How Many Is a Zillion? Sources of Number Distortion
Lance J. Rips
Northwestern University
Send correspondence to:
Lance Rips
Psychology Department
Northwestern University
2029 Sheridan Road
Evanston, IL 60208
Email: rips@northwestern.edu
Phone: 847-491-5947
Keywords: Number concepts, Numerical reasoning, Representations of mathematics
How Many Is a Zillion? / 2
Abstract
When young children attempt to locate the positions of numerals on a number line, the positions
are often logarithmically rather than linearly distributed. This finding has been taken as evidence that the
children represent numbers on a mental number line that is logarithmically calibrated. This article reports
a statistical simulation showing that log-like positioning is a consequence of two factors: the bounded
nature of the number line and greater uncertainty about the meaning of the larger, less frequent number
words. Two experiments likewise show that even college students produce log-like placements under the
same two conditions. In Experiment 1, participants identified positions on a number line for a set that
included both conventional and fictitious numbers (e.g., a zillion). In Experiment 2, participants did the
same for conventional numbers that included some larger, unfamiliar items (e.g., a nonillion). Both
experiments produced results better fit by logarithmic than by linear functions.
How Many Is a Zillion? / 3
Our most primitive indicator of numerical information may be a continuous mental quantity that
varies directly with the number of objects we perceive. This approximate number system (or analog
magnitude system) renders larger numbers of entities as larger continuous amounts: The larger the
number of perceived objects, the larger the amount of activation (e.g., Dehaene, 1997; Gallistel &
Gelman, 1992; Wynn, 1992). Thus, the amount of activation can serve as an approximate measure of the
cardinality of a set. The approximate number system can detect the number of objects in a visual array,
the number of tones in a sequence, and the number of jumps of a puppet (see Feigenson, 2007, for a
review). Evidence from many tasks suggests that this system is at work in animals, infants, and adults
(Gallistel, Gelman, & Cordes, 2006).
Studies of the approximate number system agree that the system is differentially sensitive to
small numbers of objects. For example, the system represents the difference between 4 objects and 5 as
larger than the difference between 14 objects and 15. The system appears to follow a ratio principle: For
example, humans and animals perform similarly in discriminating 5 objects from 10 as in discriminating
10 objects from 20. One way to understand this pattern is to view the system as transducing the number of
entities into an internal measure approximately equal to its logarithm. Because log(10) – log(5) =
log(20) – log(10), the logarithmic transformation correctly predicts the data on equal ratios, in accord
with classical Fechnerian psychophysics (e.g., Marks & Algom, 1998).
Results supporting a logarithmic transformation come from experiments in which American or
European children and Amazon natives (both children and adults) are asked to locate the position of
specific numerals on a number line, such as Figure 1a (for children, see Berteletti, Lucangeli, Piazza,
Dehaene, & Zorzi, 2010; Booth & Siegler, 2006; Geary, Hoard, Nugent, & Byrd-Craven, 2008; Opfer &
Siegler, 2007; Siegler & Booth, 2004; Siegler & Opfer, 2003; for Amazonians, see Dehaene, Izard,
Spelke, & Pica, 2008). On each trial, participants see a bare number line, with only the end points labeled,
for example, at 0 and 100. Participants also see a numeral, such as 73, and they must mark its position on
How Many Is a Zillion? / 4
the line. A graph of the observed positions of participants’ marks against their true positions often
produces a distribution that is better fit by a logarithmic than by a linear function. American children
show an approximately logarithmic function for the range 0-100 until second grade, when the functions
become more nearly linear. Even second graders, however, produce log-like functions for the range
0-1000, attaining linear spacing only in fourth to sixth grade (Booth & Siegler, 2006; Opfer &
Siegler, 2007; Siegler & Opfer, 2003). The function for Amazon natives is log-like even for the 0-10
range (Dehaene et al., 2008).
