# Binary to Decimal Converter

Since the early days of computers, binary has been a simple way of communicating between the biological and digital worlds. Relays represented the 1s and 0s that talked to the ancestors of today's laptops, and even though computers have gotten smaller, they still function using this very basic principle.

## Converting Between Number Systems

Most computers run large data sets using hexadecimal code (a base-16 number system) to speed up data transfer, but on the occasion you need to convert between binary (base-2) and decimal (base-10) numbers, this handy widget will help speed things along.

- To convert a binary number to a decimal, use the first section of the widget. Simply input the binary number you would like to convert and click the "calculate" button.
- To convert a decimal number into binary, use the second section and click on the corresponding "calculate" button to get your desired result.
- You will not need to fill both sentences to convert values, but you will have to click each button to convert your inputted variables.

A button beneath both sentences will allow you to erase all data from both fields to start fresh.

## Manual Conversions

The standard decimal system is so similar to binary, octal (base-8) and hexadecimal systems, it's elementary. The base-10 decimal number system is used most commonly and is most recognized as the "regular" number system, but this simple trick will help decipher all the other number systems.

### Understanding Number Systems

With base-10, you are limited to the numbers 0 through 9 in each place. Given this, consider the example of the number 1,623. This can be represented as 1,000 + 600 + 20 + 3 if it were broken down into the corresponding places for the 1,000, 100, 10 and 1 values. It can be broken down further with base-10 notation like this:

[(10^3*1) + (10^2*6) + (10^1*2) + (10^0*3)]

The following chart may be a little easier to follow, remembering that any number to the power of 0 is represented as 1.

10^3 | 10^2 | 10^1 | 10^0 |

1 | 6 | 2 | 3 |

### Binary to Decimal

Binary numbers can be understood exactly the same way as decimal numbers, except each space is a power of 2 rather than a power of 10 and each space can only have a value of 0 or 1 (rather than any number 0 to 9).

Converting the binary number 11001010111 to decimal form would take a few short steps.

- Working from right to left, denote each space as a power of 2, starting with 2^0 for the first space, 2^1 for the second space and so on.
2^10

2^9

2^8

2^7

2^6

2^5

2^4

2^3

2^2

2^1

2^0

1

1

0

0

1

0

1

0

1

1

1

- Calculate the corresponding value for each space where a "1" appears in the binary number.
- In other words, 2^10 = 1024, 2^9 = 512, and so on

- Add up all the values calculated in step 2.
- With this example, that would be 1024 + 512 + 64 + 16 + 4 + 2 + 1, which equals 1,623 in decimal form.

Notice the same number in decimal form only takes four places, compared to the eleven places taken up by the binary number. This can make decimal numbers more advantageous when doing simple calculations.

### Decimal to Binary

While the concept isn't too difficult, you will need a pen and paper to keep track of this conversion.

- Divide the base-10 number by the base equivalent you'd like to achieve. In the case of binary, that would be base-2.
- 1623 / 2 = 811 ½ (or 811 with 1 remainder)

- If you started with an even number, there will be no remainder, so put a "0" in the first spot. If it is odd, there is a remainder and a "1" is put in that spot instead. The remainder of 1 is placed in the 2^0 place.
- Divide the whole number that remains (811 in this case) and continue with all remainders being added to the adjoining placeholders to the left.
1/2

3/2

6/2

12/2

25/2

50/2

101/2

202/2

405/2

811/2

1623/2

1

1

0

0

1

0

1

0

1

1

1

The last remainder drops into the furthest left placeholder , resulting in the final binary number of 11001010111.

## Binary Beginnings

ENIAC, one of the first computers, was built in the mid-1940s to necessitate military needs. Ms. Bilas and Ms. Jennings helped design the massive machine with 18,000 vacuum tubes to represent bits. Since eight bits represent one byte, that was equivalent to 2.25 kilobytes of memory!

That may not sound like much compared to today's terabyte hard drives, but it was huge for the time. Air handling and filtration systems were still in their infancy so debugging the computer meant exactly that: removing bugs from the circuits.

## A Series of 1s and 0s

You may not encounter binary numbers of a frequent basis, but binary numbers are all around you, from the colors on this website to these words you are reading. This handy widget will make it easy to convert binary numbers to and from decimal numbers with just a couple clicks.