Decimal Number System
2026-02-28 17:59 Diff
https://brightchamps.com/en-us/math/numbers/decimal-number-system

 We use only two digits in the binary number system (0 and 1), and its base is 2. So, to convert a binary number to a decimal number, we must multiply every digit of the binary number by a power of 2. The exponent of 2 depends on the position of the binary number. The rightmost digit is multiplied by \(2^0\), the next digit is multiplied by \(2^1\), and so on. After the multiplication process is done, we add up the results to get the converted value. 

Multiply each digit of the binary number by 2 raised to the power of its position from right to left, starting at 0. Then add the results together to get the decimal number. 

The given binary number is \((1011)_2\).

\({ (1011)_2} = (1 × {2^3}) + (0 × {2^2}) + (1 × {2^1}) + (1 × {2^0})  \)
\( = (1 × 8) + (0 × 4) + (1 × 2) + (1 × 1)  \)             
\( = 8 + 0 + 2 + 1 \)
\(= 11\)

Thus, \({(1011)_2} = {(11)_{10}}\)

Octal to Decimal Conversion:

The octal number system has a base of 8. It has 8 digits, from 0 to 7, to represent numbers. To convert an octal number to a decimal number, multiply each digit by the decreasing power of 8 and add the products. Let us convert (167)8 to its decimal form. 

\({(167)_8} = {(1 × 8^2)} + ({6 × 8^1)} + {(7 × 8^0)} \)
\(= (1 × 64) + (6 × 8) + (7 × 1) \)
\(= 64 + 48 + 7  \)
\(= 119 \)

Thus, \({(167)_8 }= {(119)_{10}} \)
After the conversion, the base power changes from 8 to 10.

Hexadecimal to Decimal Conversion:

The hexadecimal number system uses 16 symbols: digits from 0 to 9 and letters from A to F. The conversion of a hexadecimal number to a decimal number happens when we multiply each digit of the hexadecimal number by the powers of 16. Once again, the rightmost digit will be multiplied by 160, the next digit by 161, and so on. After multiplying the digits by the powers of 16, we should add the results to obtain the converted value. 

\((18)_{16} = (1 × 16^1) + (8 × 16^0)\)

\(= (1 × 16) + (8 × 1) \)
\(= 16 + 8  \)
\(= 24 \)


Thus, \({(18)_{16}} = {(24)_{10}}\)