What is a Decimal Number?
As it relates to the conversion calculator on this page, a decimal number is a numerical expression that uses the base 10 system for counting and representing values.
Of course, this base 10 system -- which uses the numbers 0 through 9 -- is the number system most of us were taught from toddler age on.
In fact, the reason we don't add a subscripted 10 to base 10 numbers, is because we know it will just be assumed. This can't be said for the other bases.
Aside from the numbers used, the base 10 system assigns a power of 10 to each place value, like this:
Place Values of Base 10 SystemNow, since we are looking to convert a base 10 number into a base 2 number, let's compare the above with the place values in a binary number system, which only uses the number 0 and 1:
Place Values of Base 2 SystemSo you see, since each place value in a base 10 number is different than the corresponding place value in a base 2 system, we need a method for converting 0-9 base 10 place values into 0-1 base 2 place values.
How to Convert Decimal to Binary
To convert a base 10 number into a base 2 number, the first step is to find the first base 2 place value that is greater than or equal to the decimal number you are converting -- starting at the 20 place and working your way to the left. For example, suppose you want to convert the decimal number 15 into a binary number. In that case, you would find the first base 2 place value that is greater than or equal to 15, which would be 16:
First Base 2 Value Greater than 1510Once you have located your base 2 place value starting point, the next step is to create a conversion chart, like this:
Conversion ChartNext, attempt to divide the amount in row B into the amount in row C. If the amount in row C is greater than the amount in row B, enter a "0" in row D and move the amount in row B one cell to the right. Otherwise, if the amount in row C is less than the amount in row B, enter a "1" in row D and enter the difference between B and C (remainder) in the next open cell in row B. Then simply repeat this process for each subsequent column, like this:
Notice that the decimal you want to convert is placed in the left-most cell of Row B, just above the base 2 place value row (C).
Conversion Chart for Converting 1510 to BinaryDivide each cell in row C into the corresponding cell in row B and move the remainder of each division to the next cell in row B. Repeat for all columns. Note that the above is how the decimal to binary converter will show its work.
From the above, we can see that the base 10 number 15 converts to the base 2 number 1111 (one-one-one-one). Note that the leading zeros are dropped since they represent no value (just like the base 10 system).
As you can see, converting a decimal number to a binary number is a simple process of identifying the first base 2 place value greater than or equal to the base 10 number you are converting, and then dividing each place value into the remainder of previous division.