Binary Notation

Example
In order to understand how a number in binary notation is constructed, we will first discuss decimal notation. Let's use the decimal-notation number "231" as an example.

Decimal notation uses the base of 10, and is therefore also called Base 10 notation. The base is the exponent which increases with each place to the left. In decimal notation, each place has a value which corresponds to a power of the base 10. Numbers are formed by positioning powers of the base 10. Compare the decimal notation numbers 1, 10, 100 and 1,000. The figure "1" means "1 x 100" when placed on the far right (= 1). In the second place from the right it means "1 x 101" (= 10). In the third place from the right it means "1 x 102" (= 100), and in the fourth place from the right it means "1 x 103" (= 1,000). The power, represented by the exponent, increases by 1 with each place to the left (101, 102, 103, etc.). In decimal (base 10) notation, the power is a power of 10. The base is 10. So when the figure moves one place to the left and the power increases by 1, the figure is worth 10 times more.

The calculation of the number 231 in decimal notation would therefore be:

(2 x 10E+02) + (3 x 10E+01) + (1 x 10E+00) =

(2 x 100) + (3 x 10) + (1 x 1) = 200 + 30 + 1 = 231

Numbers in binary notation are calculated the same way, but using 2 as base, not 10, and only the figures '0' and '1' exist.

For our discussion of binary notation, we will call the power which increases by position the weight of the figure. Consider the binary figure: 1011. Converting this number into our common decimal notation is helpful for understanding what it means. The weights of the figures now are not powers of 10, but powers of 2. So when the power increases by 1, the figure is worth 2 times more.

To convert the number 1011 from binary notation into decimal notation, use this formula for each figure:

(the figure:"1" or "0") x (the base "2" to
the power of the place)

1011

(1 x 23) + (0 x 22) + (1 x 21) + (1 x 20) =

(1 x 8) + (0 x 4) + (1 x 2) + (1 x 1) =

8 + 0 + 2 + 1 =

11.


In the binary number 1011, the right "1" has the lowest weight and the left "1" has the highest weight. They are therefore called the Least Significant Bit (LSB) (the bit carrying the least weight) and the Most Significant Bit (MSB) (the bit carrying the most weight).


1 0 1 1
|     |
MSB     LSB

See also: Binary Number, Gray Code, Octal, Unsigned Binary Number.

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Subjects: Computing Mathematics