NUMBER AND GEOMETRY
Counting was not always important. In fact, it was unnecessary when keeping track of such items as sheep. All that was needed was a one-to-one correspondence. If one had fifteen sheep and kept fifteen pebbles in a bag, it was a simple procedure to remove one pebble for each sheep to see if they were all present. An extra pebble meant a lost sheep. One doesn’t have to know how many sheep or pebbles one has – only to check the one-to-one correspondence between them. Using fifteen knots on a string or fifteen knicks on a stick could also do this. (One-to-one correspondence is a precursor of counting yet pervades much of mathematics.)
It was easier to represent how many after counting words like “one, two, three” were developed. These counting words, or names for numbers, are abstract concepts – abstract meaning to “draw from.”
What can be found in common among different sets of objects and be “drawn from” the objects? If an early man needed to count fish, he might use one set of words. He might use an entirely different set of words to count something different. In other words, if he had two fish, the word "two" would only be used with fish; in fact, the word "two" alone could mean two fish. If one saw this many birds , another name besides “two” would be used to describe how many birds there were.
Now consider what is common among the following sets:
The abstract concept of four represents what is common among the sets. Each set of objects here is in one-to-one correspondence to the others. Any set of four objects will be in one-to-one correspondence with each other set. This abstract concept of using only one set of counting words or symbols to count any type of objects was a tremendous advancement for civilization.
One of the more sophisticated number systems developed was the positional system of the Babylonians, a little like our system of counting. The number 342 has the following positions:
3 in the hundreds position representing 3 hundred, |
4 in the tens position representing 4 tens, and |
2 in the ones position representing 2 ones. |
It is remarkable that it took several hundred years for them to add a zero to their system. One had to understand the size of a number in context with how it was used. If we had no zero, then 31 would represent 31 or 301 or 3010, etc.! It took a stroke of genius to discover or invent zero.
Zero is also a different concept than nothing. If you don’t take a particular math test since you are not enrolled in that class, you get nothing. If you are in that class and take the test and miss all the questions, then you get a zero! To illustrate, look at these two squares and .The first contains no numbers. The second contains a number. The number zero can represent how many numbers are in the first square, but there is one number in the second square.
Mathematics before 600 BC was mostly whatever worked and could be applied usefully to surveying, commerce, astronomy, engineering, even astrology and mysticism. Traditionally, around 600 BC, the Greek mathematician, Thales, used deductive logic to prove mathematical statements. The Greeks took math formulas and procedures from other cultures and their own and set about proving them.
Much work was done in math from 600 BC to around 300 BC culminating in Euclid’s works in deductive geometry. During this time, the Pythagoreans, a society founded by the Greek, Pythagorus, believed so much in the importance of numbers that they thought everything (i.e., objects, people, ideas, etc.) to be illusions. The only reality was the numbers these things symbolized. They believed everything to literally be numbers.
They developed the relationships between music and numbers. A tight string that is plucked at different lengths will produce a harmonic chord if the lengths are in ratios of the first few counting numbers (1 to 2, 2 to 3, and 3 to 4). This can be demonstrated by pressing a guitar string at these points and plucking the string each time: 1/2 its length, 2/3 its length, and 3/4 its length. This relationship between music and numbers was one example of why they believed more in the reality of numbers than anything else. It led them to believe that the heavenly bodies moved at distances of these same counting number ratios, which produced a sound. This sound was the harmony caused by the movement of the heavenly spheres.
There is evidence that Pythagorus also used the simile of the cave used by Plato. This simile identifies us with cave dwellers who see only the shadows of the real world cast on our cave wall from outside the cave. Reality to us is only the shadows. This may have influenced the Pythagoreans with the idea of an ideal counter-earth on the other side of the sun, an idea used in modern science fiction stories. More importantly, mathematics was to them the best connection between the ideal word and our physical experiences.
The Pythagoreans thought that the even numbers (2, 4, 6, 8, etc.) were ephemeral or changing and that the odd numbers (3, 5, 7, 9, etc.) were celestial or non-changing. They didn’t consider 1 an odd number nor did they accept negative numbers.