Controversy surrounds the exact shape of the function relating estimated position to true position
on the number line (Barth & Palladino, 2011; Cohen & Blanc-Goldhammer, 2011; Ebersbach, Luwel,
Frick, Onghena, & Verschaffel, 2008; Landy, Silbert, & Goldin, in press; Moeller, Pixner, Kaufmann, &
Nuerk, 2009). (See Barth, Slusser, Cohen, & Paladino, 2011, and Opfer, Siegler, & Young, 2011, for
debate on this issue). However, no one doubts the descriptive result that young children’s placement
exhibits compression with respect to the larger numbers. The present paper examines factors that could
contribute to this compression. In what follows, I will continue to use log-like to refer to such
distributions, with the understanding that other, non-logarithmic functions could account for them.
Two Assumptions about Number Lines
An alternative account of the number-line data begins with the idea that younger children are
more uncertain about the meaning of the larger numbers in the tested range. Kindergarteners, for example,
may be more familiar with the meanings of numerals near 0 than with those near 100, since the numerals’
frequency of occurrence decreases with their size (Dehaene & Mehler, 1992). Lack of familiarity may
translate into greater uncertainty about the position of the numerals on the number line. Increasing
variability with larger numerals is a hallmark of some theories of the approximate number system (e.g.,
Gallistel et al., 2006; Gibbon, Church, & Meck, 1984), and these models present an alternative to the
view that people represent cardinality on an internally compressed log-like scale. Cantlon, Cordes,
Libertus, and Brannon (2009) and Cohen and Blanc-Goldhammer (2011) have pointed out that these
How Many Is a Zillion? / 5
alternative systems can also help explain the number-line placement results. For present purposes,
however, we need not commit to an internal representation with increasing variability. All that is essential
is that children’s unfamiliarity with the meaning of the larger numerals can lead to greater variability in
their positioning of these numerals (see Matthews & Chesney, 2011).
One further fact is helpful in explaining the number-line data. Because the physical number line is
bounded (e.g., at 0 and 100 in Figure 1a), children are constrained to place the tested numbers within
these limits. Thus, the distribution of possible positions for a numeral is truncated by the ends of the
number line (Cohen & Blanc-Goldhammer, 2011). This truncation can lead to estimates of the numeral’s
position that are too close to the center (see Huttenlocher, Hedges, & Prohaska, 1988, for a similar model
of other bounded estimation tasks).
A Statistical Simulation of Number-Line Placement
The unfamiliarity of larger numbers and the boundedness of the number line suffice to explain the
log-like placement of the numerals. Figure 1b shows the results of a statistical simulation that
incorporates these assumptions. The individual distributions in the figure represent the possible
placements of the numerals one to nine on a number line that extends from 0 to 10. (This smaller range is
adopted for simplicity.) Each distribution was generated from a Gaussian whose mean was exactly equal
to the corresponding number. For example, the right-most distribution comes from a Gaussian with
mean 9. The standard deviations of these distributions, however, increase in proportion to the number.
This increase captures the idea that the larger numerals are associated with greater uncertainty. The
individual distributions are also truncated at 0 and 10: If a sampled number fell outside this range, it was
discarded and replaced with a new random number from the same underlying distribution.
The black vertical lines in Figure 1b show the means of the resulting truncated distributions.
Although the underlying distributions have true means at 1, 2, 3, …, 9, the means of the corresponding
truncated distributions are compressed at the top. For example, the difference between the means for the
truncated distributions for one and two is almost exactly 1, but the difference between the means for eight
How Many Is a Zillion? / 6
and nine is only 0.46. Similarly, the mean for the numeral one is at position 1.00 on the x-axis, but the
mean for nine is shifted to 8.14. Thus, the mean positions of the truncated distributions show log-like
spacing similar to children’s positioning of the numerals in the experiments cited earlier.
