"Do )'OU think it helps?" clsked Doris.
"Not very much," said Jamil. "Mr. Bockcl can't
tand him."
"Well, the feeling's mutual," Robert replied.
Robert stole cl glance at the schoolyard.
As usual, he thought. A regubr dump! So much
)r his dream. But the numbers remained. He
JLdd count on the numbers.
At that moment the door flew open c1nd in
alked the inevitable Mr. Bockel with a briefcase
10ck-full of pretzels.
1 r,s
1-he Ninth Night
•
�
Robert dreamed he w,1.., dreaming. Hl' w,1.., uc;;cd w
it b, now. \Xlhenc, er he lud ,1 dream ,1bout '>ome­
thing unple,1c;;,111t-the one ,1bout bl'ing '>tr.111ded
on ,1..,lipper� rock in the middle ot ,1 r,1ging c.;tre,1m,
for in..,t.111ce-he would think to himc.;clf, Horrible
,1.., it i,, it\ all ,1 dre,1m.
But one d,1� he c1m?;ht the flu, ,111d, lying in bl'd
,111 day with a temper,uure, he found the trick
didn't work. Be..,ide'>, fever dream, were thl' worst.
The L1st time he'd been sick in bed he'd had c1
dre,1m ,1bout ,1 volc,1110 erupting. Fire-spewing
mountains Jud flung him into the air, and he was
,1bout to descend ..,Jowh·-,lowly, how curious­
into the nuw of the volc,1no. lt gave him the creeps
ju,t to think of it. So he tried to stay awake, even
though his mother kept telling him, "The bec.;t
thing to do is to slel'p it off. Don't rc,1d so much.
lt'c;; unhe,1\th, !"
I
After the twelfth comic b o ok, howe\·er, his e\rcs
were so heav\·
. that thev
. closed bv
. themsel;·es'
and what he dreamed was as str.1nu-e
as strano-e
t'J
t,
c o uld be.
He dreamed he was in bed with the flu and the
number devil was sitting ne:\.t t o him. There was a
glass of water on his bedside table and he thought,
I'm hot. I have a temperature. I don't think I'm
asleep.
"What about me?" asked the number devil."Are
you dreaming me or am I rcall:v here?"
"I'm not sure," said Robert.
"What difference d o cs it nuke?" said the
number devil. "I'm just making ,1 sick call. And
since you're ill and must stay in bed and can't
climb trees in the dese rt or c ount rabbits in the
country, I thought we'd spend a quiet eYcning at
home. I've brought along cl few numbers to take
yo ur mind off things. They're perfectly harmless,
I assure you."
"That's what yo u always say.''
Just then there was a knock on the door.
"Come in!" the number devil called out.
And in thcv
., marched. Thev
. reminded him of
racing cyclists or marathon runners, because they
spo rted their numbers o n white T-shi rt s. The�·
came in such quantities that before he knew 1�,
Robert's r o om was packed. At first he was amazed
to sec so many squeeze into so small a space, but
then he realized that as more and more crowded in,
the do or moved farther .1nd farther away, until it
sto od at the end o f a lo ng, narrow c orridor and he
could hardlv make it out.
For a while the numbers just stood there laugh­
ino-b and chatterimi: awav. Then the number devil
shouted in his best army-sergeant voice, "Attention! First row, fall in!" .rnd they immediately
lined up, backs to the wall, one at the head, the
others fo llowing in numerical order.
"\Xlhere's zero?" Robert asked.
"Zero, front and center!" the number devil
r o .ued.
Zero had hidden under the bed and crawled out,
terribly embarrassed.
"I tl;otiu-ht
I'd not be needed. I'm not myself
b
•
to da). I must be corning down with the flu. I'm
afraid I'll have to ask for sick leave."
"Dismissed!" the number devil shouted, and
zero crept b.1ck under Robert's bed.
