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Integer numbers are whole numbers; they are always
represented exactly (without a decimal point) in the computer word:
-3, 0, 379, etc. For instance, 5 (decimal) is represented exactly
by 101 (binary) in the computer. Since that's binary, the rightmost bit
represents 2-to-the-zeroth power, or "one". Since it's "on" in this example,
we have a value of "one". The next binary digit to the left represents 2
to the 1st power, or "two"; but it's off, so its value is zero. The leftmost
bit represents 2 to the 2nd power, or "four". Since it's "on", we add four
to the value. One plus four is five, which is why "101" in binary is "5"
decimal... Exactly.
All positive, non-zero integer numbers are represented in this manner. There are differences in the way zero and negative numbers are represented, depending upon whether a ones complement , twos complement , or sign-magnitude system was implemented.
Negative X is obtained by inverting each bit of X; i.e.,
-X = not(X)
The leading bit of a number is the sign bit. Positive numbers have a zero as the leading bit; negative numbers have a one as the leading bit.
One consequence of this is that there are two values for zero: +0 and -0.
For example, for a four-bit number (remember, the leading -- that is, leftmost -- bit is a sign bit):| Binary: | 0000 | 0001 | 0010 | 0011 | 0100 | 0101 | 0110 | 0111 | 1000 | 1001 | 1010 | 1011 | 1100 | 1101 | 1110 | 1111 |
| Decimal: | +0 | +1 | +2 | +3 | +4 | +5 | +6 | +7 | -7 | -6 | -5 | -4 | -3 | -2 | -1 | -0 |
Negative X is obtained by inverting each bit of X, and then adding one; i.e.,
-X = not(X)+1
The leading bit of a number is the sign bit. Positive numbers have a zero as the leading bit; negative numbers have a one as the leading bit.
There is only one value for zero: all bits are zero (as opposed to the system used for One's Complement).
For example, for a four-bit number (remember, the leading -- that is,
leftmost -- bit is a sign bit):
| Binary: | 0000 | 0001 | 0010 | 0011 | 0100 | 0101 | 0110 | 0111 | 1000 |
1001 | 1010 | 1011 | 1100 | 1101 | 1110 | 1111 |
| Decimal: | 0 | +1 | +2 | +3 | +4 | +5 | +6 | +7 | -8 |
-7 | -6 | -5 | -4 | -3 | -2 | -1 |
| Binary: | 0000 | 0001 | 0010 | 0011 | 0100 | 0101 | 0110 | 0111 | 1000 |
1001 | 1010 | 1011 | 1100 | 1101 | 1110 | 1111 |
| Decimal: | +0 | +1 | +2 | +3 | +4 | +5 | +6 | +7 | -0 |
-1 | -2 | -3 | -4 | -5 | -6 | -7 |
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