Basics of Math
Numbers expressed using figures -- 1, 2, 3, 4, 5, 6, 7, 8, 9, and 0 are called digits. Out of these, 0 is called the ‘in significant’ digit whereas the others are called significant digits.
Numerals: A group of figures, representing a number, is called a numeral. Numbers are divided into the following types:
Natural Numbers
Numbers are used for counting the object. Natural numbers are represented by N
N = {1, 2, 3, 4, 5, ….}
Whole Numbers
When we include zero in the natural numbers, it is known as whole numbers. Whole numbers are denoted by W.
W = {0, 1, 2, 3, 4…….}
Prime Numbers
A number other than 1 is called a prime number if it is divisible only by 1 and itself.
Composite Numbers
A number, other than 1 which is not a prime number is called a composite number.
e.g. 4, 6, 8, 9, 10, 12, …etc.
Even Numbers
The number which is divisible by 2 is known as an even number.
e.g . 2, 4, 6, 8, 10, …. etc.
It is in the form 2n (where n is a whole number)
Odd Numbers
A number that is not divisible by 2 is known as an odd number.
e.g. 3, 9, 11, 17, 19, … etc.
Consecutive Numbers
A series of numbers in which each is greater than its predecessor by 1, is called consecutive numbers.
e.g. 6, 7, 8, or 13, 14, 15 or 101, 102, 103.
Integers
The set of numbers that consists of whole numbers and negative numbers is known as integers. It is denoted by me.
e.g. I = {-4, -3,-2, -1, 0, 1, 2, 3, 4}
Rational numbers
When the numbers are written in fractions, they are known as rational numbers. They are denoted by Q. e.g. rational numbers. Or, the numbers which can be written in the form a/b (where a and b are integers and b is not equal to 0) are called rational numbers.
Irrational Numbers
The numbers which cannot be written in form of p/q are known as irrational numbers (where p and q are integers and q not equal to 0).
Real Numbers
Real numbers include both rational as well as irrational numbers.
Rule of signification
(i) In simplifying an expression, first of all, the vinculum or bar must be removed. For example, we know that -8 -10 = -18 but if bar is mentioned in the above of this equation than -8 -10=2
(ii) After removing the bar, the brackets must be removed, strictly in the order (), {}, [].
(iii) After removing the brackets, we must use the following operations strictly in the order given below:
(a) of
(b) division
(c) multiplication
(d) addition and
(e) subtraction.
Note: The rule is known as the rule of ‘VBODMAS’, where V, B, O, D, M, A, and S stand for Vinculum, Bracket, Of, Division, Multiplication, Addition and Subtraction.
General Rules for solving problems in Arithmetic
(1) (a + b)(a - b) = a2 - b2
(2) (a + b)2 = a2 + 2ab + b2
(3) (a - b)2 = a2 - 2ab + b2
(4) (a + b)3 = a3 + b3 + 3ab(a + b)
(5) (a - b)3 = a3 - b3 - 3ab(a - b)
(6) a3 + b3 = (a + b)(a2 - ab + b2)
(7) a3 - b3 = (a - b)(a2 + ab + b2)
Frequently Asked Questions
VBODMAS can be defined by each letter in the term VBODMAS where V, B, O, D, M, A, and S stand for Vinculum, Bracket, Of, Division, Multiplication, Addition, and Subtraction.
There are lots of chapters and concepts used in math that have different applications, to become experts in math students must focus on concepts and their application by solving the numerical. Important chapters of Math deepened on class, every class and grade has different chapters which are important. But one must give more time to solving the basics of math and arithmetic which are used in all concepts of Maths.
The rule I: The quotient of two integers both positive or both negative is a positive integer equal to the quotient of the corresponding absolute values of the integers.
Rule II: The quotient of a positive and negative integer is a negative integer and its absolute value is equal to the quotient of the corresponding absolute values of the integers.
(i) If a and b are integers, then a Divide b is not necessarily an integer.
(ii) If a is an integer different from 0, we have a divide 1 = a.
(iii) If for every integer a, we have a divide 1 = a.
(iv) If a is a non-zero integer, then 0 Divide a = 0.
(v) If a is an integer, then a Divide 0 is not defined ()
(vi) If a, b, and c are non-zero integers, then
(i) a > b a Divide c > b Divide c, if c is positive.
(ii) a > b aDivide c < bDivide c, if c is negative.