"That zer o ! Always making problems, wanting
something special. But the others-I hope you
appreciate how well they follow order . "
He seemed tremendously pleased with a line of
perfectly ordinary numbers:
�J
�
/ 1 I 3
I � I ,
1
6
1
• I 8
I 9 / 10 / 11 / 11 / 13 J
"Second row, fall in!" he shouted, ,rnd immcdi­
,1tel> ,1 new contingent of number<; stormed in ,111d
found their pL1cc, with ,l grc,1t cLntcr ,rnd ,lrn fflc:
The, -;rood dircctl> in front of the other..,,
nuking the room look even more like an inter­
mi11.1blc tunnel. The>' were ,111 decked out in idcn­
tic.1! red T-shirt,.
"1 sec," s,1id Robert. "The odd numbers."
"Right. Now I w,rnt \ ou to guess how mam of
them there ,1re comp,u�d with �heir whitc-shi;·tcd
comrades along the wall."
"That's obv ious," s,1id Robert. "E \'en· other
number is odd, so there .1rc half ,1"> 111,111, • red .., ,1s
t I1crc ,UC \V Ilites.
. ,,
"Wlut you 're �aying is clue there arc twice ,1,
many ordinary numbers ,ls odd."
"Right."
The number devil bughcd, bur it w,1'-n 't ,1 nice
laugh. Robert thought it sounded '-,Hcastic.
"Sorn· to disappoint you, m> boy, but as you
sec, the1�e arc cxc1ctly the s,1me number of one ,1s of
the other.''
"Ridiculous!" Robert cried. "All c.rn't be the
same as half"
"\'<latch cc1rcfuJJy. ,rnd I'll show vou what I
mcc1n. ,,
He turned to the numbers and ro,ued, "First
,rnd second rows, sluke hands!"
"You don't need to scream at them, do you?''
Robert "i,1id ,rngrily. "This isn't ,111 army barracks.
Try being a little more polite."
But his protest went unheeded, bcc,msc by then
they h,1d formed pairs like tin soldiers and each
white was shaking hands with a red:
.
,'f
2
3
If-
5
6
�
8
�
10
-11
-12
13
,f
3
5
-=,.
9
11
"13
15
1J
19
2.1
Z3
Z5
"Sec? E,1ch ordi1ury number from one on has its
own odd number from one on. Can you show me
,1 single red without ,1 partner? So there is an infi­
nite quantity of ordinary numbers and an infinite
qu,111tity of odd numbers. Infinite, understand?"
1'-obert thought for .1 while.
"So if I divide .111 infinite quantit: in half
I get
two infinite quantities. But then the whole
is the
same size as the half.''
"Correct," s,1id the number de\'il. "And
not
only rlut." He pulled .1 whistl e out of his
pocket
and gave a toot. All at once ,1 new colu
mn of
numbers-this one in green T-shirts-appe
ared
our of the depths of the endless room, jigg
ling
.rnd joggling until the number devil comman
ded,
"Third row, fall in!"
In ,1 flash the greens formed a neat line in
front
of their red ,rnd white comrades:
i / 3 /
s / � / 11 /
•1 / 11 / 19 / ,�/ • •
I
3. / 3, / �·
I
''Prinu donnas," Robert concluded from the
numbers on their T-shirts.
The number devil mere!:· nodded. Then he gave
another toot on his whistle and ,mother and an­
other and another. All hell broke loose. A night­
mare! Who would have thought that so man:·
numbers could fit in ,1 single room, even if it had
by now grown .1s long ac;; the path ,1 rocket ukes to
the moon. There was no air left. Robert's head felt
like a glaring lightbulb.
"Stop! Stop! I c,rn't take an) more of this!"
I�,,
-
'
•
_J'
� ·..:.�1
, '' I
•
'
,• C()/1/l /JI.I" tlJl' 1111111/Jcr de;,,,! c,dfcd out, and Ill m,1rchcd the 1111 mbcn Ill ;lllh q11<111t1tn
•;
that bt' fore Robert !me .... ll, IJ1s ruom .... <l:, p,nkcd.
"Your flu must be getting to you," said the
number devil. "I'm sure you'll feel better tomorrow. ,,
Then he turned to the numbers and shouted
'
"Now hear this! Rows four, five, six, and seven, fall
in! On the double!"
Robert forced his drooping eyes open c1nd saw
seven kinds of numbers in white' red ' bo-reen' blue,
orange, black, and pink T-shirts standing one behind the other in neat but endless rows:
II
s
6
':J,
8
9
-10
,f,f
-f2
-13
-14
15
:,..
