### Bits and Bytes

This 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 Two

Binary numbers all consist of combinations of the two digits ‘0’ and ‘1’. These are some examples of binary numbers:

- 1

10

1010

11111011

11000000 10101000 00001100 01011101

Engineers and mathematicans sometimes call the binary numbering system a **base-two** system because binary numbers only contain two digits. By comparison, our normal decimal number system is a **base-ten** system. Hexadecimal numbers (discussed later) are a **base-sixteen** system.

### Converting From Binary to Decimal Numbers

All binary numbers have equivalent decimal representations and vice versa. Our handy Binary-Decimal Number Converter performs these calculations automatically for you. 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 lies 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

and the binary number 1111011 converts to decimal as:

**1**+ (2 *

**1**) + (2*2 *

**0**) + (4*2 *

**1**) + (8*2 *

**1**)+ (16*2 *

**1**) + (32*2 *

**1**) = 123

Therefore, the binary number 1111011 is equal to the decimal number 123.

### Converting From Decimal to Binary Numbers

To 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**

Therefore the decimal number 109 equals the binary number **1101101**.

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