Modern digital technology is dependent entirely upon the binary number system, which has only two elements, viz., 0 and 1, in it. However, we are more familiar with decimal number system, which has ten basic elements (0 to 9) in it. In this chapter, we discuss the principles of various number systems that we commonly come across in various contexts. The discussion begins with the decimal number system. This is followed by the binary, octal, and hexadecimal number systems.

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**THE DECIMAL NUMBER SYSTEM**

The decimal number system is the most commonly and widely used number system by the common man. As stated above, we have ten elemental numbers in this number system. They are, respectively, 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9. Since there are ten digits in this group, we say that the number system has a

*base*or*radix*of 10*.*In the decimal number system, numbers greater than 9 are represented by repeatedly using these digits in a definite order. As an example, consider let us consider the decimal number 123. This may be expressed in the form (123)_{10},_{ }where the subscript 10*represents the base or radix of number 123. It can be seen that 123 represents a short form of expressing numbers. This may be expanded as:* (123)

_{10}= (1 ´ 10^{2}) + (2 ´ 10^{1}) + (3 ´ 10^{0})where digits 1, 2, and 3 are the coefficients of the appropriate powers of ten (i.e., 10

We proceed in a similar fashion, as given above. This is shown in Fig.1.3.

From Fig.1.3, we get

From Fig. 1.9, we find that

^{2}, 10^{1}, and 10^{0}, respectively). Larger numbers can be expressed in similar fashion by writing them as coefficients of the powers of ten. In fact,*in any number system*,*numbers are expressed as coefficients of appropriate powers of the radix of the system.***Decimal Number to Binary Number Conversion:**We proceed in a similar fashion, as given above. This is shown in Fig.1.3.

From Fig.1.3, we get

**(201)**

_{10}= (11001001)_{2}

**Decimal Number to Octal Number Conversion:****Example:**Convert the decimal number 1234 into its equivalent octal number.

**Solution:**

*Figure 1.9 shows a conversion table prepared on the basis of the steps shown in Fig. 1.8.*

Fig. 1.9 Tabular form for converting (1234)_{10} into octal |

From Fig. 1.9, we find that

**(1234)**

_{10}**º**

**(2322)**

_{8}

Check

*:*2 ´ 8^{3}+ 3 ´ 8^{2}+ 2´8^{1}+2 = (1234)_{10}The above method is very fast in arriving at solutions connected with octal conversions