!J
1-1
'13
15
n
1CJ
21
23
2.S
2. -1-
?. CJ
11
13
n
1q
2.3
21:J
31
3l
lf.1
lt3
3
s
8
13 2-t 3¥
ss
89
.,.,,, 113
ltl-
10
15
21
28
lfS
S5
(,6
JI
2.
3
3
s
3
5
...
2
,�
3
6
36
311 610 .. .
91 105
'f 2.0
The numbers on the pink T-shirts were soon so
long that they barely fit, and it was all Robert
could do to read them.
"They get large so quickly! I'll never keep up."
...
"Vroom!" said the number devil. "The number
with the exclamation mark:
3.
:::
4-!:
Ax2x3
-1X2x3x4
And so on. Hard to keep up with them, isn't it? But
what about the others? Do you recognize them?"
"Let's see. The reds are odd, the greens prima
donnas, the blues-I don't know, but they look
familiar."
"Think rabbits."
"Oh, yes. The Bonaccis. Which would make the
orange numbers triangle numbers."
"Not bad. Flu or no flu, you're making progress."
"The blacks are obviously hopping numbers: 2 2,
2,- , _J{ , and so on. ,,
"And there is an equal quantity of each color,"
said the number devil.
"An infinite quantity," said Robert with a sigh.
"Awful, isn't it. A real mob scene."
"Row one through seven, dismissed!" the
number devil roared.
In a flurry of scraping, pushing, puffing, and
trampling, the numbers left the room. They were
replaced by an exquisite stillness. Robert's room
was as small and bare as before.
J 79
"All I need now is ,111 asp1n n and a glass of
w,1ter. ,,
"PJu..., c1 good rest, and :'ou'll be back on your feet
tomorrow," said the number de\'il, gent!:· tucking
Robert in.
"But do you think you can keep your eyes open
long enough to take care of what we ha\'e left?"
"Left of what?''
"The thing is," he said, w,wing his stick ,1gain,
"we booted the numbers out because the\' made
such ,1 mes... of \·our room. But we still 11.1\'c '>cries
to deal with."
"Series? What are series?"
"Well, you don't think number-; just stand there
like tin soldiers, do you? What happens when the:·
arc combined-that is, when thcv're ,1dded to­
gether?"
"I don't know wh,1t vou'rc ulking ,1bout,"
Robert moaned.
But bv then the number devil had dr,nvn the first
series on the ceiling with his stick.
"I thought you said I needed rest."
"This won't take much out of voL1.
All .vou haYc
.
to do is read what it says."
.
.
1moanc d agam. "Y uc 1<.' ''
"Fractions.' " Rooert
"What do you mean? What could be simpler
than fractions? Look at these!"
r
_c::,
\\ h l'n Robert looked up ,ll tin cci/1111!, ,1�:w1, the 1111mhcn ..:·e1c �mn· ,111d h,1d been
rcpl,1ud h_1 .1 /0111!, Inn.
1
-+-+--+-+-+1
4
1
�
�
El a II m m m
-
,I � , -
"One-half," Robert read, "plus one-quarter, plus
one-eighth, plus one-sixteenth, plus one-thirty­
second, and so on. Ones on top and hopping twos
on the bottom. The same as the black T-shirts: two,
four, eight, sixteen, thirtv-two ... And I'm sure I
know what comes next."
"Yes, but what comes from adding them all
together?"
"I have no idea, though if the series never ends
then what comes of it must be never-ending too.
On the other hand, one-quarter is less than one­
half, one-eighth is less than one-fourth, and so on.
So the numbers I add on will get smaller and
smaller."
W hen Robert looked up at the ceiling again, the
numbers were gone and had been replaced by a
long line:
"I think I get it," he said, after staring up at it for
half
a while. "You start with one-half, then add on
of one-half, in other words, one-quarter."
As he spoke, the numbers appeared in black and
white on the ceiling:
'
I
0
-v,"'
""!2-
0
1
/81
-
'1/2.
