Lec 2

It is obvious that the representation of a quantity in binary numeration system takes mores ciphers than in decimal system. We can therefore ask ourselves why the binary system is preferred to decimal system in computer sciences. The reason is that in electronics, it is easier to materialise two quantities “0” and “1” (by two different voltages for example) than to materialises 10 different quantities “0” ,”1”
,”2”,”3”,”4”,”5”,”6”,”7”,”8”, and ”9” (by 10 different voltages). In fact, in digital circuits, 0 and 1 are materialised by specific ranges of voltages or current; this will be discussed later.
Remark 1.1:
With n bits we can represent 2n different binary numbers. The higher H number is given using the following formula.
H ? 2n ? 1 (1)

Example 1.4:
With 4 bits we can represent 24 = 16 different binary numbers (from 0 to 15), and the higher number is H = 24 – 1 = 15.
Remark 1.2: Conversion from binary to decimal
To convert a number written in binary numeration system to its equivalent in decimal, we just have to calculate the products of the bits with their respective weights, as in example 1.3 above. For binary numbers with “binary point” (equivalent of decimal point for decimal numbers), the conversion is done as follow.
2 1 0 -1 -2 -3
A= 1 0 1. 1 0 1
A= 1×22 + 0×21+1×20 +1×2-1+0×2-2+1×2-3
A = 5.62510