“Magic Numbers” Approach to Introducing Binary Number
Representation in CS0
Gerald Kruse
Juniata College
Huntingdon, PA USA
kruse@juniata.edu
Categories and Subject Descriptors
K.3.2 [Computer and Information Science Education]:
Computer science education.
1. INTRODUCTION
This is a playful introduction to binary numbers, which can be
used in an “Introduction to Computer Science” or CS0 course.
While there is no formal assessment of this demonstration’s
effectiveness, students seem to understand binary numbers well
and are comfortable applying this knowledge when converting
them to other bases. In addition, the interaction and increase in
energy level resulting from the demonstration translates to a
fruitful follow-up discussion.
9
10 11
our familiar decimal system, with 10 digits means we start “rolling
over,” like a car odometer after the last digit is used. I end up
writing from 0 to 15 on the board, as in Figure 1 (a). Then I ask,
“What happens if we only have two digits to count with?” The
answer is that we start “rolling over” much sooner. I write from 0
to 15 in binary on the board, fairly close to the base 10 count, as
in Figure 1 (b). Next, I pad with 0’s, so each binary
0
1
10
11
100
101
110
(a)
(b)
111
1000
1001
10
1010
11
1011
12
1100
13
1101
14
1110
15
1111
Figure 1. Decimal and Binary numbers.
Copyright is held by the author/owner(s).
.
ITiCSE’03, June 30–July 2, 2003, Thessaloniki,
Greece
4
5
6
7
(b)
12 13 14 15
3
6
12 13 14 15
7
(c)
I begin by asking the students to help me count. I point out that
ACM 1-58813-672-2/03/0006.
8
(a)
2
2. THE TECHNIQUE
0
1
2
3
4
5
6
7
8
9
number is represented with 4 bits. Next, I ask for a student
volunteer. They come up to the board, write a number between 1
and 15 so everyone in the room can see it, except me, then erase
the number and return to their seat. Then I show each of the four
cards in Figure 2, asking the class to say, “Yes” when their
number appears on a card, or “No” when it does not appear. We
might iterate through this several times. As we do, I will remind
1
3
5
7
(d)
10 11 14 15
9 11 13 15
Figure 2. Four cards used to determine the number.
the students that binary (digital) computers have circuits with just
two states: ON (or 0) and OFF (or .1), so it is natural to use binary
numbers to model digital circuits. I will also point out that their
“Yes” or “No” responses can be considered as binary digits.
Finally, if the suspense has become unbearable because no one
has guessed how the demonstration works, I’ll take the most
recent number written on the board and work through it.
For example, if the number was 9, I successively produce each
card. Using the labels in Figure 2, we see that 9 prompts a “Yes”
to card (a) in Figure 2, “No” to card (b), “No” to card (c), and
“Yes” to card (d). I tell the class that whenever they said, “Yes,”
I added the smallest number on the card.
So 9 = Yes, No, No, Yes = 1001 = 8 + 0 + 0 + 1 . Finally, I
point out that each number on card (a) has a 1 in the fourth
column (from the right) in its binary representation, each number
on card (b) has a 1 in the third column in its binary representation,
each number on card (c) has a 1 in the second column in its binary
representation, and each number on card (d) has a 1 in the
rightmost column in its binary representation.
ACKNOWLEDGMENTS
Our thanks to Jennifer Davenport, of Denver, Colorado, for
introducing us to the magic trick that this tip is based on.