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Numbers: Their History and Meaning
Much in our daily lives is defined in numerical terms-from the moment we wake in the morning and look at the clock to dialing a phone or paying a bill. But what exactly is a number? When did man begin to count and record numbers? Who made the first calculating machine-and when? At what point did people first think of solving problems by equations? These and many other questions about numbers are answered in this engrossing, clearly written book.
Written for general readers by a teacher of mathematics, the jargon-free text traces the evolution of counting systems, examines important milestones, investigates numbers, words, and symbols used around the world, and identifies common roots. The dawn of numerals is also covered, as are fractions, addition, subtraction, multiplication, division, arithmetic symbols, the origins of infinite cardinal arithmetic, symbols for the unknown, the status of zero, numbers and religious belief, recreational math, algebra, the use of calculators — from the abacus to the computer — and a host of other topics.
This entertaining and authoritative book will not only provide general readers with a clearer understanding of numbers and counting systems but will also serve teachers as a useful resource. "The success of Flegg's lively exposition and the care he gives to his surprisingly exciting topic recommend this book to every library." — Choice.
Written for general readers by a teacher of mathematics, the jargon-free text traces the evolution of counting systems, examines important milestones, investigates numbers, words, and symbols used around the world, and identifies common roots. The dawn of numerals is also covered, as are fractions, addition, subtraction, multiplication, division, arithmetic symbols, the origins of infinite cardinal arithmetic, symbols for the unknown, the status of zero, numbers and religious belief, recreational math, algebra, the use of calculators — from the abacus to the computer — and a host of other topics.
This entertaining and authoritative book will not only provide general readers with a clearer understanding of numbers and counting systems but will also serve teachers as a useful resource. "The success of Flegg's lively exposition and the care he gives to his surprisingly exciting topic recommend this book to every library." — Choice.
- GenresMathematics
304 pages, Paperback
First published January 1, 1987
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Displaying 1 - 2 of 2 reviews
July 26, 2025
The ubiquity and familiarity of numbers as a medium of quantitative expression makes it easy to overlook that numbers have a history. That history, especially the early history, is one of developing a method of expression for efficiently supporting cognition across a variety of cultural, scientific, economic, artistic, and religious contexts.
Flegg starts his history of numbers with the observation that they are meant to represent some of the most basic experiential relationships that we have with the world we live in. We experience things in space, where they have magnitude, quantity, and arrangement. We also experience them in time, where things have sequence and duration. These qualities are expressible quantitatively. And early use of fingers and tally marks made it easy enough to create abstractions of those spatio-temporal qualities to make them shareable across distances and stable across time. But the phenomena we sought to describe began to transcend individual and immediate experience and methods of reckoning with those amounts began to exceed the practical limitation of mental counting systems.
There was a point in our history when perfectly suitable counting schema that went from 1 - 10 followed by “a lot” or “many” needed more accuracy, and number systems evolved to allow more abstraction, thereby allowing us to work with quantities larger than could be accommodated by our fingers (i.e., "digits"). These numerals represented regular extrapolations of those more bodily ways of reckoning amounts. As I see it, as our attention expanded beyond the self and our immediate community to more distant geographic places, to the stars, to larger social collectives, to bigger storage, to more things, and the need for larger numbers to accommodate became clearer. A similar need to consider smaller and smaller entities, below the level of perception and below the level of whole numbers likewise drove the development of decimals and fractions (123).
Processes of numerical calculation (i.e., addition, subtraction, multiplication, division, ratios, proportions) developed alongside our emerging cultural, scientific, religious, and economic needs. Flegg’s chapter on the subject offers a bewildering array of techniques used over human history for doing simple calculations. I recognize in those techniques the ways that I learned to do simple maths (e.g., long division using the “Austrian” or “contracted” method), but there other ingenious ways that have been tried and abandoned in history. None of these account for the ways that my children learned calculation, which are different still but also mediated by the cognitive affordances and constraints of numerals.
Likewise, the chapter on computation (i.e., more complex, algorithmic processing of data sets) offers a fascinating look at how mathematical concepts derive from “computers” (e.g., beads, abaci, Napier’s bones, slide rules, calculators, and modern computers). One hardly needs to read far into a newspaper to find concepts that we unquestioningly take to be part of the fabric of our lived world that are entirely numerical and computationally derived: (e.g., GDP, medical risk, consumer intent, book recommendations).