Two Experimental Simulations of Number Line Placement
College students know that the natural numbers have a linear structure: They know, at least
implicitly, that the naturals have a unique first number, 0, and that each of the remaining numbers is
generated from the immediately preceding one by adding one (see Leslie, Gelman, & Gallistel, 2008;
Rips, Bloomfield, & Asmuth, 2008). They therefore know that the difference between successive
numbers is linearly spaced and not logarithmically spaced. College students may retain an analog
magnitude system and display evidence for such a system in speeded judgments and estimation tasks
(e.g., Moyer & Landauer, 1967). But they would be unable to arrive at correct answers to even elementary
arithmetic problems (e.g., 5 – 4 = 15 – ?) unless they understood the correct spacing of the numbers.
Despite their knowledge of natural numbers, college student may yet show log-like placement of
numerals on a number line if they are unfamiliar with the larger items in the range. This prediction
follows from the same two assumptions embodied in the simulation of Figure 1b: First, the larger
unfamiliar numerals should be associated with greater uncertainty in their positions on the number line.
And, second, truncation at the ends of the line skews the resulting distributions. The two experiments in
this article test this prediction by asking college students to place numerals on a number line, following
the procedure of the developmental studies cited earlier. The present experiments, however, include
numerals whose values are unknown or unfamiliar to adults.
Experiment 1: From Zero to a Gazillion
Participants’ task in this study was to place a series of 20 numerals on number lines. The task
followed the procedure of earlier number-line placement experiments, but with a novel set of numerals.
Nine of the numerals, items a-i in Table 1, are conventional numbers, including some very large ones.
How Many Is a Zillion? / 7
The remaining numerals (items j-t in Table 1) were fictitious, such as forty-three umptillion and twentyfive, that are sometimes used to express large, arbitrary amounts. For each numeral, participants marked
the position on a number line where they thought it should go. The line itself resembled the one in
Figure 1a, but with the upper end labeled “One Gazillion” to accommodate the fictitious numerals.
Method
Procedure. Each participant received a booklet containing a page of instructions followed by 20
pages of number lines. The instructions showed how to put a mark on the line and then continued:
Although some of the numbers will be ordinary whole numbers …, other numbers will be
fictitious. For example, you may be asked about numbers like “three bagillion” that aren’t
among the usual integers. For these numbers, please do your best to decide where the
numbers should go on the line if they were true numbers.
On each of the next 20 booklet pages, one of the numerals from Table 1 appeared, centered above a blank
number line (108 mm in length), whose ends were labeled “Zero” and “One Gazillion.” Participants
marked the position of the numeral on the line with pen or pencil. The Table 1 numerals appeared in the
booklet in a new random order for each participant. (All numerals appeared in the natural-language form
of Table 1, since there is no mathematical notation for the fictitious numbers.)
Materials. All the stimulus numerals except for three have the form: x-y-number word-z-w (e.g.,
twenty-three thousand forty-five). The numbers x-y and z-w were both chosen randomly from a uniform
distribution of the integers 10 to 99. The intent was to keep the items approximately constant in length,
and about equal in the precision they convey. The conventional numerals followed the American English
system (see the dictionary entry for “Number,” 1961). The fictitious number words came from a
Wikipedia article on “Indefinite and Fictitious Numbers” (2012) and are attested in novels and other
sources (citations to these sources appear in the article).
How Many Is a Zillion? / 8
Participants. Twenty-three college students participated in this study as part of a requirement for
an introductory psychology course. Two had high school precalculus or statistics as their highest-level
math course; the rest had taken calculus in either high school or college.
Results and Discussion
Participants should be uncertain about the positions of the fictitious numerals and the larger
conventional numerals. If uncertainty can produce compression in the placement of these numerals, we
should observe this compression in the data. Figure 2a plots the mean position of the numerals on the
number line and shows this is the case. The points indicate the numerals, with labels corresponding to
those in Table 1. In seven cases, participants skipped a page or marked a position off the number line.
These instances are treated as missing data.
The first six conventional numbers (points a-f in Figure 2a) are correctly ordered in the mean
data, relative to each other. However, the ordinal positions of sixty quintillion and ten and eighteen
sextillion and forty-five (points h and i) are reversed, and several fictitious numbers precede these in the
sequence. These two numerals, along with the fictitious items, are also associated with large standard
errors, in accord with their relative unfamiliarity.
