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Symmetry in Mathematics V


From physics we learn that the "Planck length" may be the smallest amount produced by Nature. Physicists often use "scientific notation" to represent "orders of magnitude." So Planck's length is written as 10-35 meters. Remember that the diameter of an atom, for comparison purposes, can vary from 10-15 meters to 10-10 meters. Therefore, the order of magnitude of Planck's length is twenty times smaller than the diameter of an atom: 10-35 = 10-15 ' 10-20.

The key to understanding symmetry between infinitely large it's the infinitely small It is precisely the custom of physicists to represent both through scientific notation. For every very large number, say 1035, we match a very small number, in this case the number 10-35. Note that the product of these two numbers is 1 because 10-35 is, in the positional system, 0 followed by a comma and 34 zeros followed by 1, while 1035 is 1 followed by 35 zeros before the comma. As we multiply each other, each zero before the comma of 1035 causes the other's comma to move one square to the right. So in the end we'll get 10-35 ' 1035 = 1! Let's note that the account we made was as if we were simply doing 10-35+35 = 100 = 1. In words, moving comma 35 places to the left and then 35 places to the right leaves the 1 unchanged.

The comma is the watershed between the big and the small. We can produce smaller and smaller numbers simply by placing zeroes after the comma, as many as we like, followed by any number. For example, 0.000000000000000000000007. Similarly, and symmetrically, we can get an increasing number by writing a number followed by as many zeros as we like. For example, 70,000,000,000,000,000,000,000. Note that it is much easier to represent large numbers in the Decimal Positional System than to find names for them.

In physics we cannot extrapolate Planck's length to even smaller quantities, but in mathematics there are no limits to our imagination. Just as there is no known quantity smaller than Planck's length, so it is not known if there are more than 10100 atoms in our observable universe. In our imagination it's okay to conceive numbers like 101000 and 10-1000. What's more, the product of these two is equal to 1! This is the interesting part. There is a multiplicative structure It is remarkable that we will look in more detail now.

Every number of the form 10N corresponds to another of the form 10-N and the product of each other is 1. We may then ask: is it that for each large number there would also be a small number such that the product of the two is always 1? If we assume this for every positive number, then we have the following information about the positive numbers:

(a) every positive number admits another such that the product of both is 1;

(b) by multiplying three positive numbers we can do so in any order;

(c) 1 is neutral in multiplication.

Property (a) is a result of our wish that that interesting symmetry between large and small extends to all positive numbers. Property (b) is a result of our desire not to spoil a knowledge we already had about multiplication, that no matter the order in which we multiply three numbers. Finally, property (c) is a very useful and important observation. It tells us that 1 is a neutral element in number multiplication.

But a question immediately arises: Why on earth do we select precisely these three properties? Because three is the minimum number of properties we need to describe a symmetry. Numbers that satisfy these three properties form a symmetrical structure called group. This symmetrical structure called group has been widely used in physics to measure symmetries of nature, and has generated a tremendous amount of new knowledge in mathematics itself.

The positive and negative integers form a group, but in relation to the addition. That is, for integers we consider the same three properties above by changing the word multiplication by addition, which forces us to also change the neutral element that becomes zero. We say that the symmetry of the integers is additive while the symmetry of the large and small positive numbers is multiplicative. For the integers we write:

(a) every integer admits another such that the sum of the two is 0;

(d) by adding three integers we can do so in any order;

(e) 0 is neutral in addition.

One may ask now: why do we run out of negatives in the symmetry between large and small? This question is interesting and necessary right now. The answer is that there is an almost perfect symmetrical copy of the multiplicative group of large and small positives if we place the negative sign positives on the left side of 0 on a numerical line. Every large or small positive has its symmetrical relative to 0 situated at the same distance from zero. For example, 1,000 and 0,001 have their symmetric -1,000 and -0,001. We said almost perfect symmetric copying because by multiplying negatives we get a positive that falls off the negative side of the number line. However, we can “see” for each negative very far from zero a negative “symmetrical” very close to zero. For example, -1,000,000 and - 0,000001.

The final photograph of the symmetry between large and small numbers can then be described as follows: In the numerical line on the right side of 0 lie the large and small positives forming a multiplicative group. For each large positive number, that is, far from zero on the right, there is a negative very far from zero on the left. For every very small positive number, that is, very close to zero, there is a negative very close to 0 but to the left. Distant or near zero negatives do not form a group. Negatives along with positives still form a multiplicative group, as the three group properties are still satisfied.

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