About Mathematics Commentary
Chapter 8. Arithemtic
What this chapter is aboutThis chapter is about the algorithms that support arithmetic computation. And the first question that may occur to you is:
Why wasn’t this chapter the first one in this book? There are several answers to that question: (1) I don’t want these students
to see this project as a review of the roadmap they followed through school. If this chapter had come first, I think that these
students would have been far less impressed with what the course has to offer. (2) I want them to see arithmetic as co-equal to
the other topics studied here. And (3) I want them to take these algorithms seriously because they provide the underlying structure
to the calculation that they will now more often see performed for them in this technological age. (4) The chapter comes quite
naturally before Chapter 9 on exponents and logs. Finally (5) some of these students may go on to teach in elementary schools
and this chapter will serve them well in their own understanding of part of the curriculum they will teach. Note:
In addition to teaching undergraduate students with this text, I have used it in a mixed class of elementary and secondary school
mathematics teachers. I found it interesting to observe that, despite their prior preparation and differing aims, the two groups
performed equally well.
It is important to add here that this is not a number theory chapter disguised as arithmetic. Any of you who have studied or
taught that course know how challenging are its concepts. Rather, the scope of this chapter is severely limited, but some of the
ideas are both useful and interesting.
Section Notes and Activities: Section8.1 on counting is focused on the special role of n – 1 in base n . In counting it signals both the return to
zero in its own column and the upcoming change of one digit in the next column to its left. In the (now increasingly distant) past you
could illustrate this with an odometer taken from an old car or a filling station gas pump counter. These devices were made up of a
series of disks, one for each column: units, tens, hundreds, etc. On the outside of each disk appear the ten digits, 0 through 9. Thus,
as you turn that disk you see the digits in order through a frame that allows only one at a time as distance is achieved or gas pumped.
A series of these disks are mounted so that as any disk turns from 9 to 0, a hook turns the disk to its left one digit.
Unfortunately, just as digital clocks have replaced analog clocks, these devices have been replaced by digital counters and the
effect is muted if not lost entirely.
Even if you cannot find such a counter, you can replicate this counting for base 2 with a line of four or five students each
holding a poster on one side of which is a 0 and the other a 1. With all the students displaying 0, have the rightmost student turn
his card each time you count the next decimal number. And all the students to the left follow a simple rule: turn your card when
the card to your left changes from 1 to 0. This can be a fun demonstration because, despite the simplicity of the instructions,
someone will make a mistake and you’ll have to have them start over.
Some unexpected uses of different bases: (1) A mix of decimal and base 3 can simplify baseball reporting with 5.2 representing
two outs in the fifth inning; (2) Roman numerals serve as a mix of bases five and ten with V, L and D serving as base five divisions.
Note, however, the lack of symbols for 25, 125, etc. and that there is no 0 in this system, the latter a special disadvantage of
Roman numerals. (3) An indication of the many numeral systems is the Wikipedia entry: “Category: Numeral systems”.
This might also be a good time to discuss with your students some of the technical distinctions among some arithmetic words.
During the 1970s much was made of these differences. First is the case of number and numeral, two words that are considered by most
of us, mathematicians included, to be synonyms. Some University of Illinois mathematicians considered number to be a concept and
numeral the symbol for that concept. When we talk of 8 students, the 8 is an idea responding to the query, how many students? But
it is also a symbol representing that idea, thus 8 is a numeral. One way those UI professors showed that distinction was when an
innumerate correspondent responded to the question, “What is half of 8?” with the answer, “3, the right half.” He assumed that the
8 was merely a symbol that could be split down the middle to give a 3 and a backwards 3. Thus, to him that 8 was a numeral. If you
answer that same question with “4” on the other hand, you would be treating the 8 as a number. While the difference between symbol
and idea is worth pointing out, that strict use of number and numeral has (happily) gone out of style.
