CBSE Class 8 Mathematics Number System Notes Set 09

Types of Numbers

Natural numbers N = {1, 2, 3, ...}; whole numbers W = {0, 1, 2, 3, ...}; integers I = {..., -2, -1, 0, 1, 2, ...}.
A prime number is greater than 1 and has no divisors other than 1 and itself; a composite number is greater than 1 and not prime. Even numbers have the form 2n.
A rational number can be written as p/q with p, q integers and q not zero; an irrational number cannot (for example the square root of a non-square natural number). Rational and irrational numbers together form the real numbers.

VBODMAS

To simplify an expression, remove the vinculum (bar) first, then the brackets in the order ( ), { }, [ ], and then perform of, division, multiplication, addition and subtraction in that order.

Sums of Counting Numbers

\( 1+2+\cdots+n=\frac{n(n+1)}{2} \); sum of first n odd numbers = \( n^2 \); sum of first n even numbers = \( n(n+1) \).
\( 1^2+2^2+\cdots+n^2=\frac{n(n+1)(2n+1)}{6} \) and \( 1^3+2^3+\cdots+n^3=\left[\frac{n(n+1)}{2}\right]^2 \).

Divisibility Rules

By 2: last digit even. By 3 or 9: sum of digits divisible by 3 or 9. By 4: last two digits divisible by 4. By 5: last digit 0 or 5. By 6: divisible by both 2 and 3.
By 7: subtract twice the last digit from the rest and repeat. By 8: last three digits divisible by 8. By 10: last digit 0. By 11: difference between the sum of digits at odd places and at even places is 0 or a multiple of 11.

Operations on Rational Numbers

Addition and subtraction: bring to a common denominator, then add or subtract numerators. Multiplication: \( \frac{a}{b}\times\frac{c}{d}=\frac{ac}{bd} \). Division: multiply by the reciprocal, \( \frac{a}{b}\div\frac{c}{d}=\frac{a}{b}\times\frac{d}{c} \).
The additive identity is 0 and the additive inverse of a is -a. The multiplicative identity is 1 and every non-zero number has a reciprocal; 0 has none.

Standard Form, Equality and Order

A rational number p/q is in standard form when q is positive and p and q have no common factor other than 1.
Two rational numbers a/b and c/d are equal when ad = bc. To compare, write them with the same positive denominator and compare numerators; every positive rational is greater than every negative rational.

Rational Numbers Between Two Numbers

Infinitely many rational numbers lie between any two rational numbers. If a < b, the number (a + b)/2 lies between them; repeating the process gives more. To insert n rational numbers, write both with a common denominator large enough to leave at least n numerators between them.

Worked Example

Find the other number if the sum of two rational numbers is -3/5 and one of them is -9/20.
Other number = -3/5 - (-9/20) = -12/20 + 9/20 = -3/20.

Worked Example

Simplify (32 + 2 x 17 + (-6)) / 15.
Multiplication first: 2 x 17 = 34. Then 32 + 34 - 6 = 60, and 60 / 15 = 4.