THE GREATEST INTEGER SYMBOL—AN APPLICATIONS APPROACHAuthor(s): ZALMAN USISKINSource: The Mathematics Teacher, Vol. 70, No. 9 (DECEMBER 1977), pp. 739-743Published by: National Council of Teachers of MathematicsStable URL: http://www.jstor.org/stable/27961073 .

Accessed: 13/09/2014 15:34

Your use of the JSTOR archive indicates your acceptance of the Terms & Conditions of Use, available at .http://www.jstor.org/page/info/about/policies/terms.jsp

.JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range ofcontent in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new formsof scholarship. For more information about JSTOR, please contact support@jstor.org.

.

National Council of Teachers of Mathematics is collaborating with JSTOR to digitize, preserve and extendaccess to The Mathematics Teacher.

http://www.jstor.org

This content downloaded from 24.18.52.217 on Sat, 13 Sep 2014 15:34:20 PMAll use subject to JSTOR Terms and Conditions

THE GREATEST INTEGER SYMBOL?AN APPLICATIONS APPROACH

A collection of problems for classroom use.

By ZALMAN USISKIN

The University of Chicago

Chicago, IL 60637

This article has two major purposes. One is to demonstrate the surprising number of

applications of a topic that is often pre sented in books but taught without any

applications. The second is to demonstrate that real situations can be used to introduce this mathematical topic; this is what is

meant by the phrase "applications ap proach." This use of applications tends to enhance the mathematical skills and theory underlying the symbol [ ].

All elementary school students have ex

periences with the idea of rounding, both in and out of school. They know that $3.75 can "round up" to $3.80 or $4.00 and can "round down" to $3.70 or $3.00, depending on the question asked. Questions of round

ing 5 1/2 to the nearest integer require more rules (the most common being "round halves up" or "round to the even integer"), and even these rules are known by students.

The different types of rounding are easy to calculate when specific numbers are

given. But, as often is the case, these rules are a bit harder to describe and handle in their general, or symbolic, forms. These

symbolic forms are needed if we wish to understand the operation of computers or

calculators, which must round every in finite decimal because they have only finite

capacity; or if we want to symbolize the

workings of businesses, who round costs or

weights or mileage; or if we are interested in

mathematically estimating possible errors that might occur because of rounding.

The symbolization of rounding is made

easy by using the symbol [ ], commonly called the "greatest integer symbol" and

normally defined in the following manner:

[x] = the greatest integer less than

or equal to x.

For example, [ ] =

3; [73] =

73; [?J] = ?

1; [0] = 0.

The wording of the customary definition is pedagogically unfortunate, for the phrase "greatest integer less than" confuses many students and disguises the relationship to

rounding. In this article, we instead call the

greatest integer symbol [ ] the "rounding down" symbol, a name whose appropriate ness is illustrated by the examples above.

The "rounding down" (or "greatest in

teger") function, by which we mean the function with equation y

= [ ], is a simple

example of a step function, so called be cause of its graph (see fig. 1). It is also a

Fig. 1. Graph of rounding down function (Rotate 180? about origin to get graph of rounding up func

tion.)

December 1977 739

This content downloaded from 24.18.52.217 on Sat, 13 Sep 2014 15:34:20 PMAll use subject to JSTOR Terms and Conditions

good example of a function that is not con tinuous. Thus a study of the symbol [ ] can be interesting even if one is not interested in

rounding applications. But here we concen trate on applications and nontrivial prob lems?aspects not as commonly found in textbooks.

Rounding down is often used in business when someone wants to pay as little as

possible.

1. If a kitchen chair can be bought for 5 books of trading stamps, how many such chairs can be bought for books?

(Answer: 2. A salesman receives a $100 bonus for

each $1000 worth of equipment that he sells over his base requirement. How much bonus will he receive for d dollars in sales over his base requirement? (An swers to this and other questions are at the end of the article.)

It is important to realize a difference in the real-world application of questions 1 and 2,

although the mathematics is identical. To answer question 1, the symbol [ ] is prob ably never used. The consumer divides by 5 and ignores any fraction that might occur. But commissions on sales are now often calculated by computer. So in question 2, the programmer must instruct the com

puter to round down, and this is done by an

operation equivalent to the use of the sym bol [ ].

