# Rounding Methods

## What is "Rounding" ?

Rounding means reducing the digits in a number while trying to keep its value similar. *The result is less accurate, but easier to use. *

Example: 7.3 rounds to 7

Because 7.3 is closer to 7 than to 8

But what about **7.5**? Is it closer to 7 or closer to 8? It is half-way in between, so what should we do?

## Half Round Up (the common method of rounding)

The common method of rounding is to make 0.5 go **up**, so 7.5 rounds up to 8 (see Rounding Numbers).

7.5 usually rounds up to 8

But this is not a law or anything, it is just what people normally agree to do, and we get this:

- 7.6 rounds up to 8
- 7.5 rounds up to 8
- 7.4 rounds down to 7

## Half Round Down

But ** 5 could go down** if we wanted. In this case 7.5 would round down to 7, and we get this:

- 7.6 rounds up to 8
- 7.5 rounds down to 7
- 7.4 rounds down to 7

But you should let people know you are using "Half Round Down".

Why would you make 0.5 go down? Maybe you have lots of 0.5's in your numbers and want to see what rounding down would do.

Have a Play ... try different rounding methods on the Rounding Tool.

## Negative Numbers

But what about **-7.5**?

- Does it round to -8 (and is that going "up" or "down" ?),
- Or does it round to -7

Help! I am confused!

In fact the whole world is confused about rounding negative numbers ... some computer programs round -7.5 to -8, others to -7

But we can agree **here** that "up" means heading in a positive direction, like on this number line:

## Half Round Up (including negative numbers)

So we get this:

- 7.6 rounds up to 8
- 7.5 rounds up to 8
- 7.4 rounds down to 7
- -7.4 rounds up to -7
- -7.5 rounds up to -7
- -7.6 rounds down to -8

## Half Round Down (including negative numbers)

If we were rounding 0.5 down we would get this:

- 7.6 rounds up to 8
- 7.5 rounds down to 7
- 7.4 rounds down to 7
- -7.4 rounds up to -7
- -7.5 rounds down to -8
- -7.6 rounds down to -8

## "Symmetric" Rounding

But maybe you think "7.5 rounds up to 8, so -7.5 should go to -8", which is nice and symmetrical.

Well you are in luck because that is** rounding towards or away from zero**:

## Round Half Away From 0

For this method, 0.5 rounds the number so it is **further away from zero**, like this:

- 7.6 rounds away to 8
- 7.5 rounds away to 8
- 7.4 rounds to 7
- -7.4 rounds to -7
- -7.5 rounds away to -8
- -7.6 rounds away to -8

## Round Half Towards 0

Or you can have 0.5 round the number closer to zero, like this:

- 7.6 rounds away to 8
- 7.5 rounds to 7
- 7.4 rounds to 7
- -7.4 rounds to -7
- -7.5 rounds to -7
- -7.6 rounds away to -8

## But Being Consistent Can Be Bad

Choosing any of those methods **can** be bad, though!

Especially if you have lots of 0.5s, like 5.5, 7.5, 6.5, 9.5, etc. The rounding would all be going in the same direction and you would have a **bias**.

So you can decide to round towards even (or odd) numbers! It works like this:

## Round to Even

You round 0.5 to the nearest **even** digit

Example:

7.5 rounds **up** to **8** (because 8 is an even number)

but 6.5 rounds **down** to **6** (because 6 is an even number)

Other numbers (not ending in 0.5) round to nearest as usual, so:

- 7.6 rounds up to 8
- 7.5 rounds
**up**to**8**(because 8 is an even number) - 7.4 rounds down to 7
- 6.6 rounds up to 7
- 6.5 rounds
**down**to**6**(because 6 is an even number) - 6.4 rounds down to 6
- etc

## Round to Odd

Just like "Round To Even", but 0.5 heads towards odd numbers

Example:

7.5 rounds down to** 7** (because 7 is an odd number)

but 6.5 rounds **up** to **7** (because 7 is an odd number)

## Round Randomly

You could also choose to round 0.5 up or down randomly, but how? By tossing a coin? Or a computer function?

With a large list of numbers this may give good results, but would give a different answer each time. Unless you used a fixed list of random choices.

## Floor and Ceiling

There are two other methods that don't even consider 0.5. They are called Floor and Ceiling.

Floor and ceiling give you the **nearest integer** up or down.

### Example: What is the floor and ceiling of 2.31?

The Floor of 2.31 is **2**

The Ceiling of 2.31 is **3**

## Floor

Using "floor", all digits go down, no matter what the dropped digit is:

### Example: 7.8 goes down to 7

so does 7.2, 7.5, 7.9, etc.

And 7 goes to 7, too.

## Ceiling

And "ceiling" goes up:

### Example: 7.1 goes up to 8

so does 7.2, 7.5, 7.8, etc.

But **7 stays at 7**.

## Summary

Number | Half Up | Half Down | Half Away 0 | Half To 0 | Half Even | Half Odd | Floor | Ceiling |
---|---|---|---|---|---|---|---|---|

8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 |

7.6 | 8 | 8 | 8 | 8 | 8 | 8 | 7 | 8 |

7.5 | 8 | 7 | 8 | 7 | 8 | 7 | 7 | 8 |

7.4 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 8 |

7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 |

-7 | -7 | -7 | -7 | -7 | -7 | -7 | -7 | -7 |

-7.4 | -7 | -7 | -7 | -7 | -7 | -7 | -8 | -7 |

-7.5 | -7 | -8 | -8 | -7 | -8 | -7 | -8 | -7 |

-7.6 | -8 | -8 | -8 | -8 | -8 | -8 | -8 | -7 |

-8 | -8 | -8 | -8 | -8 | -8 | -8 | -8 | -8 |

## Rounding to Tens, Tenths, Whatever ...

In our examples we rounded to a whole number, but you can round to tens, or tenths, etc:

### Example: "Half Round Up" to tens:

15 rounds up to 20

14.97 rounds down to 10

### Example: "Half Round Up" to hundredths:

0.5168 rounds up to 0.52

1.41119 rounds down to 1.41