Great IdeasNature and science
Number
Counting, primes, ratios and the patterns in numbers.
The question
What are numbers, and what patterns hide inside them?
How the books answer
Oldest first. Each stop is our own short summary of what the author argued; follow the link to the rung for the whole lesson.
c. 300 BC
Euclid, The Elements, Books V–X
Euclid's books on number define primes and prove there are infinitely many of them: for any list of primes, there is always another. He also gives a way to find the greatest common divisor, still taught today, and a rule for making perfect numbers.
“Prime numbers are more than any assigned multitude of prime numbers.”
3rd century BC
Archimedes, The Sand-Reckoner
In The Sand-Reckoner, Archimedes sets out to count the grains of sand it would take to fill the whole universe as the Greeks pictured it. Greek numerals stopped at a myriad (ten thousand), so he invents a system for huge numbers. His point: no number is too big to name.
c. AD 100
Nicomachus of Gerasa, Introduction to Arithmetic
Nicomachus sorts numbers into families: even and odd, prime and composite, and perfect, abundant and deficient. A perfect number like 6 or 28 equals the sum of its smaller factors. His textbook taught arithmetic in schools for a thousand years.
“Number is limited multitude or a combination of units or a flow of quantity made up of units; and the first division of number is even and odd.”
1637
René Descartes, The Geometry
Descartes treats lengths as numbers and uses letters for them, so geometry problems become algebra. Multiplying two lengths gives a length, not an area, once a unit length is chosen. This joined number and shape in a new way.
“Often it is not necessary thus to draw the lines on paper, but it is sufficient to designate each by a single letter.”
1647–1654
Blaise Pascal, Treatises on the vacuum, the equilibrium of liquids and the weight of the air, and the arithmetical triangle
Pascal studies a triangle of numbers in which each number is the sum of the two above it. He uses it to count combinations and to solve problems about games of chance. It is now called Pascal's triangle, though others had found it earlier.
“The number of each cell is equal to that of the cell which precedes it in its perpendicular rank, added to that of the cell which precedes it in its parallel rank.”
1911
Alfred North Whitehead, An Introduction to Mathematics
Whitehead explains that mathematics began when people saw that two apples and two days share something: the number two. Mathematics is the study of such general patterns, apart from any particular things. He shows how this abstract thinking is also very practical.
“Mathematics as a science commenced when first someone, probably a Greek, proved propositions about any things or about some things, without specification of definite particular things.”
1940
G. H. Hardy, A Mathematician's Apology
Hardy, a great number theorist, writes about why he loved pure mathematics. He uses Euclid's proof of infinitely many primes as an example of mathematical beauty. He thought his number theory was useless, but it later became important in computer security.
For young readers
Numbers help us count, share and measure. But they also have secret patterns. Some numbers can be split into equal rows, like 12 (3 rows of 4). Others, like 7, can't be split evenly at all: those are called prime numbers.
Euclid proved over 2,000 years ago that primes never run out. And some numbers are 'perfect': 6 is perfect because 1 + 2 + 3 = 6. Can you check whether 28 is perfect?
Talk about it
- Find all the prime numbers up to 30. Is there a pattern?
- Archimedes counted grains of sand to fill the universe. What's the biggest number you can name?
- Build the first five rows of Pascal's triangle. What patterns do you see?
- Whitehead says two apples and two days share something. What is it?
Where else on the site
- Infinitely Many Primes
- Perfect Numbers and Mersenne Primes
- Prime and Composite Numbers
- Testing Primes and the Sieve of Eratosthenes
- The Binomial Theorem
- The Euclidean Algorithm
Every rung tagged with this idea:
These pages are our own short synthesis, written for families. They were inspired by Mortimer J. Adler's Syntopicon, the index of great ideas in volumes 1 and 2 of Great Books of the Western World, but the questions, summaries and groupings are ours, not the Syntopicon's. Short excerpts come from public-domain translations, credited where they appear.