Code_ The Hidden Language of Computer Hardware and Software - Charles Petzold [142]
00010100 00110010 01010001 00100000 00100101
or, in hexadecimal:
14h 32h 51h 20h 25h
Notice the nibble at the far left is 1 to indicate a negative value. That's the sign bit. It would be 0 if the number were positive. All the digits in the number require 4 bits each, and you can read them directly from the hexadecimal values.
If you needed instead to represent values from –99,999,999.99 through 99,999,999.99, you'd need 6 bytes—5 bytes for the 10 digits and a whole byte just for the sign bit.
This type of storage and notation is also called fixed-point format because the decimal point is always fixed at a particular number of places—in our example, at two decimal places. Notice that there's nothing actually stored along with the number that indicates the position of the decimal point. Programs that work with numbers in fixed-point format must know where the decimal point is. You can create fixed-point numbers with any number of decimal places, and you can mix and match these numbers in the same computer program. But any part of the program that does arithmetic on the numbers has to know where the decimal points are.
Fixed-point format works well only if you know that numbers aren't going to get too large for the memory location that you've mapped out and that you won't need more decimal places. Where fixed-point format utterly fails is in situations in which numbers can get very large or very small. Suppose you need to reserve an area of memory where you can store certain distances in units of feet. The problem is that these distances can range all over the place. The distance from the earth to the sun is 490,000,000,000 feet, and the radius of the hydrogen atom is 0.00000000026 feet. You'd need 12 bytes of fixed-point storage to accommodate values that can get as large and as small as these.
We can probably work out a better way of storing numbers such as these if we recall that scientists and engineers enjoy specifying numbers using a system called scientific notation. Scientific notation is particularly useful for representing very large and very small numbers because it incorporates a power of ten that allows us to avoid writing out long strings of zeros. In scientific notation, the number
490,000,000,000
is written
4.9 x 1011
and the number
0.00000000026
is written
2.6 x 10-10
In these two examples, the numbers 4.9 and 2.6 are called the fraction part, or the characteristic, or sometimes (although this word is more properly used in conjunction with logarithms) the mantissa. But to be more in tune with the terminology used with computers, I'm going to call this part of scientific notation the significand.
The exponent part is the power to which 10 is raised. In the first example, the exponent is 11; and in the second example, the exponent is –10. The exponent tells you how many places the decimal point has been moved in the significand.
By convention, the significand is always greater than or equal to 1 and less than 10. Although the following numbers are the same,
4.9 x 1011 = 49 x 1010 = 490 x 109 = 0.49 x 1012 = 0.049 x 1013
the first is preferred. That's sometimes called the normalized form of scientific notation.
Notice that the sign of the exponent indicates only the magnitude of the number and not whether the number itself is negative or positive. Here are two examples of negative numbers in scientific notation:
–5.8125 x 107
is equal to
–58,125,000
and
–5.8125 x 10-7
is equal to
–0.00000058125
In computers, the alternative to fixed-point notation is called floating-point notation, and the floating-point format is ideal for storing small and large numbers because it's based on scientific notation. But the floating-point format as used in computers employs binary numbers written in scientific notation. The first thing