# Rounding errors and estimating

## In a nutshell

Rounding errors are an application of rounding that helps find the highest and lowest possible value of a quantity. Estimating is a very useful skill when you have to do complicated calculations without a calculator.

## Rounding errors

### Upper and lower bounds

When given a rounded quantity, it's not possible to find out what the true value of the quantity was. However, errors give an upper and lower bound of what the number could have been. To find a rounding error, follow this procedure.

#### procedure

1. | Half the unit place value that's being rounded. |

2. | Add this to the rounded number for the **upper bound** (the highest possible value). Subtract this from the rounded number for the **lower bound** (the lowest possible value). |

##### Example 1

*The mass of an object is *$4650kg$, *rounded to the nearest ten. What are the upper and lower bounds for the mass?*

*First, halve the unit place value that's being rounded. In this case, it's ten.*

$10\div2=5$

*Add this number to the rounded number (which is $4650$) for the upper bound:*

$4650+5=4655$

*The upper bound is *$\underline{4655kg}$.

*Subtract $5$ from the rounded number for the lower bound:*

$4650-5=4645$

*The lower bound is *$\underline{4645kg}$.

**Note:** If you actually round $4655$* to the nearest ten, it actually rounds up to *$4660$*. The upper bound isn't actually part of the bound itself, it's more like an **upper limit**. It's more intuitively expressed through inequalities.*

### Expressing rounding errors with inequalities

The way to express rounding errors with inequalities is:

Lower bound $\leq x <$ Upper bound ($\star$)

Where $x$ is the "true" value - it represents the value of the quantity before being rounded. This deals with the fact that the upper bound isn't actually part of the bound, because the "less than" symbol $(<)$ means that $x$ cannot be equal to the upper bound.

##### Example 2

*Let $M$ be the mass of the object in the above example. Write down the rounding errors as an inequality.*

*From the above example, the upper bound is $4655kg$ and the lower bound is $4645kg$. Putting these into the expression ($\star$) gives:*

$\underline{4645kg\leq M < 4655kg}$

## Estimating

When estimating a calculation, round every number to one or two significant figures, depending upon what you think is an appropriate level of precision. Use the symbol $(\approx)$ which means "approximately equal to".

##### Example 3

*Estimate the value of $\frac{87.3-39.92}{11.4}$.*

*First, round all three numbers to an appropriate number of significant figures:*

$87.3\mapsto90$

$39.92\mapsto40$

$11.4\mapsto10$

*Now, do the calculation with these rounded numbers:*

$\frac{87.3-39.92}{11.4}\approx\frac{90-40}{10}=\frac{50}{10}=5$

$\underline{\frac{87.3-39.92}{11.4}\approx5}$