A ‘calculator’ is defined by the Oxford English Dictionary as ‘something used for making mathematical calculations, in particular a small electronic device with a keyboard and a visual display‘.
However, the definition can be broadened to cover any means that facilitates the manipulation of numbers. This could include those humans with exceptional ability for mental calculation (such as adding, subtracting, multiplying or dividing large numbers) – but this is outside the current scope – if interested go to Mental Calculator
What follows is an overview of the history of calculators, supported by occasional calculator-related posts. But first a brief look at what constitutes mathematics.
The Scope of Mathematics
Number Sense
The term “number sense” was first used in the context of mathematics education in the late 1970s and early 1980s. It refers to an intuitive understanding and familiarity with numbers and their relationships including estimating quantities and making reasonable approximation, appreciating the relative size and magnitude of numbers, estimating quantities and making reasonable approximations.
Counting
Counting, involving determining the total number of a collection of items, is one of the first numeracy skills children learn and is the basis for all later mathematical learning. But the ability to count can be traced back thousands of years where our ancestors made use of their fingers or toes or of sticks and bones marked with notches or other small objects.
The illustration below is of an Ishango Bone found in Central Equatorial Africa and thought to be over 25,000 years old. The bone has a series of notches made using a piece of quartz located at one end of the bone and is considered to be more than a simple tally. For example, rows (a) and (b) each add up to 60 and row (c) might indicate multiplication by two. A similar and older artifact, also found in Africa, is the Lebombo Bone originating some 40,000 years ago.
Counting usually relies on having a number base which is an arbitrarily chosen whole number greater than 1 in terms of which any number can be expressed as a sum of that base raised to various powers. The most common one is base 10 which aligns with our 10 fingers (including the thumb). Base 10 has two factors allowing it to be divided by 2 and 5.
Some have argued that base 12 should be preferred as it has 4 factors – 2, 3, 4 and 5 which allows greater flexibility on counting. It was used by early Egyptians and by King Charlemagne for his kingdoms’ money but never took hold given the dominance of the decimal system in science and engineering. Interestingly the word ‘eleven’ derives from Old English endleofan, literally meaning “[ten and] one left [over],” and twelve from twelf, meaning “two left”; the endings -teen and -ty both refer to ten. Nevertheless, base 12 survives in areas such as timekeeping and the system of imperial measurement – 12 inches to one foot.
In the digital age, the binary systems prevails because in computer systems it offers an efficient way to control logic circuits and to detect an electrical signal’s true (1) and false (0) states.
Mathematical operations
Beyond counting comes the ability to manipulate numbers by way of mathematical operations. An operation is a function which takes one or more inputs (named operands) and produces an output. The most important operations are addition, subtraction, multiplication, and division; others include exponentiation, extraction of roots, and logarithms.
Pure Maths
Pure maths is the study of maths, independent of any of its practical applications. It includes the study and research of theories and abstract mathematical concepts. Although theoretical in nature, what emerges can find applications in the real world. For example:
- Complex numbers (1831) enabled General Electric to simplify and rapidly scale the US electricity grid in the 1890s
- Boolean algebra (1847) provided the logical underpinnings of the first digital computers in World War 2
- Radon Transforms (1917) were crucial to the patent that enabled x-rays to see cancers and won a Nobel Prize in 1979
A BRIEF HISTORY OF CALCULATORS
Developments beyond the Ishango Bone are shown in the figure below which charts the progress of mechanical devices to the time when the advent of transistors and integrated circuits in the 1960s led to the introduction of electronic calculators.
Images and a link for further information about the inventions is provided below
Additional Sources
Alex’s Adventures in Numberland by Alex Bellos. Bloomsbury Press 2020 ISBN 978-1-5266-2399-7
Calculating Machines and Computers by Geoffrey Tweedale. Shire Album 247, Shire Publications. ISBN 0747800804
Early Calculators – an overview provided in the HP Museum
The Oughtred Society – A Society dedicated to the preservation and history of slide rules and other calculating instruments.