Of course, we can’t plot the observed position of these numerals against their true positions, since
the fictitious numbers have no true value. Still, we can see whether a linear or a logarithmic function best
predicts the observed values, just as we could ask whether the position of beads on a string are linearly or
logarithmically spaced. If oi is the obtained ordinal position of numeral i, and di is its metric distance
along the number line, the question is whether Equation (1a) or (1b) yields better predictions:
(1)
a. di = m + boi
b. di = m + b log oi
Linear and logarithmic regression evaluated these equations, and both models significantly predicted the
data (for the best-fitting linear model, F(1,18) = 128.36, p < .001, R2 = .88; for the best-fitting logarithmic
model, F(1,18) = 307.29, p < .001, R2 = .94). The logarithmic model, however, provided a better fit than
How Many Is a Zillion? / 9
the linear one, as appears in Figure 2b. A comparison of the absolute values of the residuals from these
models showed smaller deviations for the log model, t(19) = 3.03, p = .007, d = .957.
Much the same conclusion about the superiority of the logarithmic function holds, even if we
restrict attention to the conventional numbers. Figure 2a shows that these items (points a-i) are roughly
equally spaced on the number line, with the exceptions of b and c and the reversal of h and i, mentioned
earlier. For example, the obtained distance between 28,032 (point c) and 75,000,035 (point d) is about the
same as the latter is from 29,000,000,091 (point e). However, the true numeric difference between the
second pair is about 400 times the difference between the first pair! To examine this effect, we can fit
logarithmic and linear functions to the means from the nine conventional numbers, using equations like
(1a) and (1b), but replacing the ordinal values oi with the true numeric values. The linear model this time
failed to produce a significant effect (F(1,7) = 1.58, p = .24, R2 = .18), but the logarithmic model did
(F(1,7) = 197.87, p < .001, R2 = .96). This finding coincides with recent results by Landy et al. (in press).
In that study, college students located the position of numbers on a number line whose ends were labeled
“1 thousand” and “1 billion.” A substantial percentage of participants (about 35%) positioned numbers
near one million at about the midpoint of the scale, despite the fact that the difference between one billion
and one million is 1000 times greater than the difference between one million and one thousand. These
participants apparently believed that one thousand, one million, and one billion are about equally spaced
on the number line.
Note, though, that the distribution of the means for the conventional numbers appears to be loglike, even when these means are plotted against their ordinal positions, as in Figure 2b. If participants
knew the true ordering of these numbers (and took them to be less than the fictitious numbers), then the
means should instead plot as a straight line in the figure. This is true even if participants incorrectly
believed the numbers were equally spaced. Figure 2a also shows that the mean positions of these items
were not especially close to the upper endpoint. So truncation may not suffice to explain their log-like
shape. Although several reasons for this finding are possible, one simple explanation is that participants
sometimes used the middle of the number line as a kind of “don’t know” response when dealing with big
How Many Is a Zillion? / 10
conventional numbers like eighteen sextillion and forty-five, about whose value they were uncertain. (For
similar tendencies with other response scales, see Schwarz & Hippler, 1987.)
Except for three, each numeral had the form x-y-number word-z-w (e.g., seventy-five million and
thirty-five). Although these numbers were randomly assigned, the prefix number (e.g., seventy-five) or the
suffix number (e.g., thirty-five) might have affected the positioning of the numerals. However,
correlations of these values with the mean position on the number line were slightly negative and nonsignificant (for the prefixes, r(17) = -.11, and for the suffixes, r(17) = -.23, p > .10 for both).
