Accuracy and Error
Some math problems require an exact answer, while for others, an approximate answer is good enough.
When a problem involves measurement of a real-world quantity, there is always some level of approximation happening. For example, consider two different problems involving measures of time. If the problem involves sprinters running the -meter dash, you will need to use precise measurements: hundredths of a second or better. If the problem involves people’s ages, it’s usually enough to approximate to the nearest year.
Sometimes when a problem involves square roots or other irrational numbers like pi , or even a complicated fraction, it’s useful to use decimal approximations—at least at the end, when you’re reporting your answer.
Deanna is making a wire fence for her garden in the shape below. Find the length of the hypotenuse of the triangle in meters.
Using the Pythagorean Theorem, we get:
So the hypotenuse is meters long. That’s the exact answer. But having the answer in this format isn’t very useful if we’re trying to build something. How do you cut a piece of wire m long?
A good calculator will tell you the value of correct to decimal places:
Even this is an approximation. But it’s a much better approximation that you need. In this case, rounding the value to the nearest centimeter (hundredth of a meter) is probably enough.
The accuracy of a measurement or approximation is the degree of closeness to the exact value. The error is the difference between the approximation and the exact value.
When you’re working on multi-step problems, you have to be careful with approximations. Sometimes, an error that is acceptable at one step can get multiplied into a larger error by the end.
A plastic disk is the shape of a circle exactly inches in diameter. Find the combined area of such disks.
Suppose you use as an approximation for . Using the formula for the area of a circle, we get the area of one disk as:
Multiply this value by to get the combined area of disks.
This gives an answer of square inches. But wait!
See what happens when we use a more accurate value for like :
Multiply this by to get the combined area:
This is almost square inches more than our previous estimate!