Binary Calculator

The binary system is a numerical system that functions almost exactly like the decimal system, which most people are more familiar with. The base number for the decimal system is 10, while the binary system uses 10. The binary system uses 2, whereas the decimal system uses 10, while the binary system uses 1, which is called a bit. These differences aside, operations like addition, subtraction, and multiplication are all calculated using the same rules as in the decimal system.

Because of its simplicity in implementation in digital circuitry with logic gates, almost all modern technology and computers use the binary system. It is easier to design hardware that can detect only two states (on and off, true/false, or present/absent) than to see more states. Hardware that can detect ten states using a decimal system will be required, which is more complicated.

Here are some examples of conversions between decimal, hex, and binary values:

You can convert the decimal system by following this step-by-step procedure:

Every position in a binary number represents a power of 2 just like every position in decimal numbers represents a power of 10.

In order to convert to decimal, you will need to multiply each position by 2 to the power number of the position number. This is done by counting from left to center and starting with zero.

Addition follows the same rules as the addition in the decimal method except that; instead of carrying a 1, when the values added equal 10, a carry-over occurs when the result is branch equals 2. 

The only difference between binary and decimal addition is that the binary system's value 2 corresponds to the decimal system's equivalent value of 10. You will notice that superscripted 1,s denote digits that have been carried over. When performing binary addition, a common mistake is when 1 + 1 = 0. Also, 1 from the previous column to its left has a 1 that was carried over. The value at the bottom should then be 1 instead of 0. In the example above, you can see this in the third column.

Similar to addition, there is not much difference between decimal and binary subtraction, except those caused by using the digits 1 and 0. Borrowing can be used when the number being subtracted is greater than that of the original number. Binary subtraction is where one is removed from 0. This is the only instance in which borrowing is required. When this happens, the number 0 in the borrowed column becomes "2". This transforms the 0-1 to 2-1 = 1 while reducing 1 in the column being rebought from by 1. If the following column has a value of 0, borrowing will need to be done from all subsequent columns.

Multiplication can be simpler than decimal multiplication. Multiplication is simpler than its decimal counterpart, as there are only two values. Noting that each row has a placeholder 0, the result must be added and the value must be shifted to the right, much like decimal multiplication. Binary multiplication's complexity is due to tedious addition that depends on how many bits each term contains. See the example below to see more.

Binary multiplication is exactly the same process as decimal multiplication. You will notice that the 0 placeholder appears in the second row. In decimal multiplication, the 0 placeholder is typically not visible. The same thing can be done in this case, but the 0 placeholders will be assumed. It is still included because the 0 is relevant to any binary addition/subtraction calculator like the one shown on this page. If the 0 was not shown, it is possible to ignore the 0 and add the binary values above. It is important to note that the binary system considers any 0 right of a 1, while any 0 left is irrelevant.

The division is similar in process too long division using the decimal system. The dividend is still done by the divisor in exactly the same way. The only difference is that the divisor uses subtraction instead of decimal. For division, it is crucial to understand subtraction.