6 Numbers and Numerals

You will need: A-blocks (Material Cards 2A, 2B, 2C, 2D)

Most of us in the modern world pretty much take the concept of numbers and counting for granted. In fact, we take language for granted –but that topic is too involved to go into right now. Imagine what a breakthrough it was when people first thought to give names to objects. Eventually they used words, which we will call number names, to describe sets of objects (remember sets?!) in terms of their size. This was a considerable mathematical advancement. Although there are still some cultures that have not assigned names for numbers larger than three, most cultures have developed this concept. Some tribes even have different number names for different types of objects.

One difficulty with the concept of numbers as we use them today is that they are concepts and not necessarily used only for counting objects. The number four, for instance, is used in many ways –four years old, room 4, four feet high, 4 children, four hours, four minutes after eight, mail station 4, 4 fingers, a grade of 4 on a quiz, count to 4, four times bigger than something, 4th place, four wishes, fourth in line, call 444-4444, code 44, etc.

So, what is a number?

For example, the number three is the concept of three items. The symbol 3 is a numeral representing the concept three. You can’t see or touch a number. The symbols that we see and touch that are used to represent numbers are numerals. In our culture, we use the Hindu-Arabic system where the numerals are made up of one or more of these ten symbols –0,1,2,3,4,5,6,7,8 and 9. For instance, in the Hindu-Arabic system, the number two million is represented by writing the numeral two million which we write like this: 2,000,000. Notice, it took one symbol to write the number for three and seven symbols to write the number for two million (we don’t count the commas as symbols because they aren’t essential to writing the numeral). Although it took seven symbols to write the numeral two million, note that there are only two different symbols used to write that numeral –a two (2) and a zero (0).

NOTE: Two sets that match DO NOT have to match in a natural or pleasing way the way we might match the colors of our clothes together. It might be easier to think of the word pairing instead. Think of showing a matching between two sets up by taking one element in the first set and pairing it up to dance with one element in the second set. Now that they are “dancing” together, take another element from the first set and pick someone in the second set for it to dance with. Continue doing this until all elements are have a partner. This is a matching! If your first set had 5 girls and your second set had 4 boys, they wouldn’t match because one girl would be left without a partner. That is why it is necessary to have the same number of elements in each set in order to represent a matching.

Exploring Numeration Systems

Today, most but not all civilizations around the world use the Hindu-Arabic system of counting and numerals 1,2,3,4,5,6,7,8,9,10,11,12,… There are many advantages to this number system. While you might think this common, widely used system of counting the obvious choice, you might consider both the advantages and disadvantages of Hindu-Arabic as we learn about other numeration systems.

Here are three different ways to write the numeral sixty-five in the Old Egyptian system:

[latex]\bigcap\bigcap\bigcap\bigcap\bigcap\bigcap|||||\qquad\qquad\text{or}\qquad\qquad\bigcap\bigcap||\bigcap|\bigcap\bigcap||\bigcap[/latex]

or as shown below with all the symbols enclosed in a visual set

You simply add up the values of each of the symbols to get the answer.

In this additive system, the Egyptian numeral

𓁨 𓂭 𓆼 𓆼 𓆼 𓏲 𓏲 | | | |

represents 1,000,000 + 10,000 + 1,000 + 1,000 + 1,000 + 100 + 100 + 1 + 1 + 1 + 1 = 1,013,204.

Remember: The order in which the symbols are written is irrelevant in an additive system.

The Egyptian system employs the technique of grouping, where a certain number of the same symbols are grouped together and replaced with a new symbol. This is necessary in order to symbolize very large numbers efficiently. Systems that use this grouping principle are said to be of a certain base depending on how many symbols it takes to exchange to a new symbol. For instance, a system that groups six symbols together and then replaces them with a new symbol is said to be a Base Six system.

From now on, because this is how they are used today, write all Roman numerals using the least number of symbols possible and the subtractive principle where applicable unless you are otherwise instructed.

In the next section we will look at two more systems, but for now we’ll end this section by comparing the systems we have worked with so far. First, we will consider the number of symbols one has to memorize in order to understand each individual system. We will then determine how many symbols one has to write down in order to represent various numerals in the different systems.