A numeral system is a system in which numbers of a given set are represented by mathematical notations. These are the basic building stones of mathematics. This system provides a unique representation to every number and reflect the arithmetic and algebraic structure of the numbers. It not only gives unique identity to numbers, but also allows to use a number in different arithmetic operations like addition, subtraction, multiplication and division.

Various Number systems are mentioned below:

1. Decimal number system (Base-10 )

2. Binary number system (Base-2)

3. Octal number system (Base-8)

4. Hexa decimal number system (Base-16)This is a wide topic in mathematics and new research has been done based on this topic.

A number system defines a set of values that is used to represent quantity. It is the way a society records and communicates ideas about numbers. It consists of a set of elements, and basic operations. This topic have been around for thousands of years. If we talk about pre-history, the first method of counting was counting on fingers, later tallies made by carving notches in bone, stone and wood. This counting method was used for 40 thousand years. However, it can be classified into two categories as follows:

Non Positioned : 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 system to represent the numbers.

For example

V = For 5

X = For ten

L= For 50

C = For hundred and so on. By 300 BC, people started using symbols to represent 1 to 9. By AD 600, they had invented a place system and zero.Since, it was very difficult to perform arithmetic operations as no logical techniques are used in this system.

The following table compares some ancient number systems:

**Positioned : **First, it** **was invented by the Babylonians. They used a base 60 system. Now, this is used as decimal system (base 10).

It is also known as binary number system. There are only 2 binary digits: 0 and 1.

More specifically, the usual base-2 system is a radix of 2. Numbers represented in this system are commonly known as binary numbers. Binary number is only combination of 0 and 1. There is no place for 2, 3, 4,....9. For example, 100101 is a binary number.

The weighted values for each position is determined as follows:

We can easily convert any system into binary and vice versa. Lets see with the help of an example.

Example : Write $(14)_{10}$ as binary number.

Solution:

It is also called as octal number system, and uses the digits 0 to 7. It is one of the alternatives that works with long binary numbers. The digits used in this are: 0, 1, 2, 3, 4, 5, 6 and 7. In the octal system each place is a power of eight. Check the conversion process from octal to decimal through examples.

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)$_{10}$ This system 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 this system is indicated by a subscript and this will be followed by the value of the number.

2. If the number we are rounding is followed by number less than 5, then round number down.

Non Positioned :

For example

V = For 5

X = For ten

L= For 50

C = For hundred and so on. By 300 BC, people started using symbols to represent 1 to 9. By AD 600, they had invented a place system and zero.Since, it was very difficult to perform arithmetic operations as no logical techniques are used in this system.

The following table compares some ancient number systems:

It is also known as binary number system. There are only 2 binary digits: 0 and 1.

More specifically, the usual base-2 system is a radix of 2. Numbers represented in this system are commonly known as binary numbers. Binary number is only combination of 0 and 1. There is no place for 2, 3, 4,....9. For example, 100101 is a binary number.

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 | --- |

We can easily convert any system into binary and vice versa. Lets see with the help of an example.

Example : Write $(14)_{10}$ as binary number.

Solution:

It is also called as octal number system, and uses the digits 0 to 7. It is one of the alternatives that works with long binary numbers. The digits used in this are: 0, 1, 2, 3, 4, 5, 6 and 7. In the octal system each place is a power of eight. Check the conversion process from octal to decimal through examples.

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)$_{10}$ This system is always expressed in decimal numbers.

Number System | Base value | Set of digits | Example |

Base 3 | 3 |
0, 1, 2 | (123)$_3$ |

Base 4 | 4 |
0, 1, 2, 3 | (145)$_4$ |

Base 5 | 5 |
0, 1, 2, 3, 4 | (425)$_5$ |

Base 6 | 6 |
0, 1, 2, 3, 4, 5 | (225)$_6$ |

Base 7 | 7 |
0, 1, 2, 3, 4, 5, 6 | (1205)$_7$ |

Base 8 | 8 | 0, 1, 2, 3, 4, 5, 6, 7 | (105)$_8$ |

Base 9 | 9 |
0, 1, 2, 3, 4, 5, 6, 7, 8 | (25)$_9$ |

Base 10 | 10 | 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 | (1125)$_{10}$ |

The rounded number value will be same as the original value, but it will less exact. It is an approximate value of the original number. A number can be rounded to any place value. Values can be rounded to the nearest ten, the nearest hundred, the nearest thousand, and so on. There are various methods for rounding and here, we are discussing very common rules.

**Rules:**

2. 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.

Rounding a number means replacing it by another value that is approximately equal, but it has a more or less representation. There are very simple rules to round the numbers. And new rounded numbers are easy to multiply. Here, the numbers are rounded and then multiplication operation is carried on.

** Example 1 : **Round and solve 62 x 56

**Solution :**

Rounded value of 62 is 60

Rounded value of 56 is 60

Product = 60 x 60

Product = 3600

** Example 2:** Find the approximate value of 26 x 9.

**Solution:**

The product of 26 and 9 is about 300.

Because 26 rounded to the nearest 10 is 30

And 9 rounded to the nearest 10 is 10

=> 30 x 10 = 300.

The same procedure is also carried out in decimals.

This operation is similar as "rounding and multiplication". The only difference is that we need to divide dividend by divisor. There are two methods to round and divide the numbers:Method 1: First round then divide the numbers.

Method 2: The basic method for determining a division remainder is to determine the quotient of the division of the dividend and divisor, and then round that value to the nearest integer. The remainder is then the difference of the quotient multiplied by the divisor, and the dividend.

**Example 1:**

Find decimal quotients then round to a given place value.

$\frac{42}{18}$

**Solution:**

=> $\frac{42}{18}$ = 2.333

2.333 rounded to the nearest 10 is 2

The result shows that the rounding numbers are now rounding the values down to the nearest integer.

**Example 2:**

Find the approximate value of $\frac{49}{6}$

**Solution:**

Rounded value of 49 is 50

Rounded value of 6 is 10

Now dividing the rounded numbers

$\frac{50}{10}$ = 5

Rounding is a way to reduce the digits in a number while keeping its value almost same. Rounding is done with the purpose to obtain a value that is easier to write and operate than the original value. In order to find round sum, it is best to round the numbers before adding them.

**Example 1:**

Solve 33 + 56

**Solution:**

Rounded value of 33 is 30

Rounded value of 56 is 60

Sum of the numbers = 30 + 60 = 90

**Example 2:**

Round and add: 23, 34

**Solution:**

Rounded value of 23 is 20

Rounded value of 34 is 30

Sum = 20 + 30 = 50

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