The number system is defined as the use of numbers and symbols in arithmetic and algebraic expressions. It is used in arithmetic operations like addition, subtraction, multiplication, and division, and also covers Arithmetic and geometric progression. It is also used in simplification by using VBODMAS where V stands for vinculum, B stands for the bracket, O stands for of, D stands for division, M stands for multiplication, A stands for addition, and S stands for subtraction.
Types of Numbers:-
There are various types of numbers in the number system. Detail explanation is given below:
- Integers:- The numbers which are positive and negative, including 0 are termed Integers. -3,-2,-1,0,1,2,3,4,5 are integers.
- Natural Numbers:- Numbers that start with 1,2,3……so on are called Natural Numbers. Zero, negative numbers, and decimals are not counted as natural numbers.
- Whole Numbers:- Numbers start with zero (0,1,2,3…….) all whole numbers. All the natural numbers are whole numbers, but all the whole numbers are not natural numbers.
- Rational Numbers:- The numbers which are repeating, terminating, and can be written in the form of p/q form where p and q are integers and q shouldn’t be zero are called Rational Numbers. For example 0.676767
- Irrational Numbers:- The numbers which are non-repeating, non-terminating, and can’t be written in the form of p/q form where p and q are integers and q shouldn’t be zero are called Irrational Numbers. For example 3.14159…
- Prime Numbers:- The numbers which are divisible by 1 and the number itself are called Prime Numbers. For example: 2,3,5,7,11,13,17 etc.
- Composite Numbers:- The numbers which have more than two divisors of the numbers other than prime numbers are called Composite Numbers. For example: 4,6,8,9,10,12,14,15 etc.
- Co-prime Numbers:- The pair of numbers which do not have common factors except 1 are called Co-prime Numbers. For example, 3 and 5, 7 and 11, etc.
- Complex Numbers:- The numbers which are formed by a real number and an imaginary number are called Complex Numbers. The Standard Form of a complex number is A+İB where A is a real number and B is an imaginary number.
Some Important Formulas of Number System:-
- Sum of first n natural numbers:- n(n+1)/2
For example:- 1+2+3+………..+99 = 99(99+1)/2 = 4950
- Sum of first n even numbers:- n(n+1)
For example:- 2+4+6+………….+56 (28th even number) = 28(28+1) = 28*29 = 812
- Sum of first n numbers:- n2
For example: 1+3+5+………+13 = 72 = 49
- Sum of squares of first n natural numbers:- n(n+1)(2n+1)/6
For example:- 12 + 22 +…….+82 = 8*9*17/6 = 204
- Sum of cubes of first n natural numbers:- (n(n+1)/2)2
For example:- 13 + 23 +…….+73 = (7*8/2)2 = 789
Relation Between Divisor, Dividend, Quotient and Reminder in a Division Sum:
Dividend = Divisor*Quotient + Reminder
Questions Based on Number System:
1. When an integer m is divided by 6, the remainder is 4. What is the remainder when 7m is divided by 3?
Solution:- Let integer m = 10, because it satisfies the first statement of the question i.e 70/6 we get remainder 4
⇒ 7m = 7*10 = 70
now divided by 3:- 70/3 = Remainder:- 1
option d) is correct.
Alternate method (shortcut approach):- 4/3, we get remainder 1.
2. When x is added to each of 8,14,20 and 30 the numbers obtained are in proportion. what is the mean proportional between the numbers (4x-4) and (8x+1)?
Solution:- (8+x)(30+x) = (14+x)(20+x)
⇒ 240 + 38x = 280 + 34x
⇒ 4x = 40
x = 10
Mean proportion:- √(40-4).(80+1) = √(36×81) = 6×9 = 54
option (c) is correct answer.
3. Find the sum of 2+4+6+………….+38?
⇒ Sum of first 19 natural numbers:- 19×20/2 = 190
⇒ Sum:- 2×190 = 380
option (b) is correct answer.
4. Find the number of prime factors in expression (35)6 × (38)4 ?
⇒ 356 = 56 × 76
⇒ 384 = 24 × 194
Total number of prime factors in expression:- 6+6+4+4 = 20
option c) is the correct answer.
5. In number 235749, what is the face value of the numeral 5?
Solution:- The face value of 5 is 5000 as the numeral is at the thousands place.
Option a) is the correct answer.
6. In 87659_21 what is the least number which can be filled in blank so that the number is divisible by 11.
Solution:- divisibility rule of 11:- difference between the sum of odd places and the sum of even places is divisible by 0,11,22…
Let number:- x
(8+6+9+2) – (7+5+x+1) = 0,11,22
25–13–x = 0,11,22
12–x = 0,11
x = 12–11
x = 1
Hence, option a) is correct.
7. M is a prime number and (M²+3) is also a prime number. How many numbers that M can assume?
Solution:- Only one value M can assume, which is 2.
At M=2, M²+3 = 7, which is also prime.
Again at M = 3,5,7,11……. M²+3 is an even number, which can’t be prime.
Hence, option d) is correct.
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