## Linear Versus Logarithmic Thinking about Numbers

Some folks argue that humans intuitively think about numbers logarithmically versus linearly. My experience strongly suggests that this is not the case. If you’ve ever tried to explain logarithms or logarithmic scales to people, or asked them to interpret graphs with logarithmic scales, as I have often done, you probably share my belief that logarithms are not cognitively intuitive. The behaviors that are sometimes described as intuitive logarithmic thinking about numbers can be reasonably explained as something else entirely.

According to some sources, a research study found that six-year-old children, when asked to identify the number the falls halfway between one and nine, often selected three. Unfortunately, after extensive searching I cannot find a study that actually performed this particular experiment. One article that makes this claim cites a study titled “A Framework for Bayesian Optimality of Psychophysical Laws” as the source, but that study does not mention this particular experiment or finding. Instead, it addresses the logarithmic nature of perception, especially auditory perception. Keep in mind that perception and cognition are related but different. Many aspects of perception do indeed appear to be logarithmic. As the authors of the study mentioned above observed about auditory perception, “…under the Weber–Fechner law, a multiplicative increase in stimulus intensity leads to an additive increase in perceived intensity,” but that’s a different matter. I’m talking about cognition. Even if many kids actually did select three as halfway between one and nine in an experiment, I doubt that they were thinking logarithmically. At age six children have not yet learned to think quantitatively beyond a rudimentary understanding of numbers. Until they begin to learn mathematics, children tend to think with a limited set of numbers consisting of one, two, three, and more, which corresponds to the preattentive perception of numerosity that is built into our brains. With this limited understanding, three is the largest number that they identify individually, so it might be natural for them to select three as the value that falls halfway between one and numbers that are larger than three. If the numbers were displayed linearly and in sequence for the children to see when asked to select the number in the middle (illustrated below), however, I suspect that they would correctly select five.

1 2 3 4 5 6 7 8 9

You might argue that this works simply because it allows children to rely on spatial reasoning to identify the middle number. That is absolutely true. We intentionally take advantage of spatial reasoning when introducing several basic concepts of mathematics to children. This works as a handy conceptual device to kickstart quantitative reasoning. Believing that children naturally think logarithmically would lead us to predict that, if asked to identify the number halfway between 1 and 100, they would be inclined to choose 10. Somehow, I doubt that they would.

Another research-based example that has been used to affirm the intuitive nature of logarithmic thinking about numbers is the fact that people tend to think of the difference between the numbers one and two as greater than the difference between the numbers eight and nine. I suspect that they do this, however, not because they’re thinking logarithmically, but more simply because they’re thinking in terms of relative magnitude (i.e., proportions). Even though the incremental difference between both pairs of numbers is a value of one (i.e., 2 – 1 = 1 and 9 – 8 = 1), the number two represents twice the magnitude of one while the number nine is only 12.5% greater than eight, a significantly lesser proportion. I anticipate that some of you who are mathematically inclined might object: “But logarithmic thinking and proportional thinking are one and the same.” Actually, this is not the case. While logarithms always involve proportions, not all proportions involve logarithms. A logarithmic scale involves a consistent proportional sequence. For example, with a log base 10 scale (i.e., log10), each number along the scale is ten times the previous number. Only when we think of a sequence of numbers in which each number exhibits a consistent proportion relative to previous number are we thinking logarithmically. We do not appear to do that naturally.

Another example, occasionally cited, is that people tend to think of differences between one thousand, one million, one billion, one trillion, etc., as equal when in fact each of these numbers is 1,000 times greater than the previous. Is this because people are thinking logarithmically? I doubt it. I suspect that it is simply because each of these values exhibits the next change in the label (e.g., from the label “thousand” to the label “million”), and changes in the labels suggest equal distances. If people intuitively thought about numbers logarithmically, they should automatically recognize that each of these values (one billion versus one million versus one thousand, etc.) is 1,000 times the previous, but most of us don’t realize this fundamental fact about our decimal system without first doing the math.

Along linear scales, increments from one value to the next are determined by addition—you always add a particular value to the previous value to produce the next value in the sequence, such as by adding a value of one to produce the scale 0, 1, 2, 3, 4, etc. or a value of ten to produce the scale 0, 10, 20, 30, 40, etc. Along logarithmic scales, on the other hand, increments are determined by multiplication—you always multiply the previous value by a particular number to produce the next value in the sequence, such as by multiplying each value by two to produce the scale 1, 2, 4, 8, etc., or by ten to produce the scale 1, 10, 100, 1,000, etc. The concept of logarithms, when clearly explained, is not difficult to understand once you’ve learned the mathematical concept of multiplication, but thinking about numbers logarithmically does not appear to be intuitive. It takes training.