Number Systems & Binary

Section 4.1 Number Systems & Binary

You have been using the decimal number system for so long that you probably no longer often stop to think about what the individual digits mean. You see 134 and recognize it as “one hundred thirty-four”. 134 has that value because each digit in the number represents a multiple of some power of 10:

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Table 4.1.1. "134" base 10

Place value	100s	10s	1s
Power of 10	$10^{2}$	$10^{1}$	$10^{0}$
Digit	1	3	4

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Or 1 * 100 + 3 * 10 + 4 * 1 = 134

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The binary number system is the same idea - each digit has a value based on the column it is in. The two differences are:

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Each digit can only be a 1 or a 0
Each column represents a power of 2 instead of a power of 10

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Table 4.1.2. "101" base 2

Fours	Twos	Ones
$2^{2}$	$2^{1}$	$2^{0}$
1	0	1

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101 in binary means 1 * 4 + 0 * 2 + 1 * 1 = 5

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Notice that since each digit can only be a 1 or a 0, we don’t actually need to multiply. A 1 indicates we have the value for that column and a 0 indicates we don’t. So 101 in binary simply means 4 + 1 = 5 because the first 1 is in the “fours” column and the second 1 is in the “ones” column.

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Because things can get confusing when we are talking about different bases (does “101” mean the binary number that represents 5 or the decimal number 101?), we often use a subscript to indicate the base of the number being represented. Thus

101_{10}

indicates the decimal number 101 and

101_{2}

indicates a binary number with the value 5.

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If you think about it for a minute, you can probably see that the largest value that we can represent with 3 binary digits is 7. With three decimal digits, we can express values up to 999. Binary numbers take more digits to express large numbers.

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With 8 bits, we can express values up to 255; to decode them, we need to add more columns to our table - each new column will be twice the value of the last column. Here is what