Bit Shift Calculator

What the bit shift calculator does

This slides the bits of a number sideways, left or right, by however many positions you choose, and shows the result in decimal, binary, octal, and hexadecimal. Enter a number, pick a direction and an amount, and read off the result.

Shifting looks like pure bit fiddling, but it has a clean arithmetic meaning. Sliding bits one way multiplies the number, and sliding them the other way divides it, which is the heart of why shifts are so useful.

How to use it

1. Choose a bit size and data type. The bit size sets how wide the number is; the data type lets you enter it in decimal, binary, hex, or octal.
2. Enter the number and the shift. Give the number to shift, choose left or right, and enter how many positions to move it.
3. Press Calculate. The shifted result appears in all four bases.

Press Reset to clear the inputs.

Left shift: doubling

A left shift moves every bit one place to the left and fills the gap on the right with a 0. Because each column in binary is worth twice the column to its right, moving a bit up one place doubles what it is worth. So a left shift by one position multiplies the whole number by 2, a shift by two multiplies by 4, and in general a shift by n multiplies by 2 to the power of n.

Take 0011, which is 3. Shift it left by one and it becomes 0110, which is 6, exactly double. Shift the original left by two and it becomes 1100, which is 12, four times the start. One caution: if a 1 slides off the top past the width you are working in, it is lost, the same way a sum can outgrow its space, so allow enough bits for the result.

Right shift: halving

A right shift is the mirror image. Every bit moves one place to the right, and the bit on the far right drops off the end. Moving a bit down one place halves what it is worth, so a right shift by one divides the number by 2, a shift by two divides by 4, and a shift by n divides by 2 to the power of n.

Take 1100, which is 12. Shift it right by one and it becomes 0110, which is 6, half of 12. Shift the original right by two and it becomes 0011, which is 3. One thing to notice is that the bits which fall off the right are gone, so the division rounds down. Shifting 1101, which is 13, right by one gives 0110, which is 6, not six and a half, because the dropped bit took the leftover with it.

Why shifting is worth knowing

Multiplying and dividing by powers of two comes up constantly, and a shift does it in a single, cheap step. For a long time this made shifting a favourite way to speed up those particular multiplications and divisions, since sliding bits is simpler work for a processor than a full multiply or divide. Shifts also pair naturally with the masking operations: you can move a group of bits into position with a shift and then read or set them with AND or OR. Doubling, halving, and lining bits up are the everyday jobs shifting handles.

Questions people ask

What does shifting bits do?

It slides every bit of a number left or right by a set number of positions. A left shift multiplies the number by a power of two, and a right shift divides it by a power of two.

Why does a left shift multiply?

Because each place to the left is worth twice as much, so moving a bit up one place doubles its value. A shift of n places multiplies the number by 2 to the power of n.

Why does a right shift divide?

Because each place to the right is worth half as much. A right shift of n places divides by 2 to the power of n, and since the bits that fall off the end are dropped, the division rounds down.

What fills the empty space, and what is lost?

A left shift fills the new space on the right with 0s and can lose a bit off the top. A right shift drops the bit on the far right, which is what makes its division round down.

References

1. Arithmetic shift. Wikipedia, The Free Encyclopedia. https://en.wikipedia.org/wiki/Arithmetic_shift