Decimal Number System

The decimal number system can be defined as a ten-digit number system that we use on a daily basis. The notation of numbers using digits or symbols is referred to as a number system in mathematics. The binary number system, decimal number system, octal number system, and hexadecimal number system are the four primary forms of number systems. Because it was difficult to multiply and divide big numbers by hand in ancient civilizations, the decimal number system is sometimes known as the Hindu-Arabic or Arabic number system. Let’s take a closer look at the decimal number system.

The decimal number system is the one we use on a daily basis, and it uses digits ranging from 0 to 9, i.e. 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9. Because the entire number allowed in this number system is 10, the decimal number system’s base number is 10. Any number that is represented without a base has a base of 10. For instance: 7310,13210,526710 are some examples of decimal numbers.

Rules on the Decimal System

When establishing a decimal number system, we always write it in base 10, with 0 or the first nine positive integers denoting each value. Each value has a power of ten place value, meaning that the digit in the tens place is 10 times bigger than the digit in the one’s place. Here are a few pointers or rules to keep in mind when writing in the decimal system.

The numbers in the decimal number system range from 0 to 9.

When we get to 9, we make the rightmost number 0 and add 1 to the left, resulting in 10.

When we reach a digit with a 9 in it, we always add 1 to make the number go up to the next.

In the decimal number system, each number has a power of ten place value. For a better understanding, consider the following scenario:

(134)10 = 1×102 + 3×101 + 4×100       ,(78)10 = 7×101 + 8×100 

 In the decimal number system, a number with a decimal point is stated in the descending power of 10 after the decimal point. As an example,

(24.5)10 = 2×101 + 4×100 + 5×10-1 

Conversion of Other Number Systems to the Decimal System

The binary number system, octal number system, decimal number system, and hexadecimal number system are the four types of number systems. Each of them has its own set of base numbers that aid in the conversion of numbers from one system to the next. Let’s look at how to convert binary, octal, and a

Converting from binary to decimal

By expressing each digit as a product of the provided number 1 or 0 to the respective power of 2, the binary number can be transformed to a decimal number. The binary number system’s base number is 2, which is converted to 10 after conversion.  if a binary number contains n digits, A= an-1 . . . . .a3 a2 a1a0., the decimal number for it is given as, D=( (a)0 x 20 ) + ((a)1 x 21) + ( (a)2 x 22) + ….

For example : convert binary number (11101)2 to its decimal form 

Given binary number as (11101)2 .

We have to multiply each binary digit with the decreasing power of 2 and add the products.

= (1 x 24) + (1 x 23) + (1 x 22) + (0 x 21) + (1 x 21)

= 1 x 16 + 1 x 8 + 1 x 4 + 0 x 2 + 1 x 2

= 16 + 8 + 4 + 0 + 2

= 30

Therefore, (11101)2 = (30)10

Converting Octal to Decimals

The octal base number, which is 8, is used to convert octal numbers to decimal values. The number is expanded using the base of 8, with each number being multiplied by the decreasing power of 8 before being added to get the decimal number. After conversion, the decimal number system has a base of ten.

Convert an octal number, for example. (238)8 converting it to decimal form

(238)8 = 2 x 82 + 3 x 81 + 8 x 80

    = 2 x 64 + 3 x 8 + 8 x 1

    = 128 + 24 + 8

    = 160

Therefore, (238)8 = (160)10

Converting Hexadecimal to Decimal

The base number of hexadecimals is 16, and the number 16 is used to convert hexadecimal to decimal. The decimal number is obtained by multiplying each integer by the decreasing power of 16 and then adding them together. After conversion, the decimal number system has a base of ten.

Consider the following scenario: Convert a hexadecimal number to a decimal number (12)16 converting it to decimal form

(12)16 = 1 x 161 + 2 x 160

  = 1 x 16 + 2 x 1

 = 16 + 2

= 18

Therefore, (12)16 = (18)10

Conversion of Decimal Numbers into Other Number Systems

The process of converting a decimal number to another number system is identical to the process of converting any number system to a decimal number system. For conversion, each of the base numbers is necessary. Let’s have a look at some of the conversions.

Converting Decimal to Binary

To convert a decimal number to binary, divide the number by two until the quotient equals zero. During the division procedure, we set aside the leftovers. When the quotient equals zero, we write the residual together with the last numbers starting at the bottom and working our way up to get the binary number.

Converting Decimal to Octal

To convert decimal to octal, divide the decimal value by 8 and add the reminder from the previous digit. The least significant digit (LSD) is the first remainder, and the most significant digit (MSD) is the last remainder (MSD). When the quotient is less than 8, we write the remainder in reverse order to get the octal number. Let’s look at an example to better grasp the conversion.

Converting Decimal to Hexadecimal

The conversion from decimal to hexadecimal works similarly to the other two number systems. Because hexadecimal’s base number is 16, the number must be divided by 16 until the quotient reaches zero. A hexadecimal number system uses just digits from 0 to 9 and alphabets such as A, B, C, D, E, and F for numbers 10 to 15. 

Conclusion

Here we come to know that the Number System represents the value of a number in relation to its basis. A number has a unique representation based on its Base value, and different number systems have different representations of the same number.