Page 199 - ARM 64 Bit Assembly Language

P. 199

CHAPTER 7

Integer mathematics

There are some differences between the way calculations are performed in a computer versus
the way most of us were taught as children. The ﬁrst difference is that calculations are per-
formed in binary instead of base ten. Another difference is that the computer is limited to a
ﬁxed number of binary digits, which raises the possibility of having a result that is too large to
ﬁt in the number of bits available. This occurrence is referred to as overﬂow. The third differ-
ence is that subtraction is performed using complement addition.
Addition in base b is very similar to base ten addition, except that the result of each column is
limited to b − 1. For example, binary addition works exactly the same as decimal addition, ex-
cept that the result of each column is limited to 0 or 1. The following ﬁgure shows an addition
in base ten and the equivalent addition in base two.

1 11
75 01001011
=
+ 19 + 00010011
94 01011110

The carry from one column to the next is shown as a small number above the column that it
is being carried into. Note that carries from one column to the next are done the same way
in either base. The only difference is that there are more columns in the base two addition
because it takes more binary digits to represent a number than it does in decimal.

7.1 Subtraction by addition

Finding the complement was explained in Section 1.3.3.3. Subtraction can be computed by
adding the radix complement of the subtrahend to the menuend. Example 11 shows a com-
plement subtraction with positive results. When the menuend is less than the subtrahend, the
result will be negative. In the complement method, we deal with this possibility by adding a
sign digit at the beginning of the numbers. Example 12 shows complement subtraction with
negative results. Example 13 shows several more signed addition and subtraction operations
in base ten and binary.

194 195 196 197 198 199 200 201 202 203 204