We are all used to seeing numbers written down, and to drawing some meaning from them. However, a numeral such as 6 and the number that it represents are not one and the same thing. In Roman numerals, for example, we would write the number six as VI, but we realize that this stands for the same number that is written as 6 in modern notation. Both symbolize collections of the kindcorrespondingtosixtallymarks:IIIIII.Weshallfirst spend a little time considering the different ways that we represent and think about numbers.
We sometimes solve number problems almost without realizing it.For example, suppose you are conducting a meeting and you want to ensure that everyone there has a copy of the agenda. You can deal with this by labelling each copy of the handout in turn with the initials of each of those present. As long as you do not run out of copies before completing this process, you will know that you have a sufficient number to go around. You have then solved this problem without resorting to arithmetic and without explicit counting. There are numbers at work for us here all the same and they allow precise comparison of one collection with another, even though the members that make up the collections could have entirely different characters, as is the case here, where one set is a collection of people, while the other consists of pieces of paper.
What numbers allow us to do is to compare the relative size of one set with another.
In the previous scenario you need not bother to count how many people were present as you did not have to know–your problem was to determine whether or not the number of copies of the agenda was at least as great as the number of people, and the value of these numbers was not required. You will, however, need to take a count of the number present when you order lunch forfifteen and certainly, when it comes to totting up the bill for that meal,someone will make use of arithmetic to work out the exact cost,even if the sums are all done on a calculator.