A natural question arises when rounding down: Is there a symbol for rounding up? It is clear that people round up when they

want to receive as much as possible. For

example, when a parcel is weighed for mail

ing, the weight is rounded up, not down, so that the next higher cost is charged. It may

surprise students to learn that there is no

need for a new symbol; that is, rounding up can be done in terms of the rounding down

symbol. But the problem is difficult; as so often happens in problem solving, it is best to consider first a simpler problem.

3. (Original problem.) How can rounding up be done in terms of rounding down?

(a) (Simplified problem.) What would the graph of a rounding up func tion look like? (This is an easy question to answer by trial and er ror. Students must learn to look at both integer and noninteger behav ior in this function.)

(b) (Comparison with known situa

tion.) How does the graph of the

rounding up function compare with that of the rounding down function? (Answer: Rotating the

graph of the rounding down func tion 180? about the origin gives the

graph of the rounding up function, see fig. 2).

Fig. 2. Graph of rounding up function (Some

points on graph are Or, 4), (-1, -1), (1J, 2), and

(" , -

O

(c) (Continuation of this direction.) If we have an equation for a relation, how can we get an equation for an

image under a 180? rotation about the origin? (Answer: Replace by ? and y by

? y. The answer is best

obtained by considering other ex

amples.)

(d) (Application to original problem.) So if we replace by -x and y by ?y in an equation for the rounding down function, we get the desired

740 Mathematics Teacher

This content downloaded from 24.18.52.217 on Sat, 13 Sep 2014 15:34:20 PMAll use subject to JSTOR Terms and Conditions

equation: ?y =

[? ], or in its more traditional form, y

= ?[? ].

In other words, ->

~[? ] maps a num

ber onto the smallest integer greater than or

equal to the number. For example,

7 - -[-tt] =

-(-4) =

4,

-7.2- -[-(-7.2)] =

-[7.2] =

-7,

and

6_ -[-6] = 6.

Notice the use of negative numbers to round up a positive number.

Many real situations require rounding up:

4. First-class mail now (1977) costs 13? an ounce to mail. How much postage must be paid for a parcel in this class that weighs w ounces? (This is a reason

able question if a computer is weighing parcels.)

5. If you have a charge account that adds a finance charge of 1 percent each month to each unpaid bill, how much will be added in two months to a bill of

$29.95? Here the point to be made is that it is not 0.2995 that is added to the bill the first month, but the next higher cent. Indeed,

Q3Q= -[-100(0.2995)] 100

and in general, if the unpaid balance was B, the amount added would be

-[-IOO(O.OIB)] 100

6. Can rounding to the nearest integer be written in terms of rounding down? If

so, how can it be done? If not, why not?

(To simplify this problem, you might wish to assume that halves are rounded

up.) 7. To keep the number of participants

down, an interscholastic math contest

committee decides to allow each school at most one representative for each 250

students. Suppose a school has stu

dents. How many representatives is it

allowed? (Here the phrase "for each

250 students" must be interpreted. Does this mean round up? Round down? Round to the nearest 250? As with many reasonable, practical ques tions, certain decisions must be made before a mathematical picture of the

problem can be generated. The author has no single answer for this question. Each student might decide what is fair and make an appropriate expression based on that judgment.)

8. If a taxi ride costs 40? for the first 1/4 mile and 10? for each additional 1/4 mile, find an expression relating cost C

(in cents) and miles traveled m.

9. Make up a question like question 8 for

long-distance telephone rates to the

city where someone you know lives.

The problems above are only the most obvious (even though nontrivial) instances that could use the symbol [ ]. Here are some others.

10. (Truncating.) The rounding down sym bol lops off the "decimal" part of a number. This process is called truncat

ing the number. Truncating after two decimal places was required in question 5. Any calculator or computer must truncate the decimal expansion of any number after a certain number of

places because of its finite storage ca

pacity. It is natural to ask if one could

express truncation after four (or some other number of) decimal places, using the symbol [ ]; for example, can you find an expression for a function that would contain the following ordered

pairs? (log 2, 0.3010), ( , 3.1415), (-5, -5), ( /3, 1.7320), and (1.98643, 1.9864)

11. If you work with a hand calculator that has square root and squaring keys and if you press 2, the square root key, and the squaring key?in that order?why will some calculators not list 2 as the final result?

12. (A variant of truncating.) Suppose you are dealing with large numbers and wish to instruct a computer to print the

December 1977 741

This content downloaded from 24.18.52.217 on Sat, 13 Sep 2014 15:34:20 PMAll use subject to JSTOR Terms and Conditions

numbers in thousands (as is often done with census figures). For example, 345 678 would be written as 345. What

expression indicates the number onto

which should be mapped?

