One of the things we can do with binary data is shift the number of bits left or right. This gives us a very quick way of multiplying or dividing any binary number by 2.

In this lesson, we’ll learn about:

- What binary shifting is.
- Why we use binary shifting.

## 1. What Binary Shifting Is

Binary shifting is where we take any binary number and then shift it to the left or the right. We then replace the empty space with a 0.

We can see this in action by demonstrating a binary shift to the left.

### Worked Example 1 – Shifting to the Left

Let’s take the binary number 110 and shift it to the left by 2 places.

128 | 64 | 32 | 16 | 8 | 4 | 2 | 1 |
---|---|---|---|---|---|---|---|

0 | 0 | 0 | 0 | 0 | 1 | 1 | 0 |

Shifting our number to the left one place gives us an empty space. We simply put a 0 in the empty space. This gives us 1100.

128 | 64 | 32 | 16 | 8 | 4 | 2 | 1 |
---|---|---|---|---|---|---|---|

0 | 0 | 0 | 0 | 1 | 1 | 0 | 0 |

We can now shift the number by another space. Again, we’ll fill the empty space with a 0. This gives us 11000.

128 | 64 | 32 | 16 | 8 | 4 | 2 | 1 |
---|---|---|---|---|---|---|---|

0 | 0 | 0 | 1 | 1 | 0 | 0 | 0 |

So, all we’ve done is add two 0’s to the right-hand side.

We’ll now look at how a binary shift to the right-hand side works.

### Worked Example 2 – Shifting to the Right

Let’s use the same binary number, 110. This time we’ll shift it to the right by 1 place.

128 | 64 | 32 | 16 | 8 | 4 | 2 | 1 |
---|---|---|---|---|---|---|---|

0 | 0 | 0 | 0 | 0 | 1 | 1 | 0 |

Shifting to the right will knock off the far right-hand bit, leaving us with 11. We’ll make sure we add a 0 to the left-hand side of the number.

128 | 64 | 32 | 16 | 8 | 4 | 2 | 1 |
---|---|---|---|---|---|---|---|

0 | 0 | 0 | 0 | 0 | 0 | 1 | 1 |

But what if we wanted to shift to the right again? What would happen with the 1? Well, the 1 would be shifted off the end and replaced with a 0 on the left of our binary sequence.

128 | 64 | 32 | 16 | 8 | 4 | 2 | 1 |
---|---|---|---|---|---|---|---|

0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 |

### Further Thought

What do you think would happen if we did a binary left shift and a 1 was pushed off the left-hand side?

## 2. Why We Use Binary Shifting

Shifting in binary is a very quick method of multiplying or dividing any number by 2 every time you shift one place.

Let’s look at our first example.

We had the binary number 110, which in decimal is 6.

128 | 64 | 32 | 16 | 8 | 4 | 2 | 1 |
---|---|---|---|---|---|---|---|

0 | 0 | 0 | 0 | 0 | 1 | 1 | 0 |

We then shifted it 1 place to the left which gave us 1100 which is 12 in decimal.

128 | 64 | 32 | 16 | 8 | 4 | 2 | 1 |
---|---|---|---|---|---|---|---|

0 | 0 | 0 | 0 | 1 | 1 | 0 | 0 |

6 x 2 = 12!

We then shifted it another place to the left which gave us 11000, which is 24 in decimal.

128 | 64 | 32 | 16 | 8 | 4 | 2 | 1 |
---|---|---|---|---|---|---|---|

0 | 0 | 0 | 1 | 1 | 0 | 0 | 0 |

12 x 2 = 24!

Similarly, shifting to the right allows us to divide by 2 for each shift.

Again, we start with 110, which is 6.

128 | 64 | 32 | 16 | 8 | 4 | 2 | 1 |
---|---|---|---|---|---|---|---|

0 | 0 | 0 | 0 | 0 | 1 | 1 | 0 |

Then we shift it one place to the right and we get 11, which is 3 in decimal.

128 | 64 | 32 | 16 | 8 | 4 | 2 | 1 |
---|---|---|---|---|---|---|---|

0 | 0 | 0 | 0 | 0 | 0 | 1 | 1 |

6 ÷ 2 = 3!

Of course, if we shifted it to the right again, we’d be left with 1 (both in binary and decimal). We know that 3 ÷ 2 = 1.5, not 1.

The reason this has happened is because in this form of binary we’re not storing decimals, so it’s discarded.

### Further Thought

How would we multiply or divide by numbers other than 2?

(HINT: there is no divide or multiply operation inside the CPU.)

## Lesson Summary

So to summarise what we’ve learnt in this lesson:

- Binary shifting is when you move a binary number to the left or right.
- When shifting to the left, we add a new 0 onto the right-hand side of our binary sequence.
- When shifting to the right, we add a new 0 onto the left-hand side of our binary sequence.
- If you shift to the left, this is the same as multiplying your number by 2.
- If you shift to the right, this is dividing your number by 2.