Arithmetic shift

numbers are multiplied and divided using arithmetic shifts, also known as binary shifts.

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Multiplying and dividing binary numbers using arithmetic shifts, also known as binary shifts

Multiplication

To multiply a number, an arithmetic shift moves all the digits in the binary number along to the left and fills the gaps after the shift with 0:

to multiply by two, all digits shift one place to the left
to multiply by four, all digits shift two places to the left
to multiply by eight, all digits shift three places to the left
and so on

Example – calculate 00001100 (denary 12) × 2

Place value	128	64	32	16	8	4	2	1
Value					1	1	0	0

Place value	Value
128
64
32
16
8	1
4	1
2	0
1	0

Result: shifting one place to the left gives 00011000 (denary 24)

Place value	128	64	32	16	8	4	2	1
Value				1	1	0	0	0

Place value	Value
128
64
32
16	1
8	1
4	0
2	0
1	0

Example: 00010110 (denary 22) × 4

Place value	128	64	32	16	8	4	2	1
Value				1	0	1	1	0

Place value	Value
128
64
32
16	1
8	0
4	1
2	1
1	0

Result: shifting two places to the left gives 1011000 (denary 88)

Place value	128	64	32	16	8	4	2	1
Value		1	0	1	1	0	0	0

Place value	Value
128
64	1
32	0
16	1
8	1
4	0
2	0
1	0

Remember that shifting outside of the available heading values may result in an overflow error.

Division

To divide a number, an arithmetic shift moves all the digits in the binary number along to the right:

to divide by two, all digits shift one place to the right
to divide by four, all digits shift two places to the right
to divide by eight, all digits shift three places to the right
and so on

Example: 100100 (denary 36) ÷ 2

Place value	128	64	32	16	8	4	2	1
Value			1	0	0	1	0	0

Place value	Value
128
64
32	1
16	0
8	0
4	1
2	0
1	0

Result: shifting one place to the right gives 10010 (denary 18)

Place value	128	64	32	16	8	4	2	1
Value				1	0	0	1	0

Place value	Value
128
64
32
16	1
8	0
4	0
2	1
1	0

Example: 1111 (denary 15) ÷ 2

Place value	128	64	32	16	8	4	2	1
Value					1	1	1	1

Place value	Value
128
64
32
16
8	1
4	1
2	1
1	1

Result: shifting two places to the right gives 111 (denary 7). Note - 15 ÷ 2 = 7.5. However, in this form of binary, there are no decimals, and so the decimal is discarded.

Place value	128	64	32	16	8	4	2	1
Value						1	1	1

Place value	Value
128
64
32
16
8
4	1
2	1
1	1

Example: 110110 (denary 54) ÷ 4

Place value	128	64	32	16	8	4	2	1
Value			1	1	0	1	1	0

Place value	Value
128
64
32	1
16	1
8	0
4	1
2	1
1	0

Result: shifting two places to the right gives 1101 (denary 13)

Place value	128	64	32	16	8	4	2	1
Value					1	1	0	1

Place value	Value
128
64
32
16
8	1
4	1
2	0
1	1

Negative numbers and binary shifts

There are two different types of binary shift which work differently for negative numbers: arithmetic and logical.