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Binary numbers are easy to calculate and perform numerical computation. A computer understands binary numbers which are in form of bits of 0 and 1 and performs all the arithmetic and logical calculations using these binary digits. We though decipher numbers in decimal format a computer understands binary therefore a computer converts decimal numbers to binary and then perform the calculations and then again the binary results are converted to decimal for the user to understand and interpret.

There are various methods through which we can convert a binary number to decimal yet the most simple format or method is by placing the decimal numbers in a table which corresponds to the binary equivalent. To elaborate a table is drawn from left to right in the form of power of digit 2 starting with 1. So the right most column of the table has the digit 1 and then to its left the digit 2 then follows 4, 8, 16, 32, and 64 and so on and so forth.

Just beneath this decimal number table we then place the binary numbers. The decimal number which corresponds to 1 is written separately and the decimal number which corresponds to 0 is dropped. Then the total of all the decimal numbers is made to finally find the decimal number for the binary digit.

*Let us further explain this with an example.*

Say we have to convert a binary digit 10001110 into decimal.

For this conversion first we would place all the binary digits into a table.

128 | 64 | 32 | 16 | 8 | 4 | 2 | 1 |

1 | 0 | 0 | 0 | 1 | 1 | 1 | 0 |

So here we observe that first we begin from the extreme right of the binary digit 10001110 and we take 0 and place it to the extreme right of the table corresponding to 1. Then we take the next binary digit 1 and place it corresponding to 2, and then we again take 1 and place it beneath 4 and continue this sequence till we place the beginning digit of the binary and place it corresponding to 128.

Now the next step would be to consider all those decimal numbers which correspond to 1 of the binary digit series.

In this example the decimal numbers which correspond to binary digits are;

128, 8, 4 and 2.

Rest all the numbers correspond to binary 0 so we do not consider them.

Now to find the binary equivalent of the binary digit 10001110, we would finally add the four decimal numbers we selected from the conversion table.

128+8+4+2 = 142

So it can be finally deduced that the binary equivalent of 10001110 will be 142.

Thus it is very clear from this example that to convert a binary digit into a decimal number format is very simple and convenient. All the user has to do is to place the binary number into the conversion table and ascertain the value for each binary digit. The decimal number which corresponds to 1 is selected and the decimal number which corresponds to 0 is dropped.

The numeric system that we use in general is known as the Decimal System. The decimal system uses 10 as its base. The number 10 is used as the base, since any given number is a combination of digits ranging from 0 to 9 (10 digits). The value of the digits is assigned as per their relative position in the number. This place value increases in the multiples of 10 as we go from right hand side towards the left. Hence, every digit can be represented as a multiple of 10 with an appropriate power. As a general rule, any number with the power of 0 is always 1. For example, the number 5269 can be represented as:

(5×10^{3}) + (2×10^{2}) + (6×10^{1}) + (9×10^{0}) = 5269

As we can see in the above example, each digit is multiplied by 10 and assigned an appropriate power according to its position in the number which increases as we go from right towards left.

As the base is 10 for Decimal Numbers, similarly the base is 2 for the Binary system. The Binary system uses only ‘1′ or ‘0′ to represent all numbers. Since we use only ‘1′ and ‘0′ (two digits), 2 acts as the base for binary numbers. The concept of place value in the binary system is very similar to that of Decimal system. The place value of digits in a number increases as we go from right towards left. The value of each digit is twice that of its previous digit but is represented only by ‘1′ or ‘0′.

Let us consider the following illustration:

Decimal Digit Value |
256 | 128 | 64 | 32 | 16 | 8 | 4 | 2 | 1 |

Binary Digit Value |
1 | 0 | 1 | 0 | 0 | 1 | 0 | 0 | 1 |

We don’t need to consider the values represented by 0. We will only add up the values represented by 1. So, the equation is:

256 + 64 + 8 + 1 = 329_{10}

From the above illustration, we understand that the number 101001001_{2} (binary number) is equivalent to 329_{10 }in the decimal system. When binary number system is used in computers or the digital system, ‘1′ represents ‘ON’ and ‘0′ represents ‘OFF’.

Let us consider the following example where the decimal number 240 is converted into its binary number equivalent.