Experiment 2: From Zero to a Google
The results from Experiment 1 show that college students place numerals in a log-like pattern on
a number line if the larger numbers are unfamiliar. That experiment ensured unfamiliarity by using
fictitious numbers, but the same prediction should emerge if the numbers are conventional but sufficiently
large. The present study repeats the procedure of Experiment 1, but replaces the fictitious numbers with
conventional ones, such as eleven decillion and seventy-one, which the participants have probably not
often encountered. One advantage is that each numeral has a true value, so we can assess observed versus
correct positions over the full range of stimuli. Table 2 lists the stimulus numerals, along with their
numeric values. Participants placed each numeral on a number line that was labeled with zero on the left
and one google on the right. The expectation is that participants’ placement of these numerals will
continue to display log-like structure.
Method
The procedure in this experiment duplicated that of Experiment 1. The only change concerned the
materials. First, the number word stimuli were the natural-language terms in Table 2 (see the dictionary
entry for “Number,” 1961). Second, the number lines were anchored at zero and one google. One google
(= 10100) was taken as the upper anchor because it is larger than the Table 2 numbers but not
morphologically related to them. Of course, uncertainty about the meaning of the anchor, as well as
How Many Is a Zillion? / 11
uncertainty about the meaning of the numerals, can contribute to log-like placement (Matthews &
Chesney, 2011). The line itself was 118 mm.
Participants were 25 college students, all of whom had taken at least a high school calculus class.
One participant made a mark that fell off the number line, and this point is treated as missing in the
analyses.
Results and Discussion
The number placements from this study show much the same crowding at the top as in the first
experiment. The mean of the first seven numerals are in correct order, but the remaining numerals are
compressed and often incorrectly sequenced. Figure 3a plots these trends, with labels corresponding to the
items in Table 2. As in Experiment 1, the standard errors of the means typically increase from left to right,
suggesting greater uncertainty about the positions of the larger numbers. Participants may have based
some of their decisions on the English word forms, since the numerals containing quintillion and
quindecillion and those containing sextillion and sexdecillion appear (incorrectly) in adjacent positions.
However, this was not always the case: Septillion and septdecillion are not especially close.
The main question about these data is whether a linear or a logarithmic function gives a better
account of the numerals’ spacing. Regressions based on Equations (1a) and (1b) show that both functions
significantly predict the means, but that the log function produces the better fit. For the linear function,
F(1,18) = 74.52, p < .001, R2 = .80. For the log function, F(1,18) = 323.65, p < .001, R2 = .95. A paired ttest of the absolute values of the residuals shows smaller deviations from the logarithmic model,
t(19) = 5.85, p < .001, d = 1.85. Figure 3b displays the predicted values from the two equations, and the
results appear quite similar to those from Experiment 1, despite the new, conventional values for the
larger numbers. We can also examine the relative accuracy of linear and log equations based on the
numerals’ true numeric values, this time using all the stimulus items. As in Experiment 1, the linear
model does not significantly fit the data (F(1,18) = 1.17, p = .29, R2 = .06), but the log model is highly
significant (F(1,18) = 44.14, p < .001, R2 = .71).
How Many Is a Zillion? / 12
General Discussion
Despite the fact that college students know the structure of the natural numbers, their placement
of numerals on a number line is not always linear. When the set of numerals includes both large items and
fictitious ones, as in Experiment 1, they assigned positions to the numbers that are better fit by a log than
by a linear function. Similarly, when all the stimulus numerals are conventional, but some are very large,
placement was again log-like rather than linear. These conclusions hold whether the functions are based
on the obtained ordinal positions of the numerals or on the numerals’ true values. The data thus agree
with the statistical simulation: Uncertainty about the meaning of unfamiliar numerals, together with a
bounded scale, can produce log-like positioning on a number line.
These statistical and experimental findings imply that the log-like placements aren’t conclusive
evidence for an internal, logarithmically-calibrated representation of the natural numbers. They therefore
provide a skeptical challenge to conclusions from earlier studies. However, a proponent of log-calibration
might question the basis of this challenge. Although college students know the organization of the
numbers, they may nevertheless resort to a logarithmic representation when faced with unfamiliar
numerals, such as those in the present experiments. Under speeded conditions, adults show size effects in
comparing the values of single-digit numbers (e.g., Moyer & Landauer, 1967)—for example, they are
faster at deciding that 3 is less than 4 than that 7 is less than 8. These results have been taken to support
the existence of an approximate number system that coexists with the linear representation people need
for official arithmetic. College students might likewise fall back on a logarithmic representation because
of their lack of knowledge of the numbers.
