Multiplication and Division

UMBC CMSC 211

Multiplication and Division

Multiplication is truly a shortcut for addition, while division is a shortcut for subtraction. Four times three is equal to four plus four plus four. Division is 1) twelve minus four 2) eight minus four and 3) four minus four. Forunately, the hardware designers now let us do it with a single instruction.

However, it is still a little more complex than before. First of all, it takes longer. There is also the problem with the size of numbers (even more so than addition and subtraction!) 99 times 99 is 9801. A byte times a byte can result in a word! Multiplying two n-digit numbers will probably result in a 2n-digit number. Division goes the other direction, but is even more difficult because we have to keep track of the remainder. Then comes the wrinkle of signs!

So when you multiply two sixteen bit numbers, expect the answer to require more bits! The 80x86 CPU requires special registers for this:

Operand Size Multiplicand Multiplier Product

BYTE AL Reg or Memory AX

WORD AX Reg or Memory AX (low) and DX (high)

Operand Size	Multiplicand	Multiplier	Product
BYTE	AL	Reg or Memory	AX
WORD	AX	Reg or Memory	AX (low) and DX (high)

Division is similar:

Operand Size Dividend Divisor Quotient Remainder

BYTE AX Reg or Memory AL AH

WORD AX and DX Reg or Memory AX DX

Operand Size	Dividend	Divisor	Quotient	Remainder
BYTE	AX	Reg or Memory	AL	AH
WORD	AX and DX	Reg or Memory	AX	DX

You can not use a constant operand with multiplication or division. Use the mov instruction to put a constant into the appropriate register.

Did I mention something about the sign? There are different instructions based on whether or not it is a signed or unsigned operation. That leads us to the following:

Instruction Description

mul unsigned multiplication

div unsigned division

imul signed multiplication

idiv signed division

Instruction	Description
mul	unsigned multiplication
div	unsigned division
imul	signed multiplication
idiv	signed division

Yes, but....

Since the multiplicand and dividend must be in the AX or AL or AX and DX, how do I keep them straight. Simply, let the computer do it! It is implied.
When you need a word (or double word) and have a byte (or word), you must convert the value. There is a special set of instructions for that, because we need to make sure the number will continue to have the proper sign (without us having to write a lot of code.)
cwd ; convert the signed word in ax to a double word in ax and dx
cbw ; convert the signed byte in al to a word in ax
OK, lets put it together:
To do integer division with signed variables (word-sized)of miles driven, divided by number of gallons of gas used:
```
mov ax, miles
cwd             ;ax is assumed
idiv gallons
mov mpg, ax
```
To find out how many weeks and days are in a period of days, we could use unsigned bytes and do:
```
mov al, nrDays
sub ah, ah      ;don't sign extend because that 
                ;  would change the value
mov bl, 7
div bl          ;can't use a constant here!
mov weeks, al
mov days, ah    ;the remainder is the portion 
                ;  of a week
```
Very Important Reminders
- Multiplication and Division can only take place in the ax/dx registers in exactly the form described!
- You cannot multiply or divide directly by a constant. Move the constant to a register first.
- You must convert 8-bit dividends to 16-bit and 16-bit dividends to 32-bit.
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Multiplication and Division

Yes, but....

Very Important Reminders