A totally different collection of uses for the symbol [ ] is found by examining the intersection of the graphs of y

= [ ] and

y = . As pictured (fig. 3), it is obvious that

the set of first coordinates of the points of intersection is the set of integers.

Fig. 3

13. (Rewriting statements.) The English statement "jc is an integer" can be de noted mathematically as [x]

= x. What about "jc is not an integer"?

14. Denote each of these statements by a

single equation with the aid of the sym bol [ ]. (Answers at end of article.) (a) is an even integer. (b) is an odd integer. (c) m is divisible by 3. (d) m is divisible by n. (e) = 7 (mod 41)

15. (Sentences for graph paper (!).) Or

dinary, commercial graph paper has all the lines printed for integer values of or If we think of this as the graph of

a relation, an English sentence for that relation is "jc is an integer or y is an

integer." An equation can easily be de rived:

(M-*)-(M-jO = o.

16. What is a sentence for the lattice of all

points having integers for both coordi nates?

Finally, we come to the granddaddy of all applications of the symbol, which I first found in A Source Book of Mathematical

Applications (NCTM 1942).

17. (Day of the week.) Number the days of the week with Sunday

= 1, Monday

=

2, and so on, allowing Sunday =

8,

Monday =

9, and so on. Number the months of the year with March =

3,

April =

4, . . ., December = 12, Janu

ary =

13, February = 14. Let W = the

number of the day in the week, let D =

the day of the month, let M = the number of the month as above, and let

= the year. For January and Febru

ary, replace by - 1. Then

[*]-[&] [&]+* Test this formula by verifying that Pearl Harbor was bombed on a Sun

day, 7 December 1941.

18. All parts of the formula above are ex

plainable. The first use of the symbol [ ] is a correction for the differing num bers of days in the months. The last three uses are caused by leap years, which occur every fourth year except at the beginning of the century, unless the

century number is 2000, 2400, and so on. Why isn't the formula useful for

finding out the day of the week for a

date before 1700?

Notes, Answers, and Hints

742 Mathematics Teacher

This content downloaded from 24.18.52.217 on Sat, 13 Sep 2014 15:34:20 PMAll use subject to JSTOR Terms and Conditions

4. -13? [-H>]. Sales tax is a related notion but usually much more difficult to de scribe mathematically.

6. If halves are rounded up, [x + ?] rounds to the nearest integer.

8. Assume that the meter clicks on the

quarter-mile. C = 40 + 10[4m] 9. Car rental or other rates could also be

used.

10. y = ^ \^n^wf

^ ' Y?u should generalize

this.

11. For example, the SR-51A, SR-40, TI

30, and Monroe 99 are calculators that would give 2 as the answer, but the Novus and Omron are calculators in which the operations yield an answer

less than 2. The Novus and Omron cal culators truncate the decimal expan sion of \?2 and square this approxima tion.

12. 1000 .With the use of negative ex

ponents, exercises 10 and 12 are seen to use the same mathematical idea.

13. [ ]

14. (a)

(c)

(e)

2.

m L3

2

m 3

- 7

. 41

(b)

(d) -

1

41

-h 1

m

+ 1

m

16. ([ ] - + (

- yY = 0

18. In the eighteenth century, the calendar was moved up eleven days to correct for errors due to too many leap years.

REFERENCES National Council of Teachers of Mathematics. A

Source Book of Mathematical Applications. Seven teenth Yearbook. New York: Bureau of Pub

lications, Teachers College, Columbia University, 1942.

Graduate Comt?qs i?? irrjatrje?v^tics desdad

for- -tuacfr&ns of secondary

tecjoxred -for- MST d?gH&. COURSE OFFERINGS 8oi-??Z h?rt)*R)?ribs at>d,

2?3-8?f H?rfher Algebra

^ 3 credits Modern -Appnoacr;

3,25 Selected "Tipies Analysis <3 credits

8?<? Directed r\a3din? 3 credit?0

Dsres: ??v)Q>2b -to

f>'rector ummer Jrjsf?t te Dipt of rtarhcn)5f iob

purl]?n) AM Hampshire,

December 1977 743

This content downloaded from 24.18.52.217 on Sat, 13 Sep 2014 15:34:20 PMAll use subject to JSTOR Terms and Conditions