Numbers are ubiquitous. Not a day passes when we don’t use numbers in one context or another, whether in paying a bill at the store or counting the change, in telling time or noting the date, in reading about the stock market or watching the polls.
Many have wondered where numbers come from. Are they products of our minds? That is not very likely since, even without humans, there can be three birds, four eggs, and a thousand and seven stars.
Homo sapiens managed without numbers for many millennia. Anthropologists tell us that even today there are anumeric cultures (cultures without the concept of numbers) in Amazonia. On the other hand, biologists have found that some birds and animals show signs of counting cognition. In the September 2009 issue of Scientific American there was a lengthy article entitled, “More animals seem to have some ability to count.”
If one has to think of a number, most people will think of 3, 23, 5687, etc. These are what mathematicians call positive integers. They are certainly the most common numbers we use. Every integer is unique: Thus, 60 is the smallest number with five consecutive factors (2, 3, 4, 5, 6); 73 is the smallest two-digit number which is one less than twice its reverse: 73 = 2×37 – 1; 9973 is the largest four-digit prime. Some numbers have unexpected properties. It is interesting to multiply 37,037 by any number from 1 to 9 and see the result. [TRY IT!]
But the human mind has uncovered a variety of other types of numbers and created some too. The Indic Sulba Sutras (800 – 500 BCE) used irrational numbers, and Pythagoras showed that the square root of two cannot be expressed as the ratio of two integers (fraction). Hence such numbers came to be called irrational numbers.
The idea of negative numbers seems fairly simple to the modern mind. But it was quite intriguing to people for a long time, mainly because they do not represent countable things. Indian arithmeticians handled negative numbers freely already in the seventh century. As Morris Kline wrote, “The first known use of negative numbers is by Brahmagupta about 628; he also states the rules for the four operations with negative numbers.” In Europe, negative numbers came into use only from the sixteenth century with the work of Jerome Cardan and Thomas Harriot.
We talk of even and odd numbers and have defined (given names to) countless other numbers with special properties: prime number, perfect numbers, Mersenne Primes abundant numbers, amicable numbers, palindrome numbers, etc. We have also invented imaginary, complex, and hypercomplex numbers.
We cannot imagine the largest number, because no matter how large a number we think of, we can always add one or a few billion to it and get a larger number. This frustration leads to the definition of infinity as that to which we may add anything, from which we may subtract anything, which we may multiply or divide, without affecting it in any way. We talk of infinity as that which is much larger than any number we can imagine. The Isopanishad articulated the concept of infinity in a metaphysical sort of way:
purnamadah purnamidam purnát purnam udatchyate
purnasya purnamadáya purnameva vashisyate.
Whatever is produced of that which is complete, is complete in itself.
Even if many complete units emanate from the complete,
the latter still remain complete.
Replacing the term purnam by infinity, this statement expresses the mathematical idea that infinity times anything is infinity, and infinity minus infinity is infinity.
Indic thinkers extended the mathematical idea of infinity, which they called ananta (endless) to other things as well. This is also the etymology of the English word from the Latin infinitus: that which is without end or boundary. Thus, they spoke of nominal infinity (referring to boundless greatness), epistemic infinity (referring to boundless knowledge), one dimensional infinity (observation along an uninterrupted line of sight), numeric infinity (fraction with zero in the denominator) and temporal infinity (eternity).
Aside from the usual numbers, there is a vast class of numbers that are very different. Known as transcendental numbers they include the familiar pi and e.
The usual number system has ordinal numbers, numbers that can be ordered as 1, 2, 3, … A cardinal number just tells us how many (entities) there are in a collection. There are different orders of infinity. The number of points on a straight line is infinite, but it is greater than the number of ordinal numbers. It is a higher order of infinity. All numbers are collectively known as transfinite numbers.
That there are different orders of infinity was a major 19th century discovery in number theory. The magnitude of (number of elements in) infinite sets is a transfinite number. The cardinality of 1, 2, 3, … is referred to as aleph-null: . This is the first transfinite cardinal number. The next cardinal number is . There are higher order cardinal numbers.
Efforts to define a number in the 19th century (the Peano postulates) is a fascinating chapter in humanity’s intellectual history. Like the notion of God, we use numbers without actually knowing its correct definition. In this sense too, Number may be regarded as a reflection of the abstract Divine.
Spurts of Thought 