Most of us have little trouble working out how many millilitres are in 2.4 litres of water (it’s 2,400). But the same can’t be said when we’re asked how many minutes are in 2.4 hours (it’s 144).

That’s because the Indo-Arabic numerals we often use to represent numbers are base-10, while the system we often use to measure time is base-60.

Expressing time in decimal notation leads to an interaction between these two bases, which can have implications at both the cognitive and cultural level.

Such base interactions and their consequences are among the important topics covered in a new issue of the Philosophical Transactions of the Royal Society journal, which I co-edited with colleagues Andrea Bender (University of Bergen), Mary Walworth (French National Centre for Scientific Research) and Simon J. Greenhill (University of Auckland).

The themed issue brings together work from anthropology, linguistics, philosophy and psychology to examine how humans conceptualize numbers and the numeral systems we build around them.

What are bases, and why do they matter?

Despite using numeral bases on a daily basis, few of us have reflected on the nature of these cognitive tools. As I explain in my contribution to the issue, bases are special numbers in the numeral systems we use.

Because our memories aren’t unlimited, we can’t represent each number with its own unique label. Instead, we use a small set of numerals to build larger ones, like “three hundred forty-two.”

That’s why most numeral systems are structured around a compositional anchor — a special number with a name that serves as a building block to form names for other numbers. Bases are anchors that exploit powers of a special number to form complex numerical expressions.

The English language, for example, uses a decimal system, meaning it uses the powers of 10 to compose numerals. So we compose “three hundred and forty-two” 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 overloading our cognitive resources.

Languages affect how we count

Despite the abstract nature of numbers, the degree to which numeral systems transparently reflect their bases has very concrete implications — and not just when we tell time. Languages with less transparent rules will take longer to learn, longer to process and can lead to more calculation and dictation errors.

Take French numerals, for example. While languages like French, English and Mandarin all share the same base of 10, most dialects of French have what could politely be called a quirky way of representing numbers in the 70-99 range.

Read more: How counting by 10 helps children learn about the meaning of numbers

Seventy is soixante-dix in French, meaning “six times 10 plus 10,” while 80 uses 20 as an anchor and becomes quatre-vingts, meaning “four twenties” (or “four twenty,” depending on the context). And 90 is quatre vingt dix, meaning “four twenty ten.”

French is far from being alone in being quirky with its numerals. In German, numbers from 10 to 99 are expressed with the ones before the tens, but numbers over 100 switch back to saying the largest unit first.

Even in English, the fact that “twelve” is said instead of “ten two” hides the decimal rules at play. Such irregularities spread far beyond languages.

How bases shape learning and thought

Base-related oddities are spread out across the globe and have very real implications for how easily children learn what numbers are and how they interact with objects such as blocks, and for how efficiently adults manipulate notations.

For example, one study found that lack of base transparency slows down the acquisition of some numerical abilities in children, while another found similar negative effects on how quickly they learn how to count.

Another study found that children from base-transparent languages were quicker to use large blocks worth 10 units to represent larger numbers (for example, expressing 32 using three large blocs and two small ones) than children with base-related irregularities.

While Mandarin’s perfectly transparent decimal structure can simplify learning, a new research method suggests that children may find it easier to learn what numbers are if they are exposed to systems with compositional anchors that are smaller than 10.

In general, how we represent bases has very concrete cognitive implications, including how easily we can learn number systems and which types of systems will tend to be used in which contexts.

A group of people in white protective suits and head protectors stand in front of a robotic spacecraft
Technicians lower the Mars Climate Orbiter onto its work stand in the Spacecraft Assembly and Encapsulation Facility-2 in 1998. (NASA)

At a cultural level, base representation influences our ability to collaborate with scientists across disciplines and across cultures. This was starkly illustrated by the infamous Mars Climate Orbiter incident, when a mix-up between metric and imperial units caused a $327 million spacecraft to crash into Mars in 1999.

Why understanding bases matters

Numeracy — the ability to understand and use numbers — is a crucial part of our modern lives. It has implications for our quality of life and for our ability to make informed decisions in domains like health and finances.

For example, being more familiar with numbers will influence how easily we can choose between retirement plans, how we consider trade-offs between side-effects and benefits when choosing between medications or how well we understand how probabilities apply to our investments.

And yet many struggle to learn what numbers are, with millions suffering from math anxiety. Developing better methods for helping people learn how to manipulate numbers can therefore help millions of people improve their lives.

Research on the cognitive and cultural implications of bases collected in the Philosophical Transactions of the Royal Society journal can help make progress towards our understanding of how we think about numbers, marking an important step towards making numbers more accessible to everyone.

This article is republished from The Conversation, a nonprofit, independent news organization bringing you facts and trustworthy analysis to help you make sense of our complex world. It was written by: Jean-Charles Pelland, University of Bergen

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Jean-Charles Pelland's work has been made possible by financial support from the ‘QUANTA: Evolution of Cognitive Tools for Quantification’ project, which has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant agreement No 951388).