# Quick – What’s closer to 1/150: 1/50 or 1/1000?

If you said 1/1000, you’ve given the answer provided more often by second graders than by undergraduates. And you’re also right.

Evidence from functional neuroimaging and developmental psychology indicate that the human brain possesses an abstract system for magnitude comparison, perhaps relying on the intraparietal sulcus of the posterior parietal cortex. One “feature” of this system is that it follow’s Weber’s Law – larger quantities are less discriminable from one another than smaller quantities. Thus, in this part of the brain numerical representations are not linear but logarithmic.

This system seems to be what drives the numerical processing of very young children. For example, first graders will routinely locate “50” as closer to 100 on a number line than older children, and this shift is well described by a shift from logarithmic to linear representations of quantity during the elementary school years.

In general, this bias is detrimental – as anyone who has tried to teach basic arithmetic to a first-grader can attest. However, it leads to an advantage in one carefully crafted situation: the comparison of fractional values with identical numerators, for which the distance between the resulting values is actually a power function of the denominator.

Opfer & Devries demonstrated this by asking 24 second graders and another 24 undergraduates questions to place a hatch mark indicating the location of \$1/60 minutes on a number line spanning \$1/1 minute to \$1/1440 minutes. Astonishingly, second graders were more accurate than undergraduates on this task, perhaps owing to their use of a logarithmic magnitude system. That is, children may perceive 60 as being closer to 1440 than 1, which leads to a bias in placing the hatch mark closer to the correct location when dealing with fractional values.

Children also outperformed adults on a version in which the denominator was specified in more familiar units of duration, such as \$1/hour or \$1/day. This is a more interesting effect that isn’t discussed much in the paper – one idea is that children’s perception of time is logarithmic as well, such that children perceive one hour as relatively more similar to one full day than adults. (As discussed previously…)

1. #1 Leonid Kovalev
September 30, 2008

In the sequence 1/1440, 1/60, 1 the second number is closer to the first one both on the standard scale and on the logarithmic scale. I don’t see how the preference for 1/1440 indicates that the person is thinking “logarithmically”.

The title of this blog post has a different sequence: 1/1000, 1/150, 1/50. Here the second number is closer to the first one on the standard scale, but it is closer to the third one on the logarithmic scale. A person using the logarithmic system (as second graders are presumed to do) should choose 1/50 as the closer number.

2. #2 CHCH
October 1, 2008

Leonid – you bring up an interesting point. I think the authors believe that the denominators themselves are placed on a logarithmic or linear scale, in which case 50 seems “closer” to 150 on a linear scale than it does on a logarithmic one. So the issue is not exactly which denominator is closer, but rather what is the difference between linear and logarithmic representations, and how might that skew the results (ever so slightly) in one direction or another. forgive me if I’m not understanding your question.

3. #3 Leonid Kovalev
October 1, 2008

Okay, let’s look at the denominators rather than fractions themselves. I’ll use logarithms with base 10; any other base would give the same conclusions.

The logarithms of the numbers 1, 60, 1440 are 0, 1.778 and 3.158 respectively. Thus, 60 is closer to 1440 than to 1 on the logarithmic scale (but not on the linear scale).

The logarithms of the numbers 50, 150, 1000 are 1.699, 2.176, and 3 respectively. Thus, 150 is closer to 50 than to 1000 on the logarithmic scale (and on the linear scale too).

Either way you look, the numbers used in the title of the blog post follow a different pattern than the numbers in the quoted study.

4. #4 CHCH
October 1, 2008

leonid, you’re absolutely right, and I’m tempted to change the title now.

But just in case my (somewhat moot) point didn’t come through, 150 is closer to 1000 on a logarithmic scale than 150 is to 1000 on a linear scale! So although the closest answer is 1/50 on both scales, you may be *more likely* to answer “1/1000” using a logarithmic scale.

5. #5 Amiya Sarkar
October 5, 2008

While our bodies compute mostly in linear terms, some things like sound are processed in the brain in logarithmic terms.

6. #6 eddie
October 11, 2008

This sounds like a job for –