⇩ BASE N TO DECIMAL CONVERTERS ⇩

BINARY TO DECIMAL CONVERTER (WITH STEPS)

Enter a binary number.

(101011)2 = (43)10

SOLUTION

We multiply each binary digit by its place value and add the products.

(101011)2 = (1 × 25) + (0 × 24) + (1 × 23) + (0 × 22) + (1 × 21) + (1 × 20)

= (1 × 32) + (0 × 16) + (1 × 8) + (0 × 4) + (1 × 2) + (1 × 1)

= 32 + 8 + 2 + 1

= (43)10

OTHER INFORMATION

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⇩ BASE N TO DECIMAL CONVERTERS ⇩

INFORMATION

BINARY AND DECIMAL NUMBERS

Binary and decimal numbering systems are commonly used in mathematics, computer science and electrical engineering.

Binary Numbering System (Base 2): In the binary system, numbers are represented using one of two digits: 0 and 1. Each digit is called a bit. For a whole number, the right-most bit has a place value of 20 = 1, the next bit has a place value of 21 = 2, then 22 = 4, 23 = 8, and so on, doubling with each position to the left. For a binary number with a radix point, the place value of the left-most digit in the fractional part is 2–1, the next bit on the right has a place value 2–2, and so on.

Decimal Numbering System (Base 10): In the decimal system, numbers are represented using ten digits: 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9. Each digit has a place value of 10 raised to a power depending on its position in the number.

BINARY TO DECIMAL CONVERSION

Each binary number has a unique representation in decimal system. Binary to decimal (bin to dec) conversion is the process of converting a binary number into its equivalent decimal representation. To convert a binary number to decimal we multiply each digit by its place value and add the products.

PLACE VALUES OF A BINARY NUMBER

Each place value in a binary number can be represented using an exponential number with a base of 2. The exponent of the ones digit is zero, and it increases by 1 for each digit moved to the left. In other words,

• the place value of the ones digit is 1,
• as we move to the left, the place value is doubled and
• as we move to the right the place value is halved.

Each digit of the binary number above is marked with its place value. The place value of the ones digit is 1, the next digit's place value is 2, which is twice that of 1, and the subsequent digit's place value is 4, which is twice that of 2, and so on.

The place value of the ones digit is 1, and as we move to the right, the place value halves, resulting in place values in fraction form such as ..., ..., ..., and so on.

BINARY TO DECIMAL CONVERSION EXAMPLES

EXAMPLE:

Find the decimal representation of the binary number 10112.

The place values, from left to right, are 8, 4, 2, and 1, respectively. To find the decimal equivalent, we multiply each place value by the corresponding digit and then add the products together.

10112 = 1 · 8 + 0 · 4 + 1 · 2 + 1 · 1

= 8 + 2 + 1

= 11

So, the decimal representation of 10112 is 1110.

EXAMPLE:

Find the decimal equivalent of the binary number 110.112.

The place values of 110.112 are shown below.

Multiply those values with the corresponding digits and add the products.

... ... ...

...

...

...

EXAMPLE:

Find the decimal equivalent of the largest 8-digit binary number.

The highest possible value of a binary digit is 1. Therefore, the largest 8-digit binary number is 111111112. Its decimal equivalent is:

111111112 = 1 · 27 + 1 · 26 + 1 · 25 + 1 · 24 + 1 · 23 + 1 · 22 + 1 · 21 + 1 · 20

= 128 + 64 + 32 + 16 + 8 + 4 + 2 + 1

= 25510

Alternatively, we can find the decimal equivalent of the smallest 9-digit binary number and subtract 1 to reach the result faster. The smallest 9-digit binary number is 1000000002.

1000000002 = 28

= 25610

That number is 1 more than the largest 8-digit binary number.

111111112 = 256 - 1

= 25510

EXAMPLE:

If 1a0012 = 2b10, find the sum a + b.

Let's find the decimal equivalent of the binary number 1a0012.

1a0012 = 16 + 8 · a + 1

= 17 + 8a

There are only two possible values for a: 0 and 1. If a were 0, the decimal equivalent of this number would be 1710. However, since 2b10 cannot equal 1710, a cannot be 0. Therefore, a = 1. When we calculate the decimal equivalent for this value:

17 + 8 * 1 = 2510

Consequently, b = 5. The sum of the numbers a = 1 and b = 5 is 6.

BINARY TO DECIMAL CONVERSION TABLE

The table below provides the decimal equivalents of the binary numbers up to 100.

 02 = 010 12 = 110 102 = 210 112 = 310 1002 = 410 1012 = 510 1102 = 610 1112 = 710 10002 = 810 10012 = 910 10102 = 1010 10112 = 1110 11002 = 1210 11012 = 1310 11102 = 1410 11112 = 1510 100002 = 1610 100012 = 1710 100102 = 1810 100112 = 1910 101002 = 2010 101012 = 2110 101102 = 2210 101112 = 2310 110002 = 2410 110012 = 2510 110102 = 2610 110112 = 2710 111002 = 2810 111012 = 2910 111102 = 3010 111112 = 3110 1000002 = 3210 1000012 = 3310 1000102 = 3410 1000112 = 3510 1001002 = 3610 1001012 = 3710 1001102 = 3810 1001112 = 3910 1010002 = 4010 1010012 = 4110 1010102 = 4210 1010112 = 4310 1011002 = 4410 1011012 = 4510 1011102 = 4610 1011112 = 4710 1100002 = 4810 1100012 = 4910 1100102 = 5010 1100112 = 5110 1101002 = 5210 1101012 = 5310 1101102 = 5410 1101112 = 5510 1110002 = 5610 1110012 = 5710 1110102 = 5810 1110112 = 5910 1111002 = 6010 1111012 = 6110 1111102 = 6210 1111112 = 6310 10000002 = 6410 10000012 = 6510 10000102 = 6610 10000112 = 6710 10001002 = 6810 10001012 = 6910 10001102 = 7010 10001112 = 7110 10010002 = 7210 10010012 = 7310 10010102 = 7410 10010112 = 7510 10011002 = 7610 10011012 = 7710 10011102 = 7810 10011112 = 7910 10100002 = 8010 10100012 = 8110 10100102 = 8210 10100112 = 8310 10101002 = 8410 10101012 = 8510 10101102 = 8610 10101112 = 8710 10110002 = 8810 10110012 = 8910 10110102 = 9010 10110112 = 9110 10111002 = 9210 10111012 = 9310 10111102 = 9410 10111112 = 9510 11000002 = 9610 11000012 = 9710 11000102 = 9810 11000112 = 9910 11001002 = 10010

WHAT IS BINARY TO DECIMAL CONVERTER?

Binary to decimal converter,

• Computes the decimal equivalent of the entered binary number,
• Describes the solution step by step and
• Illustrates the place values.

HOW TO USE BINARY TO DECIMAL CONVERTER?

You can use binary to decimal converter in two ways.

• USER INPUTS

You can enter a binary number into the input box and then click the 'CONVERT' button. The result and explanations will appear below the calculator.

• RANDOM INPUTS

You can click on the DIE ICON next to the input box to generate a random binary number, which will be automatically entered into the calculator. The result and explanations will then appear below the calculator. You can also create your own examples and practice using this feature.

• CLEARING THE INPUT BOX

To find the decimal equivalent of another binary number, click on the CLEAR button to clear the input box.

• You can copy the generated solution by clicking on the 'Copy Text' link located below the solution panel.