Number System
Category : 9th Class
NUMBER SYSTEM
Learning Objectives
Introduction
Numbers are the basic unit of Mathematics, after all, it with numbers that we perform the various functions which constitute Mathematics. For example: Addition, Subtraction,
Multiplication & Division. The Number system is the backbone of any competitive exam. The correct understanding will help you to solve different and complex problems that appear in these examinations. First and for most, let us have a look at the basic classification of numbers and its various kinds.
Classification of Numbers
Natural Numbers
Natural numbers are all of the whole numbers EXCEPT zero. 1, 1, 3, 4, 5, 6, 7, 8, 9, 10, 11…....
They are also called counting numbers. The lowest natural number is 1.
Whole Numbers
Whole numbers are those numbers which start by 0 or we can say when 0 is Included in the list of natural numbers then we call it whole numbers, for example 0, 1, 2, 3, 7, 4, 5.......
Integers
It is the series of both positive and negative numbers lying on the number line. It is the combination of both positive and negative whole and natural numbers.
Rational Numbers
Rational numbers are those numbers that can be written in the form of a ratio of x and y, where the denominator is not zero.
Real Numbers
The number which lies on the number line is a real number .The number can be positive or negative in nature, for example it may be like as 3, 4, 5, 6, -6, -5, -4, -3, -2.....
Irrational Numbers
Irrational numbers are those which are not rational, that is those numbers that cannot be written in the form of a ratio.
Counting Numbers
Counting numbers are those numbers which are well managed on the number line with the difference of 1. The smallest counting number in the number line is 1.
Complex Numbers
Includes real numbers and imaginary numbers are called complex numbers, eg. a + ib.
Prime numbers
The numbers which don't have any factor other than 1 or itself.
For example: 2, 3, 5, 7, 9, 29, 31, 43 .................. or we can say that the numbers which are not divisible by any number are called prime numbers. There are 24 prime numbers between 1 and 100.
List of Prime Numbers
Number Range |
Number of Primes |
1 - 100 |
25 |
101 - 200 |
21 |
201 - 300 |
16 |
301 - 400 |
16 |
401 - 500 |
17 |
501 - 600 |
17 |
601 - 700 |
16 |
701 - 800 |
14 |
801 - 900 |
15 |
901 - 1000 |
14 |
Co-prime Numbers
Two natural numbers are called co-prime numbers if they have no common factor other than
Composite Numbers
All whole numbers that are not prime are composite except for 1 or 0.
The number which is the product of two or more than two distinct or same prime numbers. Example 4, 6, 8, 9, 10, 12, 14, 15, 16, 18, 20, 21............
Some points to remember
Number system at a glance
N |
Natural |
0, 1, 1, 3, 4,... or 1, 2, 3, 4,... |
Z |
Integers |
..., ? 5, ? 4, ? 3, ? 2, ? 1, 0, 1, 2, 3, 4, 5,... |
Q |
Rational |
a/b where a and b are integers and b is not 0 |
R |
Real |
The limit of a convergent sequence of rational numbers |
C |
Complex |
a + bi or a + ib where a and b are real numbers and i is the square root of ? 1 |
Some Properties of Nature Numbers
LCM
LCM of a given set of numbers is the least number which is exactly divisible by every number of the given set.
HCF
HCF of a given set of numbers is the highest number which divides exactly every number of the given set.
Combined Formula of LCM and HCF
Commonly Asked Questions
(a) 8216 (b) 8968
(c) 8316 (d) 8896
(e) None of these
Answer: (c)
Explanation: Product of the numbers \[= LCM\times HCF=9\times 924=8316\]
(a) 15 (b) 13
(c) 17 (d) 19
(e) None of these
Answer (c)
Explanation: \[3965=391\,;\,\,434~\text{ }9425;\]
\[540-13=527\]
\[391=23\times 17\]
\[425=25\times 17\text{ }and\text{ }527=31\times 17\]
Hence the required number will be 17
(a) 24480 (b) 7560
(c) 24485 (d) 7565
(e) None of these
Answer (c)
Explanation: First we have to find the LCM of 32, 45 and 68
\[32=2\times 2\times 2\times 2\times 2\]
\[45=3\times 3\times 5\]
\[68=2\times 2\times 17\]
Therefore required number will be:
\[\left( 2\times 2\times 2\times 2\times 2\times 3\times 3\times 17 \right)+5=24485\]
(a) \[\frac{n+1}{2}\]
(b) \[\left( \frac{n+1}{2} \right)\left( \frac{2n+1}{3} \right)\]
(c) \[\left( \frac{n+1}{2} \right)\left( \frac{2n+1}{3} \right)\]
(d) \[\left( \frac{n+1}{2} \right)-1\left( \frac{2n+1}{3} \right)\]
(e) None of there
Answer: (b)
Explanation: \[Average\,area\,=\frac{{{1}^{2}}+{{2}^{2}}+{{3}^{2}}+.....+{{n}^{2}}}{n}=\frac{\left( \frac{n\,(n+1)(2n+1)}{6} \right)}{n}\]
\[=\frac{(n+1)(2n+1)}{6}=\left( \frac{n+1}{2} \right)\left( \frac{2n+1}{n} \right)\]
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