These rules of calculation and computation allow us to not only understand the world that we perceive ourselves to be living in but also the world that we infer must be there, even if it lies outside the realm of our perception. The regularities with which we can describe the world numerically suggest the conditions that must be true of the world that we cannot perceive and those unknown quantities can also be computationally derived (221). All of this cognitive work is summed up nicely in a closing chapter on how people think with numbers.
One of the main contributions I see in doing a history of numbers is that Flegg makes a strong argument for why the development of number systems, arithmetic, computation, etc. was cultural (50) and numbers and numerical expressions are derived from verbal expressions (54). Numbers followed culture and that historical connection reminds us of what is still true, but overlooked and misunderstood, today: numbers reference our world and our lived experience. The more we overlook that connection, the easier it is to misunderstand numbers and to over attribute to them explanatory powers without considering how much they may deceive or misrepresent. Flegg’s book highlights the importance of a move today toward developing greater quantitative reasoning, which has generally atrophied just at a time when it is increasingly necessary.
There really some fascinating bits of history in here, like that of the Roman method of finger counting allowing counts into the thousands, Pythagoras’s work on harmonics, and Georg Cantor’s concept of infinity.
Flegg starts his history of numbers with the observation that they are meant to represent some of the most basic experiential relationships that we have with the world we live in. We experience things in space, where they have magnitude, quantity, and arrangement. We also experience them in time, where things have sequence and duration. These qualities are expressible quantitatively. And early use of fingers and tally marks made it easy enough to create abstractions of those spatio-temporal qualities to make them shareable across distances and stable across time. But the phenomena we sought to describe began to transcend individual and immediate experience and methods of reckoning with those amounts began to exceed the practical limitation of mental counting systems.
There was a point in our history when perfectly suitable counting schema that went from 1 - 10 followed by “a lot” or “many” needed more accuracy, and number systems evolved to allow more abstraction, thereby allowing us to work with quantities larger than could be accommodated by our fingers (i.e., "digits"). These numerals represented regular extrapolations of those more bodily ways of reckoning amounts. As I see it, as our attention expanded beyond the self and our immediate community to more distant geographic places, to the stars, to larger social collectives, to bigger storage, to more things, and the need for larger numbers to accommodate became clearer. A similar need to consider smaller and smaller entities, below the level of perception and below the level of whole numbers likewise drove the development of decimals and fractions (123).
Processes of numerical calculation (i.e., addition, subtraction, multiplication, division, ratios, proportions) developed alongside our emerging cultural, scientific, religious, and economic needs. Flegg’s chapter on the subject offers a bewildering array of techniques used over human history for doing simple calculations. I recognize in those techniques the ways that I learned to do simple maths (e.g., long division using the “Austrian” or “contracted” method), but there other ingenious ways that have been tried and abandoned in history. None of these account for the ways that my children learned calculation, which are different still but also mediated by the cognitive affordances and constraints of numerals.
Likewise, the chapter on computation (i.e., more complex, algorithmic processing of data sets) offers a fascinating look at how mathematical concepts derive from “computers” (e.g., beads, abaci, Napier’s bones, slide rules, calculators, and modern computers). One hardly needs to read far into a newspaper to find concepts that we unquestioningly take to be part of the fabric of our lived world that are entirely numerical and computationally derived: (e.g., GDP, medical risk, consumer intent, book recommendations).
These rules of calculation and computation allow us to not only understand the world that we perceive ourselves to be living in but also the world that we infer must be there, even if it lies outside the realm of our perception. The regularities with which we can describe the world numerically suggest the conditions that must be true of the world that we cannot perceive and those unknown quantities can also be computationally derived (221). All of this cognitive work is summed up nicely in a closing chapter on how people think with numbers.
One of the main contributions I see in doing a history of numbers is that Flegg makes a strong argument for why the development of number systems, arithmetic, computation, etc. was cultural (50) and numbers and numerical expressions are derived from verbal expressions (54). Numbers followed culture and that historical connection reminds us of what is still true, but overlooked and misunderstood, today: numbers reference our world and our lived experience. The more we overlook that connection, the easier it is to misunderstand numbers and to over attribute to them explanatory powers without considering how much they may deceive or misrepresent. Flegg’s book highlights the importance of a move today toward developing greater quantitative reasoning, which has generally atrophied just at a time when it is increasingly necessary.
There really some fascinating bits of history in here, like that of the Roman method of finger counting allowing counts into the thousands, Pythagoras’s work on harmonics, and Georg Cantor’s concept of infinity.
February 4, 2018
Well written, well organized and structured, informative book on numbers and mathematics.
Displaying 1 - 2 of 2 reviews