The present results, however, do not question the idea that people have both an approximate
number system and an official arithmetic system. The issue is whether the number-line placement data
provide evidence for a log-calibrated internal representation (one particular instantiation of the
approximate system). What the simulations highlight are other ways to explain the same data (see, also,
Cantlon et al., 2009; Cohen & Blanc-Goldhammer, 2011; Landy et al., in press; Matthews & Chesney,
How Many Is a Zillion? / 13
2011). Moreover, fictitious numbers presumably don’t have positions on an internal line, and the same
may well be true of rarely encountered conventional numbers. Participants might temporarily assign these
numbers to points at the high end of an internal logarithmic line, but it is unclear how or why they would
do so.
A stronger argument in favor of a mental log-calibrated number line might show that the
conditions leading to log-like functions in the simulations don’t apply to previous studies. One of these
conditions—the boundedness of the number line—is built in to such experiments, but the other
condition—greater uncertainty associated with the position of the larger numerals—is more vulnerable.
Variability clearly increased in the present experiments (see Figures 2a and 3a), but other studies may not
exhibit this relation. Most published studies do not report variability as a function of the size of the
number, but Siegler and Booth (2004) note that variability failed to correlate with the size of numerals
between 0 and 100.
Caution is required, however, in interpreting this finding (beyond the usual caution associated
with null results). The statistical simulation reported earlier assumes increasing variability in the
underlying distributions of the numbers, but this increase need not translate directly into increasing
variability in the observed truncated distributions. The boundaries of the number line impose restrictions
on variability, just as they impose restrictions on the means. Thus, if the standard deviations of the
underlying distributions increase rapidly, the resulting standard deviations of the truncated distributions
quickly reach an asymptote, resulting in minimal change. It is possible to find values of the distributions’
parameters that produce both log-like positions of the means and quite small increases in standard
deviations (e.g., less than 12% over the range 1-9 in simulations like those reported earlier). The effect of
truncation also predicts that if participants received an unbounded number line—one in which no upper
bound exists—increasing standard deviations would reappear. This prediction is confirmed by Cohen and
Blanc-Goldhammer (2011).
In sum, statistical and experimental simulations suggest that people’s logarithmic placement of
numerals is not necessarily the result of a log-calibrated mental representation. It could instead reflect
How Many Is a Zillion? / 14
uncertainty about the meanings of the larger number words, together with limits imposed by the
boundaries of the number line. This suffices to explain children’s shift from a log to a linear pattern as
they learn the meanings of the larger words. These assumptions also explain how the same child could
show a linear pattern for 0-100 but a log pattern for 0-1000, since the child may know the meanings for
the larger numbers in the former range but not in the latter. The central goal for research in this area is to
determine what people know about the organization of the integers. Particular tasks, such as number-line
placement, may be helpful in this respect, but we need to inspect carefully the bridge they provide
between data and theory.
How Many Is a Zillion? / 15
Acknowledgments
Thanks to Julie Booth, David Landy, Robert Siegler, members of my lab group, and audiences at
New York University, Northwestern University, and the University of Notre Dame for their feedback. For
their help with the experiments, I thank Samantha Thompson and Antonia Yang. This research was
supported by IES grant R305A080341.
How Many Is a Zillion? / 16
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How Many Is a Zillion? / 19
Table 1.
Stimulus Numerals Used in the Number Placement Task of Experiment 1
a.
Three
b
Seventy-one hundred and ninety-three
c.
Twenty-eight thousand and thirty-two
d.
Seventy-five million and thirty-five
e.
Twenty-nine billion and ninety-one
f.
Forty-seven trillion and sixty-four
g.
