Bits and BytesThis article series assumes a basic understanding of computer bits and bytes. Binary and hexadecimal numbers are the natural mathematical way to work with the data stored in bits and bytes.
Binary Numbers and Base TwoBinary numbers all consist of combinations of the two digits '0' and '1'. These are some examples of binary numbers:
11000000 10101000 00001100 01011101
Engineers and mathematicians call the binary numbering system a base-two system because binary numbers only contain the two digits '0' and '1'. By comparison, our normal decimal number system is a base-ten system that uses the ten digits '0' through '9'. Hexadecimal numbers (discussed later) are a base-sixteen system.
Converting From Binary to Decimal NumbersAll binary numbers have equivalent decimal representations and vice versa. Our handy Binary-Decimal Number Converter performs these calculations automatically for you as do most calculators. To convert binary and decimal numbers manually, you must apply the mathematical concept of positional values.
The positional value concept is simple: With both binary and decimal numbers, the actual value of each digit depends on its position ("how far to the left") within the number.
For example, in the decimal number 124, the digit '4' represents the value "four," but the digit '2' represents the value "twenty," not "two." The '2' represents a larger value than the '4' in this case because it is positioned further to the left in the number.
Likewise in the binary number 1111011, the rightmost '1' represents the value "one," but the leftmost '1' represents a much higher value ("sixty-four" in this case).
In mathematics, the base of the numbering system determines how much to value digits by position. For base-ten decimal numbers, multiply each digit on the left by a progressive factor of 10 to calculate its value. For base-two binary numbers, multiply each digit on the left by a progressive factor of 2. Calculations always work from right to left.
In the above example, the decimal number 123 works out to:
3 + (10 * 2) + (10*10 * 1) = 123
1 + (2 * 1) + (2*2 * 0) + (4*2 * 1) + (8*2 * 1)+ (16*2 * 1) + (32*2 * 1) = 123
Converting From Decimal to Binary NumbersTo convert numbers in the opposite direction, from decimal to binary, requires successive division rather than progressive multiplication. Our Binary-Decimal Number Converter also performs these calculations automatically for you.
To manually convert from a decimal to a binary number, start with the decimal number and begin dividing by the binary number base (base "two"). For each step the division results in a remainder of 1, use '1' in that position of the binary number. When the division results in a remainder of 0 instead, use '0' in that position. Stop when the division results in a value of 0. The resulting binary numbers are ordered from right to left.
For example, the decimal number 109 converts to binary as follows:
109 / 2 = 54 remainder 1
54 / 2 = 27 remainder 0
27 / 2 = 13 remainder 1
13 / 2 = 6 remainder 1
6 / 2 = 3 remainder 0
3 / 2 = 1 remainder 1
1 / 2 = 0 remainder 1
→ Learn by example - Binary-Decimal Number Converter