Martin measures the length of a shelf that he wants to put up in his living room. The length, 𝑙, of the shelf is 82 centimetres to the nearest centimetre. Write down the error interval that results from rounding.
The error interval represents the range of values that a number could have been before rounding. Let’s consider a number line. If 82 centimetres is what the result was after rounding to the nearest centimetre, if a value is halfway between 81 and 82, that value is rounded up. Halfway between 81 and 82 is 81.5, 81 and a half, anything greater than or equal to. The values between 81 and a half and 82 all round to 82 centimetres. And that also tells us that anything that equals 82 and a half rounds up to 83.
But what about the values between 82 and 82 and a half? These values round down to 82. We can write this in the form of an inequality. We let 𝑙 be the length of the shelf. It could be any value greater than or equal to 81 and a half, but less than 82 and a half.
We can read the inequality like this. 81 and a half centimetres is less than or equal to 𝑙, the length of the shelf, which is less than 82 and a half centimetres. We can also say that rounding to the nearest centimetre is plus or minus a half a centimetre. 82 minus a half gives the lower bound, and 82 plus one-half gives the upper bound.
And then you have to remember that when dealing with the lower bound, it’s equal to or greater than, while the upper bound is only less than, not equal to.