Sixty-five quadrillion and sixty-nine
h.
Sixty quintillion and ten
i.
Eighteen sextillion and forty-five
j.
Thirty dillion and forty-twoa
k.
Eighty-three robillion and seventy-sixa
l.
Twenty-one jillion and fifty-onea
m.
Eleven bajillion and seventy-onea
n.
Twenty-one skillion and thirty-twoa
o.
Thirty katrillion and fifty-fivea
p.
Twenty-eight fantillion and fifty-foura
q.
Forty-two bazillion and forty-ninea
r.
Fifteen kabillion and seventy-twoa
s.
Forty-three umptillion and twenty-fivea
t.
Ninety-two zillion and fiftya
a
Fictitious numbers
How Many Is a Zillion? / 20
Table 2.
Stimulus Numerals Used in the Number Placement Task of Experiment 2
a.
Three
3
b
Seventy-one hundred and ninety-three
7,193
c.
Twenty-eight thousand and thirty-two
28  103 + 32
d.
Seventy-five million and thirty-five
75  106 + 35
e.
Twenty-nine billion and ninety-one
29  109 + 91
f.
Forty-seven trillion and sixty-four
47  1012 + 64
g.
Sixty-five quadrillion and sixty-nine
65  1015 + 69
h.
Sixty quintillion and ten
60  1018 + 10
i.
Eighteen sextillion and forty-five
18  1021 + 45
j.
Ninety-two septillion and fifty
92  1024 + 50
k.
Twenty-one octillion and fifty-one
21  1027 + 51
l.
Forty-three nonillion and twenty-five
43  1030 + 25
m.
Eleven decillion and seventy-one
11  1033 + 71
n.
Fifteen undecillion and seventy-two
15  1036 + 72
o.
Twenty-one duodecillion and thirty-two
21  1039 + 32
p.
Forty-two tredecillion and forty-nine
42  1042 + 49
q.
Twenty-eight quattuordecillion and fifty-four
28  1045 + 54
r.
Thirty quindecillion and forty-two
30  1048 + 42
s.
Thirty sexdecillion and fifty-five
30  1051 + 55
t.
Eight-three septendecillion and seventy-six
83  1054 + 76
a.
0
100
b.
0
1
2
3
4
5
6
7
8
9
10
Position on Number Line
Figure 1. (a) A number line of the type used in experiments on children’s placement of numerals. (b) The results of a
simulation of number-line placement, as described in the text. Each distribution is a smoothed curve, based on a sample of
106 randomly generated values.
Standard Error of Mean (mm)
a.
14
12
10
8 a
bc
d
e
f
j
kg l i mno hp q r s
t
6
4
2
Zero
One Gazillion
0
0
20
40
60
80
100
Mean Position on Number Line (mm)
b.
Mean Position on Number Line (mm)
100
80
60
40
20
0
1
2
3
4
5
6
7
8
9 10 11 12 13 14 15 16 17 18 19 20
Ordinal Position of Numeral
Figure 2. (a) Mean placement of numerals on the number line. Blue circles denote conventional numbers; red squares,
fictitious numbers. Labels on the points correspond to the listing in Table 1. The error bars (1 SEM) appear in a vertical
orientation for legibility. (b) Best-fitting logarithmic (solid line) and linear function (dashed line) to the mean number
positions, Experiment 1.
Standard Error of Mean (mm)
a.
14
12
10
a
bc
d
e
f
g
rhi spnkjlqm o t
8
6
4
2
One Google
Zero 0
0
20
40
60
80
100
Mean Position on Number Line (mm)
Mean Position on Number Line (mm)
b.
100
80
60
40
20
0
1
2
3
4
5
6
7
8
9
10 11 12 13 14 15 16 17 18 19 20
Ordinal Position of Numeral
Figure 3. (a) Mean placement of numerals on the number line. Labels on the points correspond to the listing in Table 2.
(b) Best-fitting logarithmic (solid line) and linear function (dashed line) to the mean number positions, Experiment 2.