number_systems - Number Systems Introduction Binary Number...

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Number Systems Introduction Binary Number System The goal of this handout is to make you comfortable with the binary number system. We Binary means base 2 (the prefix bi ). Based on our earlier discussion of the decimal will correlate your previous knowledge of the decimal number system to the binary number system, the digits that can be used to count in this number system are 0 and 1. number system. That will lay the foundations on which our discussion of various The 0,1 used in the binary system are called bi nary digi t s ( bits ) representation schemes for numbers (both integer and real numbers) will be based. The bit is the smallest piece of information that can be stored in a computer. It can have one of two values 0 or 1. Think of a bit as a switch that can be either on or off. For Decimal Number System example, Bit Value Counting as we have been taught since kindergarten is based on the decimal number 0 OFF / FALSE system. Decimal means base 10 (the prefix dec ). In any number system, given the base 1 ON/ TRUE (often referred to as radix ), the number of digits that can be used to count is fixed. For example in the base 10 number system, the digits that can be used to count are Table 1. Interpreting Bit Values 0,1,2,3,4,5,6,7,8,9. From the hardware perspective, ON and OFF can be represented as voltage levels Generalizing that for any base b, the first b digits (starting with 0) represent the digits that (typically 0V for logic 0 and +3.3 to +5V for logic 1). Since only two values can be are used to count. When a number b has to be represented, the place values are used. stored in a bit, we combine a series of bits to represent more information. Again the concept of place values is applicable here as well. Example 1. Consider the number 1234. It can be represented as Example 3. Consider the binary number 1101. It can be represented as 1*10 3 + 2*10 2 + 3*10 1 + 4*10 0 (1) 1*2 3 + 1*2 2 + 0*2 1 + 1*2 0 (3) Where: - 1 is in the thousand’s place This expanded notation also gives you the means of converting binary numbers directly - 2 is in the hundred’s place into the equivalent decimal number. - 3 is in the ten’s place - 4 is in the one’s place. 8 + 4 + 0 + 1 = 13 The equation (1) is an expanded representation of 1234. The expanded representation has Example 4. Consider the binary number 1101.101. It can be represented as: the advantage of making the base of the number system explicit. 1*2 3 + 1*2 2 + 0*2 1 + 1*2 0 + 1*2 -1 + 0*2 -2 + 1*2 -3 (4) Example 2. Consider the number 1234.567. It is represented as The same notation is applicable to real numbers represented in binary notation. The equivalent decimal number is 1*10 3 + 2*10 2 + 3*10 1 + 4*10 0 + 5*10 -1 + 6*10 -2 + 7*10 -3 (2) Where: 13 + 0.5 + 0 + 0.125 = 13.625 - 5 is in the tenth’s place - 6 is in the hundredth’s place - 7 is in the thousandth’s place To represent larger numbers, we have to group series of bits. Two of these groupings are of importance: In equation (2), the representation includes digits both to the left and to the right of the - Nibble A nibble is a group of four bits decimal point.
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