Code_ The Hidden Language of Computer Hardware and Software - Charles Petzold [141]
4 x 104 +
2 x 103 +
7 x 102 +
0 x 101 +
5 x 100 +
6 x 10-1 +
8 x 10-2 +
4 x 10-3
Some rational numbers aren't so easily represented as decimals, the most obvious being ⅓. If you divide 3 into 1, you'll find that ⅓ is equal to
0.3333333333333333333333333333333333333333333333333333…
and on and on and on. It's common to write this more concisely with a little bar over the 3 to indicate that the digit repeats forever:
Even though writing ⅓ as a decimal fraction is a bit awkward, it's still a rational number because it's the ratio of two integers. Similarly, is
0.1428571428571428571428571428571428571428571428571428571…
or
Irrational numbers are monsters such as the square root of 2. This number can't be expressed as the ratio of two integers, which means that the decimal fraction continues indefinitely without any repetition or pattern:
The square root of 2 is a solution of the following algebraic equation:
x2 – 2 = 0
If a number is not a solution of any algebraic equation with whole number coefficients, it's called a transcendental. (All transcendental numbers are irrational, but not all irrational numbers are transcendental.) Transcendental numbers include π, which is the ratio of the circumference of a circle to its diameter and which is approximately
3.1415926535897932846264338327950288419716939937511…
Another transcendental number is e, which is the number that this expression approaches:
as n gets very large, or approximately
2.71828182845904523536028747135266249775724709369996…
All the numbers we've been talking about so far—rational numbers and irrational numbers—are called real numbers. This designation distinguishes them from the imaginary numbers, which are square roots of negative numbers. Complex numbers are combinations of imaginary numbers and real numbers. Despite their name, imaginary numbers do show up in the real world and are used (for example) in solving some advanced problems in electronics.
We're accustomed to thinking of numbers as continuous. If you give me two rational numbers, I can give you a number between those two numbers. In practice, all I have to do is take an average. But digital computers can't deal with continuums. Bits are either 0 or 1, with nothing in between. So by their very nature, digital computers must deal with discrete values. The number of discrete values you can represent is directly related to the number of bits you have available. For example, if you choose to store positive integers using 32 bits, the values that you can store are the whole numbers from 0 through 4,294,967,295. If you need to store the value 4.5, you must rethink your approach and do something different.
Can fractional values be represented in binary? Yes they can. The easiest approach is probably binary-coded decimal (BCD). As you might remember from Chapter 19, BCD is a binary coding of decimal numbers. Each decimal digit (0, 1, 2, 3, 4, 5, 6, 7, 8, and 9) requires 4 bits, as shown in the following table:
Decimal Digit
Binary Value
0
0000
1
0001
2
0010
3
0011
4
0100
5
0101
6
0110
7
0111
8
1000
9
1001
BCD is particularly useful in computer programs that work with money amounts in dollars and cents. Banks and insurance companies are just two obvious industries that deal with money a lot; in computer programs used by these sorts of companies, many of the fractional numbers require just two decimal places.
It's common to store two BCD digits in 1 byte, a system that's sometimes called packed BCD. Two's complements aren't used with BCD. For this reason, packed BCD also usually requires an extra bit to indicate whether the number is positive or negative. This is called the sign bit. Because it's convenient to have a particular BCD number stored in a whole number of bytes, that one little sign bit usually involves sacrificing 4 bits or 8 bits of storage.
Let's look at an example. Suppose the amounts of money that your computer program needs to work with never get as high as $10 million in either the positive or negative