Number | 240 | Whenever a number is Divided by “2″, a result and a remainder are derived. MSB represents the most significant bit and LSB represents least significant bit. The binary number is derived going forward from MSB towards LSB. | |||

divide by 2 | |||||

result | 120 | remainder | 0 (LSB) |
||

divide by 2 | |||||

result | 60 | remainder | 0 |
||

divide by 2 | |||||

result | 30 | remainder | 0 |
||

divide by 2 | |||||

result | 15 | remainder | 0 |
||

divide by 2 | |||||

result | 7 | remainder | 1 |
||

divide by 2 | |||||

result | 3 | remainder | 1 |
||

divide by 2 | |||||

result | 1 | remainder | 1 |
||

divide by 2 | |||||

result | 0 | remainder | 1 (MSB) |

So, when we move from MSB towards LSB i.e. from bottom to upwards, the binary number formed is 11110000. The binary number 11110000_{2} is equivalent to the 240_{10} decimal number.

| 2042052361 septenary | p2l6g duotrigesimal | 3be7cf base31 | 11jbmm septemvigesimal | 2npbre base29 | 1201010012210211 ternary | 8071380 undecimal | 101011100100011100000100010 binary | 11101210102021111 ternary | 442423313 senary | 31i671 base21 | pde31 hexavigesimal | df6107 vigesimal | 61a2f0 hexadecimal | 43131124244 quintal | 3031232310331 quaternary | 2elske base33 | 16a91a7 tridecimal | 1d6o68 base35 | 720961 duodecimal | 142ovj base33 | 4420232425 senary | 693po duotrigesimal | 3t1hkb trigesimal | dghj3h base21 | 1523623612 septenary | b7cf90 base17 | 109cw3 hexatrigesimal | c7jnee tetravigesimal | 457472263 octal | 1b3pvr duotrigesimal | 4jdn77 octovigesimal | 59jb4b vigesimal | 4f7sq7 base29 | 3h0aoi hexavigesimal | 313200303111 quaternary | 154pf6 base31 | ejp0o base35 | 8a313e8 pentadecimal | 12101111222112012 ternary | 26okji pentavigesimal | 137bg7e octodecimal | 3ntrqs trigesimal | idh4h3 vigesimal | 32750807 undecimal | 2221333221002 quaternary | 206b66g base19 | viefw base34 | c8kf4 pentavigesimal | 9tgx8 base35 |

| 144214303224 Quintal to Base21 | 3f7gb8c Base17 to Ternary | 3kh07a Base22 to Base21 | 5692ab1 Tridecimal to Base33 | 86muw Hexatrigesimal to Trigesimal | 1s0l7 Hexatrigesimal to Binary | 7j5hed Pentavigesimal to Base31 | 17u8ej Duotrigesimal to Duodecimal | 22f4948 Hexadecimal to Base21 | 21ed1 Base29 to Trigesimal | 1100333015 Septenary to Decimal | 3b58oj Base31 to Septemvigesimal | 2kdgib Base29 to Pentadecimal | a035kk Tetravigesimal to Senary | 7g95ic Base19 to Duodecimal | 148cb34c Tridecimal to Senary | 111011110011000101000011 Binary to Senary | 1d49cbe Pentadecimal to Hexavigesimal | 133424242111 Quintal to Octodecimal | 186367657 Nonary to Octodecimal | 12121020011121 Ternary to Pentavigesimal | 29359659 Undecimal to Tetravigesimal | 2cb57d2 Base17 to Septenary | 7dk5le Pentavigesimal to Vigesimal | 3a431749 Undecimal to Quintal | 111ca4d Base17 to Base21 | 3ad883 Base23 to Base35 | 2qtedn Duotrigesimal to Quintal | 1kbse7 Hexatrigesimal to Octal | q6mc4 Base31 to Tetradecimal | 20ehi0b Base19 to Base33 | 545406406 Septenary to Quintal | 2tmpkg Base31 to Decimal | c8167a Base22 to Octodecimal | 16bojj Base34 to Quintal | 186026523 Nonary to Base21 | 8dfd54 Base23 to Base33 | 553347 Octodecimal to Quintal | 55493058 Decimal to Octovigesimal | 563g0j Base22 to Base33 | i1bdk3 Base22 to Ternary | 23567d Tetravigesimal to Tetradecimal | ca4bcdd Tetradecimal to Quintal | sd6w0 Base34 to Ternary | 300121311201 Quaternary to Binary | 33212243313 Quintal to Senary | ulsl8 Duotrigesimal to Octovigesimal | 142c891 Vigesimal to Base29 | 889899 Hexavigesimal to Base29 | ihfb2d Base22 to Hexavigesimal |

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