisible by 9.
Or if the digits of a number is reversed and subtracted from the number the difference that arrived is divisible by 9.
9. By 10 => If the last digit is Zero.
10. By 11 => If the difference of sums of the alternate digits of the number is either ‘0’ or a multiple of 11.
Or => A number is divisible by 11, if the difference of the sum of its digits at odd places and The sum of its digits at even places, is either ‘0’ or a number divisible by 11.
11. By 12 => Any number which is divisible by both 3 and 4, is also divisible by 12.
12. By 14 => Any number which is divisible by both 2 and 7, is also divisible by 14.
13. By 15 => Any number which is divisible by both 3 and 5, is also divisible by 15.
Sum or difference of two even numbers is Even.
[Even – Even = Even] or [Even + Even = Even].
Ex: 8 - 18 = 10, 8 + 18 = 26 [both are Even].
Sum or difference of two Odd numbers is Even.
[Odd + Odd = Even] [Odd – Odd = Even].
[15 – 5 = 10] [33 – 15 = 18] both 10, 18 are even.
Sum or difference of one Even one Odd is Odd.
[odd + even = odd] ex: 4 + 5 = 9 [odd].
Product of two even numbers is Even.
Even x Even = Even. Ex: 8 x 8 = 64.
Product of two odd numbers is Odd.
Odd x Odd = Odd. Ex: 9 x 9 = 81.
Product of one even and one odd is Even.
Even x Odd = Even, Ex: 8 x 15 = 120.
Sum of difference between one Odd and one Even is Odd.
[Odd + Even = Odd] [Odd – Even = Odd].
Let ‘a’ and ‘b’ are two integers and if ‘a’ is exactly divisible by ‘b’ then b is a factor of ‘a’, and, if a is multiplied with c then ‘ac’ is also divisible by ‘b’
The number to be divided is called Dividend, the number that divides the other number is called Divisor, and the result obtained is called Quotient and un divided part is called Remainder.
Dividend = Divisor x Quotient + Remainder.
Divisor = $\displaystyle \frac{\text{Dividend} - \text{Remainder}}{\text{Quotient}}$
Progression is sequence of numbers, and each term differs by a constant term [d] ‘n’th term = Tn = a + (n – 1).
Sum of the terms Sn = $\displaystyle \frac{n}{2} \left(2a + (n – 1)\right) $ or $\displaystyle \frac{n}{2} \left(\text{First term} + \text{First term}\right) $
Sum of Natural numbers = $\displaystyle \frac{n(n + 1)}{2}$
Ex: sum of 1,2,3,4,5,6 = n = 6 sum = $\displaystyle \frac{6(6 + 1)}{2} = \frac{42}{2}$ = 21.
Sum of Even numbers = n( n + 1).
Ex: sum of 2,4,6,8,10 = n = 5, sum = 5(5 + 1) = 5 x 6 = 30.
Sum of Odd numbers = n2
Ex: sum of 1,3,5,7,9,11 = n= 6, then sum = 62 = 36.
Sum of the squares of Natural numbers = (12 + 22 + 32 + .... + n2) = $\displaystyle \frac{n(n + 1)(2n + 1)}{6}$
Sum of the Cubes of Natural numbers = (13 + 23 + 33 + ... + n3) = $\displaystyle \frac{n^2(n + 1)^2}{4}$
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