A calculator won’t work very well to solve this problem. Try it and see! You can’t get the
right answer by any straightforward arithmetic operation on twenty-four and six. If you attack
this problem as I would, you’ll count out loud starting with tomorrow, July twenty-fifth
(under your breath): “twenty-five, twenty-six, twenty-seven, twenty-eight, twenty-nine, thirty,
thirty-one, one, two, three, four, five, six!” While jabbering away, I would use my fingers to
count along or make “hash marks” on a piece of paper (Fig. 1-1). You might use a calendar and
point to the days one at a time as you count them out. However you do it, you’ll come up with
thirteen days if you get it right. But be careful! This sort of problem is easy to mess up.
Don’t be embarrassed if you find yourself figuring out simple problems like this using
your fingers or other convenient objects. You’re making sure that you get the right answer
by using numerals to represent the numbers. Numerals are tailor-made for solving number
problems because they make abstract things easy to envision.Toothpicks on the table
Everyone has used “hash marks” to tally up small numbers. You can represent one item by
a single mark and five items by four marks with a long slash. You might use objects such as
toothpicks to create numerals in a system that expands on this idea, as shown in Fig. 1-2. You
can represent ten by making a capital letter T with two toothpicks. You can represent fifty
by using three toothpicks to make a capital letter F. You can represent a hundred by making
a capital letter H with three toothpicks. This lets you express rather large numbers such as
seventy-four or two hundred fifty-three without having to buy several boxes of toothpicks
and spend a lot of time laying them down.
In this system, any particular arrangement of sticks is a numeral. You can keep going this
way, running an F and H together to create a symbol that represents five hundred. You can
run a T and an H together to make a symbol that represents a thousand. How about ten
thousand? You could stick another T onto the left-hand end of the symbol for a thousand, or
you could run two letters H together to indicate that it’s a hundred hundred! Use your imagi-
nation. That’s what mathematicians did when they invented numeration systems in centuries
long past.4 Counting Methods
(24) 25 26 27 28 29 30 311234 5 6JulyAugustFigure 1- 1 How many days pass from the afternoon of July 24
until the afternoon of August 6? You can make marks
on a piece of paper and then count them to figure out
the answer.