Error Intervals
Neil Trivedi
Teacher
Error Intervals
When a number has been rounded, the original value may not be exactly equal to the rounded value.
For example, if a length is given as to the nearest the real length could be or even because all these round to
Let’s look at a number line to visualise all the numbers that round to to the nearest
The above number line is only true when talking about the integers. However, there are numbers between and which also round to to the nearest We already stated above that rounds to to the nearest and so does In fact, we can keep attaching more to the end of the number and we will still produce numbers that round to to the nearest
If you take your calculator and type in and keep pressing until it goes off your calculator screen and then press it will give you This is because mathematically, and are the same number.
However, we cannot say that is the largest number that rounds to as we know that rounds to to the nearest Therefore, we say is the upper bound, but it cannot equal this. remains the smallest possible number that rounds to
We can summarise this using inequalities. Let’s use to represent the possible values that can round to to the nearest then we can write the interval as:
This says that is between (included) and (excluded). This is called an error interval. Since we were looking at numbers that round to to the nearest we say that is the error. Notice how the difference between the upper bound and the lower bound is (the error).
Therefore, there is a quick way to finding the lower and upper bound of an error interval summarised below.
An error interval shows the range of possible values before a number was rounded.
To find an error interval:
1) Identify the error.
2) Find half of the error.
3) Subtract this from the rounded value to find the lower bound.
4) Add this to the rounded value to find the upper bound.
The lower bound is included, but the upper bound is not included and the inequality has the form
Lower bound Upper bound
Example 1:
Find the error interval for each value.
a) to the nearest metre.
Step 1: Identify the error.
The value has been rounded to the nearest metre which means the nearest metre, so the error is
Step 2: Find half of the error.
Step 3: Subtract and add to get the lower and upper bounds.
Lower bound:
Upper bound:
Therefore,
b) seconds to the nearest tenth of a second.
Single step: Add and subtract half of the error.
The error here is a tenth which is equal to
Half of the error is
Adding and subtracting half the error, we get
Lower bound:
Upper bound:
Therefore,
c) to the nearest
Single step: Add and subtract half of the error.
The error is which when halved is
Therefore, our interval will be
which simplifies to
d) correct to significant figure.
Step 1: Identify the error.
The value is correct to significant figure.
The first significant figure is which is in the thousands column.
So, the value has been rounded to the nearest
Step 2: Add and subtract half of the error.
Half of is
Therefore,
Which simplifies to
Truncation and Error Intervals
To truncate means to cut off a number after a certain point without rounding.
For example, if then,
rounded to decimal places is since the rounds the up.
truncated to decimal places is because we simply cut off everything after the second decimal place.
It looks like this:
Let’s say we did not know what the actual value of was and we were just given truncated to decimal places. When thinking about the lower and upper bound here it is a little different since there is no “error”.
We simply think about the smallest and largest digits that could come after the
The smallest number would be so just followed by infinite Remember, we are ignoring those zeros anyway! So, the lower bound is which is the original number.
The largest would be so just followed by infinite However, just like we said at the beginning that is the same as here we would say that is the same as however, it cannot equal
For a truncated decimal:
1) Keep the lower bound the same as the given number
2) Add to the last digit kept, in that same decimal place, to get the upper bound
3) The upper bound is not included.
For example: if truncated to decimal place, then
Example 2:
Find the error interval for in each case.
a) when truncated to decimal place.
Single step: Find the lower and upper bounds.
If when truncated to decimal place, the number must start with So, the lower bound is
To find the upper bound:
Add to this digit
We get as the upper bound. The number can get very close to but it cannot equal
Therefore,
b) when truncated to decimal places.
Single step: Find the lower and upper bounds.
If when truncated to decimal place, the number must start with So, the lower bound is
To find the upper bound:
Add to this digit
We get as the upper bound. The number can get very close to but it cannot equal
Therefore,
Example 3:
A science club is making slime for a school open day.
They expect to make pots of slime, to the nearest
Each pot needs of solution, correct to significant figures.
Each bottle contains litres of solution, correct to the nearest
Work out the minimum number of bottles the science club should buy to make sure they do not run out.
Step 1: Decide which bounds are needed.
The science club wants to make sure they do not run out. So, we need the worst case:
1) to make the greatest possible number of pots,
2) each pot needing the greatest possible amount of solution,
3) each bottle of solution containing the smallest possible amount.
Step 2: Find the upper bound for the number of pots.
The number of pots is to the nearest
The error here is which when divided by we get
The error interval is therefore,
However, since we want the greatest number of pots, which is an integer, the largest number we can have is
Step 3: Find the upper bound for the solution per pot.
Each pot needs correct to significant figures.
The second significant figure is which is in the ten's column, so the error here is which when divided by we get
The error interval is therefore,
Therefore, the greatest possible amount of solution per pot is
Step 4: Find the lower bound for each bottle.
Each bottle contains litres correct to the nearest
The error here is which when divided by we get
The error interval is therefore,
Therefore, the smallest possible amount in each bottle is
Step 5: Calculate the number of bottles needed.
The greatest possible amount of solution needed is pots of solution
Each bottle could contain as little as
So, the number of bottles needed is
Since the science club cannot buy part of a bottle, they must round up.
Therefore, the science club should buy bottles.
Practice Question