The other distinction is between cardinal numbers and ordinal numbers. Cardinal numbers are those that count the number of things
like 5 oranges, 3 triangles; ordinal numbers are used to order things like George will batter third, I will do that first. A personal
experience will illustrate when the difference can be important. Four of us went out to dinner at a crowded restaurant and, one of us
in order to simplify things for the waitress, wrote down what each of us wanted — 1: hamburger, fries, coke; 2: BLT and chips, water;
3:… — to be served. He handed the list to the waitress, who came back a few minutes later with ten meals: she had taken his ordinal
numbers for the four of us: 1, 2, 3 and 4, to be cardinal numbers and brought 1 of the first, 2 of the second, 3 of the third and 4 of
the fourth order.
Section8.2 on addition extends the idea of regrouping to groups of digits that is applied in the following sections as well. One way to
address this idea is to discuss the possibility of memorizing not just the sums to 9 + 9, but to 99 + 99. Instead of memorizing 100
addition “facts” (as my teachers referred to them), they must learn 10,000. (Of course, symmetry, or alternatively commutativity,
reduces those numbers to 50 and 5000.) Given that extra memorization, addition could be done two columns at a time and would go much
faster. To place this in a historical context: today we rarely add columns of figures but in the past bookkeepers, astronomers and
mathematicians did so regularly and often with many-digit numbers. Without calculators, such computational shortcuts could reduce
the time required to calculate.
Easily the most famous such memorization was that of Leonard Euler whose mind was crammed with the first 100 prime numbers and
the squares, cubes and fourth powers of the first 100 numbers. And like a computer, he could hold 50 or more digits of accuracy in
mind while calculating. Although there is a kind of magician’s trick about such information, Euler made good use of what he knew to
publish over 500 books and articles during his lifetime, leaving another 350 to be published after he died. His production was 1/3
of all published mathematics during his lifetime.
But the point here is quite different: calculating devices today calculate so quickly and easily that they represent the equivalent
of having not just 2-digit sums memorized, but 10-digit sums immediately at hand. Thus, in effect, they have 1010
x 1010/2 = 50,000,000,000,000,000,000 sums memorized and ready to apply. And these calculators will also have an
equal number of products at hand. (Of course, many sums and even more products will have more than 10 digits and will be reported
rounded in scientific notation.) What is so remarkable is the fact that this power is not just available to users of high-powered
computers but to cheap four-bangers (so-named because they only add, subtract, multiply and divide) devices that cost today three or
four dollars. (Okay, so some of them only report eight digits. The calculation then is
108 x 108/2, still a rather large brain working for you.)
Section8.3 about subtraction introduces the idea of subtracting a value by adding the complement of that value and adjusting the result.
This algorithm is almost certainly different from the subtraction method your students learned in school, but it would be worth taking
class time to determine the methods your students do use. You could ask a student to perform a subtraction that involves regrouping,
telling the class what was done at each step. Then ask the class if any of them were taught a different method. You might have no
takers, but in one of my classes I had students demonstrate three additional methods. You can then compare their methods with
exercise (83.6).
Section8.4 about multiplication returns to the point about more complex calculation that was made in Section 8.2.
The duplation-mediation multiplication method will prove useful in the next chapter when a method is developed to
carry out powers by a shortcut. For that reason, it is a good idea to stress it here.
Section8.5 is on division and here we meet the bane of school students. (You should recognize moving into this section that this is the
most widely hated of all arithmetic procedures with second place awarded to fractions.) It is important to note that there is no
new algorithm developed here. Just as with addition and multiplication, the recipe followed is the one most commonly employed in
schools. Note: I need to qualify that claim. You might find a few students who employ a repeated subtraction method. When
dividing, say, 1449 by 63, you see how many 63 x 10 = 630s you can subtract (you can even do so one at a time). Here you can
subtract two of them and are left with 189. Now you subtract 63s and find you can subtract three of them. Thus, your answer is
2 x 10 + 3 = 23. This interesting algorithm can lead to a very different program from the one employed in this chapter.