BINARY SHIFT
Multiplication and Division
Binary Shift


As you know a computer can only add, not subtract.
For the same token it can still only add, not multiply
or divide.
Multiplication is usually achieved by repeatedly
adding, and division is usually achieved by
repeatedly 'subtracting' (or as we know, adding
through the use of two's complement).
Binary Shift

There are occasions when you can 'cheat' in
multiplication/division, but only when the number
you are multiplying/dividing by is a number on the
binary scale eg. 2, 4, 8 etc.
We'll stick with whole numbers
Binary Shift


Firstly, you need to remember the binary scale
25
24
23
22
21
20
32
16
8
4
2
1
Now you need to pay close attention to two things:
 the
number in each column
 The exponent above the 2, that corresponds to each
number
 Eg. The number 8 has the exponent 3
Binary Shift: Left Shift

Now lets use a multiplication example:
5

x8
We can start by determining the exponent that
corresponds to 8, and re-write the question
5
x 23
Binary Shift: Left Shift



Remember the problem is: 5 x 23
Now use your columns to write down 5 in binary
Below you original value of 5, move all the bits 3
places to the left, and pad the holes with 0’s
1
25
24
23
22
21
20
32
16
8
4
2
1
0
1
1
0
1
0
0
0
Binary Shift: Left Shift


5 x 23 (or if you prefer 5 x 8) is equal to 40, this
can be easily achieved by shifting the whole
number 3 places up the binary scale.
Try the following two multiplications:
7
x4
 3 x 16
Binary Shift: Right Shift

Now lets look at a division example:
 35

/ 16
We can start by determining the exponent that
corresponds to 16, and re-write the question
 35
/ 24
Binary Shift: Right Shift



Remember the problem is: 35 / 24
Now use your columns to write down 35 in binary
Below you original value of 35, move all the bits 4
places to the right
25
24
23
22
21
20
.
2-1
2-2
2-3
2-4
32
16
8
4
2
1
.
.5
.25
.125
.0625
1
0
0
0
1
1
.
.
1
0
. 0
0
1
1
Binary Shift: Right Shift


35 x 24 (or if you prefer 35 / 16) is equal to
2.1875, this can be easily achieved by shifting the
whole number 4 places down the binary scale.
Try the following two divisions:
 17
/4
 131 / 32
Binary Shift

At this point you should know that:
 Multiplying
is a Left Shift (because you are making the
number bigger)
 Dividing is a Right Shift (because you are making the
number smaller)
 If one of the numbers in the multiplication/division is a
number on the binary scale, check the exponent of that
number and move the other number up/down the scale
by the number of places in the exponent