posted 18 years ago
Machines can't manage the purely abstract concept of a number. It has to know how to store the number, and it has to know how much space it needs. What makes the most sense at the machine level is binary (or base 2) representation.
At the lowest level of a calculating machine, a byte value of 42 is represented in binary digits as 00101010. That is, you have 8-digits to represent a number, but each column is a power of 2, not a power of 10. Instead of having columns for 1s, 10s, 100s, 1000s, etc., you have 1s, 2s, 4s, 8s, 16s, 32s, 64s, 128s. The only allowed values in each column are 0 and 1. So for 42, the columns with 1 in them are 'on': 32, 8, and 2. The rest are 'off'.
The data type supplied by a programming language defines all this for your convenience. A byte on most computer systems is 8-bits wide. If you turned all the bits on, you'd have the sum of all its columns, or 256. However, for the Java byte, short, int and long types, the leftmost bit expresses positive (0) or negative (1) value. This leads to some other necessary rules: after all, you don't want to call byte value 1000000 "negative zero." There are some other important conventions to know, but I'll leave them out here.
So an 8-bit number can store the range of numbers from -2^7 to (2^7 - 1), or -128 to 127. I guess you could say zero is 'positive,' if you wanted to say this is an even division of positive and negative values.
A short is 16-bits wide, so you can cover from -2^15, or 32768, to 2^15-1, or 32767.
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