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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.

| i8r8f base35 | 100000010000010000111111111 binary | b57bne tetravigesimal | 16550b5 base19 | dei02e base23 | 823ad9e pentadecimal | 43fcaj octovigesimal | 100001111111110011101011010 binary | 10011111101001100011111111 binary | ucui8 base35 | ee008c octodecimal | 5i2ad7 base23 | 1rr5hn base33 | 1841dm base31 | 1cdg4b6 octodecimal | 4hj465 octovigesimal | 28dd54 octodecimal | 1crjdh base29 | 1012021201221220 ternary | 4807864 tridecimal | 2pj6f3 base31 | bmmefm base23 | 20000101020121210 ternary | d1b762 tetradecimal | 34a26e4 hexadecimal | 195hgg tetravigesimal | 7de178 hexavigesimal | 8g8hhd base21 | 13232041321 senary | 213200212201 quaternary | 34f01e base21 | ovinw base33 | 323ae tetravigesimal | 12220303243 senary | 18d5ia8 vigesimal | 550437271 octal | aa1280 tetravigesimal | 73d704d tetradecimal | 117aa774 duodecimal | 2sa85m duotrigesimal | 3p52q3 octovigesimal | 121a764 base21 | 24f0go hexavigesimal | 43412424433 quintal | c76sl base33 | fdj9ic vigesimal | fegi8f base19 | 2mprsb base29 | e81892 hexadecimal | 7f16pn hexavigesimal |

| 50d917 Octodecimal to Tridecimal | bfibae Base21 to Hexadecimal | 11k1542 Base21 to Binary | 1muovv Base34 to Nonary | 13g363 Base19 to Base17 | 345331157 Octal to Base34 | 1a4ii3 Hexavigesimal to Pentadecimal | 13gnid Tetravigesimal to Septenary | 1bf7tb Hexatrigesimal to Binary | 2da8c16 Octodecimal to Quaternary | tn84e Trigesimal to Octovigesimal | 1nwwu4 Base33 to Hexavigesimal | 11111221000100002 Ternary to Senary | 120002010010011 Ternary to Nonary | 5446120 Nonary to Septemvigesimal | 1tt03d Base33 to Septemvigesimal | 7687a Base21 to Base29 | vgs93 Hexatrigesimal to Tetradecimal | 102021240143 Quintal to Base21 | nfucv Base34 to Trigesimal | 466257203 Octal to Nonary | aa1b61 Duodecimal to Tridecimal | 67357415 Octal to Ternary | 5a19897 Tridecimal to Ternary | 2143206414 Septenary to Base19 | 20020211212100222 Ternary to Duotrigesimal | tbnu4 Base34 to Base31 | 16ba382 Octodecimal to Senary | 14dac8b Octodecimal to Tridecimal | 8kiod Base31 to Undecimal | 29004921 Undecimal to Pentadecimal | 90111051 Decimal to Duodecimal | 40620db Hexadecimal to Hexavigesimal | 1eukn3 Base35 to Octovigesimal | acb78bc Tridecimal to Octodecimal | 2uf2lo Duotrigesimal to Quaternary | 1f2c38e Hexadecimal to Quaternary | 10354567 Tridecimal to Octovigesimal | 29846560 Undecimal to Base17 | 1ild94 Septemvigesimal to Base22 | 2a9h08h Octodecimal to Hexadecimal | 658o0f Septemvigesimal to Pentavigesimal | cd8730 Tetravigesimal to Hexadecimal | 111110210020212 Ternary to Hexatrigesimal | 5n1c0l Tetravigesimal to Octodecimal | 256dj1 Octovigesimal to Tetradecimal | oog00 Base34 to Base29 | 1g3563c Base19 to Pentavigesimal | dkd28b Base21 to Ternary | 129a034 Duodecimal to Base22 |

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