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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.

| 5hknp2 hexavigesimal | 1hc5cf7 octodecimal | 6ac1g0 pentavigesimal | 149esf base29 | gvg50 duotrigesimal | 11cc877 hexadecimal | 2eqe01 octovigesimal | 50894198 undecimal | 28854434 decimal | 1f01377 base19 | 166f2f9 base17 | 1001100001001111100000001 binary | 51282394 decimal | 69cl90 tetravigesimal | 23352038 nonary | 10100111000000010011111 binary | eh3kb6 base21 | 34734674 octal | 82c12b0 pentadecimal | 5512445522 senary | efillg base22 | 2f8ac92 base17 | 23313124430 quintal | c2400 tetravigesimal | g4calj base22 | 1je1lj base23 | 1523230230 septenary | 2188609 base17 | 1000211120120201 ternary | 11331030131111 quaternary | 6aca5a6 tetradecimal | 1h21jf hexavigesimal | 6vhtq base34 | 13587161 tridecimal | 114311402030 quintal | 1110011110000010101100011 binary | 440433333 septenary | 1221325062 septenary | 6787607 nonary | 97221036 decimal | 536512276 octal | 3kevh duotrigesimal | 156a315 tetradecimal | 47nejo base29 | 2230310541 septenary | a90cbd base23 | 1o7j4a duotrigesimal | eg0h82 base21 | 1ubf4i base31 | 11110011110011111011110011 binary |

| sithf Base34 to Tetravigesimal | 3ddi12 Vigesimal to Base19 | 1545535345 Senary to Duodecimal | knm51 Trigesimal to Base19 | 25f46db Hexadecimal to Tridecimal | 7d9im Base33 to Quintal | 1330332132133 Quaternary to Pentavigesimal | 10001001000010000001001011 Binary to Base22 | 29a9f2b Octodecimal to Base23 | mepdm Base35 to Septemvigesimal | 48d8e Base19 to Duodecimal | 2020221201012220 Ternary to Hexavigesimal | 4h5375 Base21 to Base19 | 4knn8 Tetravigesimal to Vigesimal | 1ab69b42 Duodecimal to Octal | 553626334 Septenary to Hexavigesimal | 8g8947 Octodecimal to Base21 | 2ds0ms Base31 to Hexatrigesimal | 660828 Decimal to Base31 | 203tx2 Base34 to Base23 | a5ag6o Pentavigesimal to Trigesimal | 24a88629 Undecimal to Base23 | 5e3624 Hexavigesimal to Vigesimal | 212143886 Nonary to Base21 | qatcs Hexatrigesimal to Base31 | 49qd8n Base29 to Duotrigesimal | 865052d Pentadecimal to Pentavigesimal | 2k2lrj Trigesimal to Septenary | 10525322354 Senary to Vigesimal | t8xfg Hexatrigesimal to Base19 | 21fb200 Base17 to Pentadecimal | bvu3v Base34 to Hexavigesimal | 1idetb Base35 to Senary | 4kkg04 Tetravigesimal to Duotrigesimal | 24044132244 Quintal to Base19 | 11110110010100010101000110 Binary to Nonary | 420b4f5 Base17 to Quintal | 80e0s Base33 to Nonary | a88cc9c Tridecimal to Decimal | 1i2360 Base31 to Base35 | 170b89 Hexatrigesimal to Duotrigesimal | g9af0 Base22 to Octal | 9373735 Tetradecimal to Base35 | 59fb502 Hexadecimal to Tetradecimal | 30n28b Base31 to Tetradecimal | 2glh66 Duotrigesimal to Base21 | 31bc765 Hexadecimal to Base29 | 6dd234a Pentadecimal to Base22 | 268761b6 Duodecimal to Septemvigesimal | 5027j2 Base22 to Octodecimal |

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