Even people who claim (sometimes proudly ) to be bad at mathematics can usually do addition or multiplication of multi-digit numbers, using pencil and paper. Such feats would have astonished our Roman forbears, engineers and all, because the Hindu-Arabic number notation system that allows the easy expression and manipulation of large numbers did not even exist until about 800 AD. Nor did the concepts and tools for fractions and decimals that support modern technology. It required the concept of zero as a number to make this notation system possible, and that concept occurred about 300 AD.
The earliest attempts to symbolize quantities of things that it was important to record took the form of hash marks on a malleable substrate- e.g. a clay tablet, stone, or etc… one hash mark per item. The next step up is probably one that is still used today- groups of five. That is, four vertical hash marks, followed by a diagonal line that cuts through them. This is a numbering system with only two symbols, for five and one, and the choice of five as the quantity for the next symbol is obvious.
It is only a short step to realize that five can also be represented by any symbol different than the one used for the one. For example, a one can be represented by a finger on the left hand, with a right-hand finger representing a five. This provides a handy way to represent any number up to 55 by a show of fingers.
We could also extend the range of representation by using multiple hash marks until we reach 9, then use a different symbol to represent 10. This was the approach used by the Babylonians; each “Y” symbol represented one, while each “<” symbol represented ten. By piling up combinations of these two symbols, they would count to sixty. Thus, for 51 the symbol is <<<<<Y ( 5 tens and one one).
To count even larger numbers, the Babylonians introduced the concept of positional notation, a key feature of modern numbers. When the count reached sixty, the Babylonians started over with the symbols, adding a Y to the left to represent one group of sixty, followed by another group to the right to represent the excess over sixty. After sixty groups of sixty, yet another column to the left was provided. There is some potential confusion with this system that requires the symbols to be carefully grouped and spaced- see:
The Babylonian choice of sixty as a number base is still with us, in measures of times and angles. The important concept here is that of position of a symbol being important as well as the shape of the symbol. The Babylonians were thus able to represent any number with a combination of only two types of symbols. This feature became important in the development of the abacus.
The South American Incas also caught onto positional notation as a way to represent numbers, and they did it with a base 10 representation. In this case, inventories were recorded as knots along a string, in a bundle called a quipu. There was basically only one symbol, i.e. a knot, but the position along the string determined whether the number of adjoining knots represented the number of ones, tens, hundreds, or etc. within the represented quantity. This representation was purely positional. See: http://www-history.mcs.st-and.ac.uk/HistTopics/Inca_mathematics.html
This point was lost on the Romans. Their contribution was to extend the number of symbols used, to represent groups of different size, e.g. I for ones, V for fives, X for tens, L for fifties, C for hundreds, and etc, but the symbols were never re-used to represent different quantities in different positions. A Roman numeral representation of large numbers is hard to read and impossible to use for calculations. Their obscurity is useful for movie makers who don’t want to date their movies with copyright numbers that are easy to read, e.g: MCMLXXXIX= 1989.
The modern Hindu-Arabic notation has much to do with positional notation, and has a symbol that represents zero, as a place-holder to avoid potential confusion. With this notation, we can use any desired number N as the basis of the representation, where N is usually 10 (for decimal notation), but N=16 (hexadecimal) and N=2 (binary) are also useful and popular. For base N notation, we require N-1 separate symbols and 0 (zero).
In a base 10 number system, a three symbol number (say abc, where the letters represent digits) is interpreted as follows:
The number of ones is c.
The number of tens is b.
The number of hundreds is a. And so forth.
Each position to the left represents another power of N (here, 10). The symbol sequence abc means:
a x 102 + b x 101 +c x 100
This sequence could also be interpreted in base 16 as:
a x 162 + b x 161 +c x 160
Or in binary as:
a x 22 + b x 21 + c x 20
For fractional numbers, we just continue beyond the decimal as follows (in base 10):
abc.de = a x 102 + b x 101 + c x 100 + d x 10-1 + e x 10-2
Clearly, we can also represent fractional numbers with any other number base.
Hindu-Arabic notation using base 10 representation has become the accepted language for counting things, but there are some numbers that are just too big for it, e.g. national debts, distances to stars, or the number of angels on the head of a pin. For really big numbers, the most important symbols are the ones in the three leftmost columns. Anything to the right of them is relatively small potatoes.
