Number System
Number system is defined as the proper understanding and usage of the numbers in the various places. Numbers are the basic building stones of mathematics. Number system is a way of counting using a particular base.There are different types of numbers exist, we will learn about them in brief at number system.
It includes:
- Ability to find the relative values of a number.
- How to use a number in different arithmetic operations like addition, subtraction, multiplication and division
- Finding the problem solving strategies
Non Positioned Number System: In ancient times, people used to count on their fingers, When fingers became insufficient for counting, other body parts, pebbles, stones or sticks were used to indicate the values. 'Roman number system' is the most common non positioned number system to represent the numbers, for example I, X, L, V (for fifty) and C (for hundred) and so on. By 300 BC, Indians had already begun using symbols to represent 1 to 9. By AD 600, they had invented a place system and zero. Moreover, since it was very difficult to perform arithmetic operations, no logical techniques are used in this system.
The following table compares some ancient number systems
Positioned Number System: The first positioned number system was invented by the Babylonians. They used a base 60 system. Now this system used as decimal system (base 10).
The base-2 number system is also known as binary number system. There are only 2 binary digits in the binary number system: 0 and 1. More specifically, the usual base-2 system is a radix of 2.
The weighted values for each position is determined as follows:
| --- | 2$^6$ | 2$^5$ | 2$^4$ | 2$^3$ | 2$^2$ | 2$^1$ | 2$^{-1}$ | 2$^{-2}$ | 2$^{-3}$ | --- |
| --- | 64 | 32 | 16 | 8 | 4 | 2 | 0.5 | 0.25 | 0.125 | --- |
Base 8 number system is also called as octal number system, and uses the digits 0 to 7. It is one of the alternatives to working with long binary numbers. The digits used in the 'base 8 number system' are: 0, 1, 2, 3, 4, 5, 6 and 7.
In the octal system each place is a power of eight. Let us see with the help of example
To change (436)$_8$ to base 10
(436)$_8$ = 4 $\times$ 8$^2$ + 3 $\times$ 8$^1$ + 6 $\times$ 8$^0$
= 256 + 24 + 6
= (286)$_{10}$
=> (436)$_8$ = (286)$_{10}$ The base tells the number of symbols used in the system and is always expressed in decimal numbers. The base of the decimal system is 10. This implies that there are 10 symbols, 0 to 9. Similarly, the system using three symbols 0, 1, 2 will be of base 3, four symbols will be of base 4, five symbols will be of base 5 and so on. The base of a number system is indicated by a subscript and this will be followed by the value of the number.
| Number System | Base value | Set of digits | Example |
| Base 3 number system | 3 |
0, 1, 2 | (123)$_3$ |
| Base 4 Number System | 4 |
0, 1, 2, 3 | (145)$_4$ |
| Base 5 Number System | 5 |
0, 1, 2, 3, 4 | (425)$_5$ |
| Base 6 Number System | 6 |
0, 1, 2, 3, 4, 5 | (225)$_6$ |
| Base 7 Number System | 7 |
0, 1, 2, 3, 4, 5, 6 | (1205)$_7$ |
| Base 8 Number System | 8 | 0, 1, 2, 3, 4, 5, 6, 7 | (105)$_8$ |
| Base 9 Number System | 9 |
0, 1, 2, 3, 4, 5, 6, 7, 8 | (25)$_9$ |
| Base 10 Number System | 10 | 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 | (1125)$_{10}$ |
The rounded number value will be same as the original number value but it will less exact. It is an approximate value of the original number.
Rules:
- If the number we are rounding is followed by number greater than or equal to 5, then round the number up.
- If the number we are rounding is followed by number less than 5, then round number down.
Example 1:
Round the number 46 to nearest ten
Solution:
Here 6 is greater than 5, so it is rounded up
46 is rounded to near ten digit 50
Example 2:
Round the number 43 nearest ten
Solution:
Here 3 is less than 5, so it is rounded down
43 is rounded down to near ten digit 40
Example 3:
Round the number 1975 to nearest thousand
Solution:
Here it is rounded to 2000
Here, the numbers are rounded and then multiplication operation is carried on.
Example1:
62 x 56
solution :
Rounded value of 62 is 60
Rounded value of 56 is 60
Product = 60 x 60
Product = 3600
Example 2:
25 x 9
solution:
Rounded value of 25 is 30
Rounded value of 9 is 10
Product = 30 x 10
Product is 300
The same procedure is also carried out in decimals.
Here, first round up the given numbers and then division of numbers are performed.
Example 1:
$\frac{42}{18}$
Solution:
Rounded value of 42 is 40
Rounded value of 18 is 20
Now dividing the rounded numbers
$\frac{40}{20}$
= 2
Example 2:
$\frac{49}{6}$
Solution:
Rounded value of 49 is 50
Rounded value of 6 is 10
Now dividing the rounded numbers
$\frac{50}{10}$
= 5
The same procedure is carried out for decimal rounding also.
Number system is a wide topic in mathematics and new research has been held based on number system frequently.
Here, we have to round the number and we have to perform the addition.
Example 1:
33 + 56
Solution:
Rounded value of 33 is 30
Rounded value of 56 is 60
Sum of the numbers = 30 + 60 = 90
Example 2:
23 + 34
Solution:
Rounded value of 23 is 20
Rounded value of 34 is 30
Sum = 20 + 30 = 50