Most people can easily calculate that 2.4 liters of water equals 2,400 milliliters. However, when asked how many minutes are in 2.4 hours, the answer—144 minutes—often eludes them. This discrepancy arises because the Indo-Arabic numerals we commonly use are based on a base-10 system, whereas time is typically measured using a base-60 system.
The interaction between these two numerical bases can have significant implications at both cognitive and cultural levels. This topic is thoroughly explored in a new issue of the Philosophical Transactions of the Royal Society journal, co-edited by researchers Andrea Bender, Mary Walworth, Simon J. Greenhill, and myself. The issue compiles research from anthropology, linguistics, philosophy, and psychology to examine how humans conceptualize numbers and the numeral systems we create.
Understanding Numerical Bases
Despite their everyday use, few people reflect on the nature of numeral bases. As detailed in my contribution to the journal, bases are special numbers that serve as building blocks in numeral systems. Due to the limited capacity of human memory, we cannot assign a unique label to every number. Instead, we use a small set of numerals to construct larger numbers, such as “three hundred forty-two.”
Most numeral systems are organized around a compositional anchor—a special number that helps form names for other numbers. Bases exploit powers of this special number to create complex numerical expressions. For instance, the English language uses a decimal system, relying on powers of 10 to construct numerals. Thus, “three hundred and forty-two” is composed using three times the second power of 10 (100), four times the first power of 10 (10), and two times the zeroth power of 10 (one).
This base structure allows us to represent numbers of all sizes without overwhelming our cognitive resources.
The Impact of Language on Numerical Understanding
Although numbers are abstract, the transparency of numeral systems in reflecting their bases has concrete implications. Languages with less transparent rules can be more challenging to learn and may lead to calculation and dictation errors.
Consider French numerals. While French, English, and Mandarin share a base of 10, French uses a unique system for numbers 70-99. For example, seventy is expressed as soixante-dix, meaning “sixty-ten,” and eighty as quatre-vingts, or “four twenties.” Ninety is quatre-vingt-dix, or “four twenty ten.”
French is not alone in its numerical quirks. In German, numbers from 10 to 99 place the ones before the tens, but numbers over 100 revert to stating the largest unit first. Even English has irregularities, such as using “twelve” instead of “ten two.”
How Bases Influence Learning and Thought
These base-related peculiarities can significantly affect how children learn numbers and interact with objects like blocks, as well as how efficiently adults manipulate numerical notations. Research indicates that a lack of base transparency can slow the acquisition of numerical abilities in children and affect how quickly they learn to count.
For example, children from base-transparent languages are quicker to use large blocks worth 10 units to represent larger numbers than those from languages with base-related irregularities. While Mandarin’s transparent decimal structure can simplify learning, new research suggests that smaller compositional anchors might aid in understanding numbers more effectively.
Overall, how we represent bases has concrete cognitive implications, influencing how easily we learn number systems and which types are preferred in various contexts.
The Cultural Significance of Bases
At a cultural level, base representation affects our ability to collaborate across disciplines and cultures. This was starkly illustrated by the Mars Climate Orbiter incident in 1999, where a mix-up between metric and imperial units caused a $327 million spacecraft to crash into Mars.
Numeracy, or the ability to understand and use numbers, is crucial in modern life, impacting our quality of life and decision-making in areas like health and finance. Familiarity with numbers influences how we choose retirement plans, weigh medication side effects, and understand investment probabilities.
Yet, many struggle with math anxiety, highlighting the need for better methods to teach numerical manipulation. Research in the Philosophical Transactions of the Royal Society journal advances our understanding of how we think about numbers, marking an important step toward making numbers more accessible to everyone.