It is nearly universal, in such cases, to round off to the nearest number of millions or billions or septillions or googles (1 google =10100). Notice that we are again borrowing the Roman practice of giving separate names to groups of different size: (1000 = “thousand”, 1000 thousand= “million”, 1000 million = “billion, and etc.)
The symbolic way to handle this is scientific notation, where the appropriate power of 10 is specified, e.g. 123 million is written as 123x106. But it is usually expressed with only one digit before the decimal point, i.e.
123 million = 1.23 x 108
123 billion = 1.23 x 1011
This notation lends itself to simple algorithms for multiplication, division, exponentiation, etc., and extends as well into the range of really small numbers, e.g. a length of 5 billionths of a meter is represented as 5 x 10-9 meters.
For scientists, the exponential notation is usually sufficient to represent any physical quantities that arise in the study of the things in the universe: The number of hydrogen atoms in 1 gram of the substance is about 6 x 1023. The number of atoms in the (visible) universe is about 1080
(considerably less than a google). The smallest unit of distance that it makes sense to talk about (the Plank length) is a little over 10-35 meters.
(considerably less than a google). The smallest unit of distance that it makes sense to talk about (the Plank length) is a little over 10-35 meters. To be sure, exponential notation is not actually required to display such large numbers, but only more convenient than writing down all the zeros. Even a google (10100) could actually be written down as “1” followed by 100 zeroes. But some numbers of interest are far too large to be handled this way: In string theory, calculations can be made about the possible number of universes that could be generated with different force laws, compared with the combination of force laws that we actually measure. In such calculations we arrive at numbers such as 10googol.
This is a “1” followed by a google of zeroes (called a googleplex). This cannot be written in conventional notation. If we could write one “0” on each atom in the known universe, we would run out of atoms before we could write down the entire number. And it would take a long time.
Physicists deal with some large numbers but, for mathematicians, numbers never stop. They recognize- and can prove- that some things are infinite in size, e.g. the number of integers, the number of prime numbers, the number of points on a line, and etc. They questioned whether some infinities are bigger than other infinities. One might think that an infinite number of things would be hard to count, but a mathematician named Georg Cantor figured out ways to do it. And the amazing result was that he proved that some infinities are of the same size, while other infinities are much bigger- infinitely so.
For example, the “rational” numbers are composed of all the integers (positive and negative), plus all the numbers that can be represented as ratios of two integers. There are an infinite number of such numbers. But Cantor showed that the number of rational numbers can- in principle- be “counted”. Meaning that you can put every possible rational number into a one-to-one correspondence with an integer. So the (infinite) number of integers has the same size as the (infinite) number of all rational numbers; these are countable infinities.
But there also exist all those number that are irrational- they cannot be represented as a ratio of two integers. Their decimal representation extends forever, with the digits (or groups of digits) never repeating [1]. Many numbers fit this category, for example π=3.1415926535…, where the “… “means that the digits go on forever without repeating (and it can be proved that they don’t). The natural logarithm base e= 2.718281828... and the square roots of many integers. e.g. square root(2) =1.414125623…. are also irrational. Any random, non-repeating string of an infinite number of digits is an irrational number. In fact, between every two possible rational numbers is an infinite number of irrational numbers. Cantor proved it. The size of the set of irrational numbers is not countable- it is much larger than size of the set of rationals.
Where scientists are usually satisfied with using the symbol ∞ to represent any old infinity, Cantor realized that we need different symbols to represent infinities of different sizes. The symbol typically used to represent the size of an infinity is the Hebrew letter aleph (not available on my keyboard), followed by an integer. Aleph followed by a 0 (called “aleph null”) is the smallest infinity, and successively larger infinities are represented with successively larger integers following aleph.
Mathematicians have since been amusing themselves by classifying mathematical infinities into the various sizes, and determining whether the number of possible infinities is discrete and countable, or continuous and uncountable. Normal minds tend to spin out when contemplating such issues.
The result is that humans have now progressed to having a language and a written representation for the expression and manipulation of numbers sufficient to handle all humanly imaginable and unimaginable quantities. It remains an open question whether there are an actual infinity of universes, but physical reality has a long history of accurate correspondence with the imaginations of mathematicians.
GPR: 3/21/11
[1] Any number that has an endlessly repeating series of digits can always be expressed as a ratio of two integers. E.g. 1.125125125… =1124/999. It is a neat trick of math to do this calculation.
