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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.

| 3101322322311 quaternary | 279dgn trigesimal | 10184737 tridecimal | 513311223 septenary | 1475yy base35 | 28mgid hexavigesimal | 15bhd69 base19 | 8cb179 base22 | 37g90ab base17 | 207d746 base17 | ibdd82 base21 | 6mnkg5 pentavigesimal | 3ec1x base35 | 27703416 duodecimal | 1bc95f vigesimal | 16172334 octal | 44dc1m septemvigesimal | 1000000011111010011111010 binary | 33g88f3 base17 | 3pcsq duotrigesimal | 5ce643d hexadecimal | 1ipo22 hexavigesimal | 1f7kjo septemvigesimal | 1d610ha base19 | 4di7sm base29 | 3a08fbe hexadecimal | 2211630554 septenary | 80i864 base22 | 194gi7d base19 | 4okn8c hexavigesimal | 422250240 senary | o7bb4 base29 | 1a1ad7g octodecimal | 88o874 hexavigesimal | 26940859 duodecimal | 34200125 undecimal | 5472ci tetravigesimal | 1bf7ph base34 | 12eqti base31 | 100101101100011101011011011 binary | 18mjoq trigesimal | 10012011111201120 ternary | 58f701 pentavigesimal | gcfked base22 | 790oja pentavigesimal | 5mhgl3 hexavigesimal | 2b193b base19 | 11230323321330 quaternary | 2chp11 duotrigesimal | 6356mi septemvigesimal |

| 10j3mr Base33 to Base34 | 34324033 Undecimal to Quaternary | 2271d45 Base19 to Septemvigesimal | 1o6e98 Octovigesimal to Base21 | 100001010110111100101010101 Binary to Hexavigesimal | 10101101110010110010101000 Binary to Hexadecimal | m7mnn Septemvigesimal to Trigesimal | 1tq17y Base35 to Pentadecimal | 51a6908 Duodecimal to Quaternary | 1c2d3b3 Octodecimal to Base35 | 1qceod Duotrigesimal to Base33 | 1aorx Base34 to Trigesimal | abc47d7 Tetradecimal to Duotrigesimal | 1311311233020 Quaternary to Octovigesimal | 11111101111011000101000111 Binary to Pentadecimal | 9632a45 Tetradecimal to Base23 | w2r2s Base34 to Ternary | 586m7f Pentavigesimal to Base23 | 3kjdnb Tetravigesimal to Base21 | 2415521452 Senary to Pentavigesimal | 1542688a Undecimal to Tridecimal | 1045010440 Septenary to Ternary | 9hci75 Base22 to Septemvigesimal | 1r4cpt Base35 to Base23 | 1210122022012122 Ternary to Tetradecimal | 45041614 Undecimal to Tridecimal | mi70c Pentavigesimal to Quintal | 2ega571 Base17 to Base22 | 1ea1hr Hexatrigesimal to Base21 | 15101364 Duodecimal to Ternary | 104011122241 Quintal to Undecimal | 101001100001110100000100011 Binary to Base33 | 2hb65n Base33 to Base29 | 1635532053 Septenary to Duodecimal | 2u3l90 Duotrigesimal to Binary | owvew Hexatrigesimal to Binary | 104003031412 Quintal to Base34 | 7hnk7o Hexavigesimal to Base22 | 3c9bb69 Tridecimal to Base33 | 11010100001100111011110101 Binary to Trigesimal | cb787 Hexatrigesimal to Octovigesimal | 107668143 Nonary to Undecimal | 85bbd08 Pentadecimal to Trigesimal | j81a62 Base22 to Quintal | 3fqhpq Octovigesimal to Base31 | 106565762 Nonary to Tridecimal | 13203441404 Quintal to Ternary | 16c0u5 Base34 to Vigesimal | 2510600 Base17 to Pentavigesimal | 147c06cb Tridecimal to Quintal |

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