./ \,,, eor......r
1/4-
3/4
I
'1
"And you just keep going, adding on half of the
­
previous number. Half of one-quarter is one
so
eio-bhth' h,11f of one-eigV hth is one-sixteenth, and
so
on. The pieces will get smaller and smaller,
es
small that they 'll soon be invisible, like the piec
of chewing gum you divided up that first night.
t
0
_. _.,.� ...... ·- . � --r-..
I
\.-
3/t+
1/2
�
1/4
/83
I µ
1
....._,_..,,
.._,,�
"18 "'lib AIJ2. • • ·
You can keep going till you're blue in the face
r
and vou'll never reach the one. Almost, but neve
. "
qmte.
"Well, I want you to keep going."
"And I don't want to. I'm in bed with the flu,
remember?"
.I
1/2.
�
·· l hat's just the point. You
may get tired, but
numbers don't," the numbe
r devil said. "Thcv can
go on and on forever."
St� ddenly the line on the ce
iling was replaced by
the following:
-
D
"Well done!" cried the nu
mber devil. "Excel­
lent! Keep going.,,
"But I'm tired . I need to sle
ep."
"SI eep .;i" sc11'd tI1e number devil.
.
"You are sleepmg. You're dreaming of m
e, aren't you? And \'OU
can only dream if you're sle
eping." There was nothing Robert
could say to that,
though he felt his brain was
turning into jelly.
"All right, I'll go along wi
th on� more of vour
crazy ideas, but then I've go
t to rest."
The number devil raised hi
s stick. snapped his
..
f u��ers, and a whole new
series appeared on the
cedmg:
+
:1.
4-
-1
- +
1
IS-I
- +-�,.1,_
"It\ just like the last one," Robert s,1id. "I can go
on ,1dd ino- till the cows come home, but since each
number is smaller than the one before it, they'll
never add up to one."
"Is tlut what you think? Then let's look a bit
closer. At the fir;t two numbers, for instance.''
Now onlv the first two numbers of the series
were left on the ceiling:
"What's the answer?"
"I don't know," Robert muttered .
"Don't ,Kt stupid now. Which is more? One-half
or one-third?"
"One-half, of course!" Robert said, ,1nnoyc d.
"What do vou take me for?"
"Now, 1�ow. Just tell me this: Which 1s more?
One-third or one-fourth?"
"One-third, of course."
"So we have two fractions, both of which arc
more than one-fourth. And what do two-fourths
,
ma I{C.) ,
"What a dumb question! Two-fourths make a
half."
"Good.
/85
�
+
is therefore
more than
And if we take the next four terms of the series and
add them together, they too come out to be more
than one-half. Look:
,1
--
1
+5+ b -t­
"That's too complicated for me," Robert grum­
bled.
"Nonsense!" cried the number devil. "Which i
more? One-fourth or one-eighth?"
"One-fourth."
"Which is more, one-fifth or one-eighth?"
"One-fifth."
"Right. And the same hold for one-sixth and
one-seventh. Sec these following fractions?
They are all more than one-eighth. And what do
:our-eighths make?"
Robert did not even want to answer, but he
·inally said, "Four-eighths make exactly one-half."
I Sr,
"Excellent. So now we have
r 'l- -t- �'
�
more
than Y2
,1
,1 + ... -1 f- ·- , �
t- 1- -r- t /I- -r- '• t -·
l 8 q 10 11 12.
1
'--
more
than Y2
more
than Y2
And so on. Till the cows come home. You'll notice
that if we add the first six terms of the series to­
o-ether thev• come to more than one. And we can go
t,
on like this as long as we like."
"No, no!" Robert cried. "Please!"
"But if we went on-don't worry, we won'twhere would it take us?"
"To infinity, I suppose," said Robert. "How
devilish of y ou."
"Except that it would take forever. Even if we
worked at lightning speed, we wouldn't reach the
first thousands till, say, the end of the world.
That's how slowlv the series increases."
"Then let's leave well enough alone."
"Yes, let's leave well enough alone."
And with that the writing on the ceiling began
to fade, the number devil grew thinner and
thinner, and time moved on.
Robert did not wake up until the sun was tickling his nose.
JS
"Thank God the fever is gonC'," his mother �aid,
putting her hand on his forehead.
By then he had forgotten how easy and how
init\'.
slow it can be to slide from one
/ 8,\'