Number |
Represented |
1 |
Reason |
2 |
Woman |
3 |
Man |
4 |
Justice - since it is the product of equals (2 times 2) |
5 |
Marriage – the union of the first masculine number, 3, and the first feminine number, 2. |
One of the methods of proving characteristics about numbers was by the use of dots arranged in a geometric pattern. This method could also be used to illustrate their philosophy. As an example of the changeable character of even number, consider any rectangular array of dots that has one side longer than the other.
The example on the right begins with 2 dots across and 3 down. It would take an even number of dots to expand this figure. Adding 6 dots would form a new rectangle that is 3 across and 4 down. Again, it would take an even number of dots, 8, to expand this again. Adding even numbers of dots will continue to form a rectangle, but the rectangle changes shape – its sides changing in ratio from 2/3 to 3/4 to 4/5 and so forth. This illustrates the changeable character of an even number of dots added to a rectangle. |
If we use a square array of dots, we have to add an odd number of dots to expand the square. In this case, as on the right, the ratios are 2/2, then 3/3, then 4/4, etc. (all equal to one). |
Adding an odd number of dots illustrates the unchanging character of odd numbers. The square grows, but continues to have the same shape where the ratio of one side to the other is the constant 1.
We still use dots to teach children about numbers. Of course, today we avoid the philosophical aspects attributed by Pythagorus, but their philosophical concerns about numbers led to one of the most unusual conflicts in mathematics. The only numbers they recognized were positive integers, also called natural numbers (1, 2, 3, 4, etc.) and the ratios of these numbers. The positive integers and ratios of positive integers were called rational numbers (today we include 0, negative integers, and negative rational numbers). The Pythagoreans had professed that the only numbers that existed were ratios of natural numbers that we call rational numbers. They examined the numbers assigned to points on a line as in the figure on the right. |
A point is labeled 0 and another point labeled 1. Then 2, 3, 4, and so forth could be marked off as well as 1/2, 1/3, 1/4, 3/2, 4/3, etc. It seems reasonable that between any two points representing rational numbers on the line, there always lies another point representing a rational number. All the points on the line would be used up labeling them with rational numbers. |
Astonishingly, though, the Pythagoreans could prove that a square (1 unit on each side) would have a diagonal length OP on the number line, and length OP could not represent a rational number. The result is that , the square root of 2, is not a rational number. |
In other words, it can be proven that no rational number exists such that if you multiply it by itself the product will be two (2). Since there is no rational number equal to the , this must be a number that is not rational. For some time, there was a strict cover-up with threat of death since this contradicted their mathematics; but more importantly, it contradicted their philosophy. As mentioned before, we call these numbers that are not ratios of integers irrational (meaning not a ratio).
Another interesting problem that was over two thousand years old even to the Greeks was that of squaring the circle. Is it possible with a ruler (unmarked) and compass to construct, for a given circle, a square of the same area?
Given a circle of area A, | can we construct a square of area A, of the same area A? |
In attempting a solution to this problem, it was determined that the ratio of the circumference of a circle – C – to its diameter – D – was a constant. In fact, C / D = this constant for any size circle whether it is one inch in diameter or a billion miles in diameter. (Some ratios do remain constant such as the previously mentioned square array of dots or just a square. By the definition of a square, the length of one side divided by the length of the other side is always l (for “small” or “large” squares).
It was probably Euclid who showed this constant for a circle to also be the same as the ratio of the area “A” to the square of the radius “r^{2}” of a circle. A / r^{2} = this constant for any size circle. Therefore, to construct the square of area “A”, this constant was needed since “A” would equal “r” times this constant. Repeating, for large circles, small circles, any size circles, the constant remains unchanged ( i.e., C / D = A / r^{2} = this constant).
Over two thousand years after Euclid, this constant was proven to be an irrational number and also a “transcendental” number. Transcendental numbers were proven to be impossible to construct with only a straight edge and a compass. This is why the ancients were unable to construct a square of the same area as a given circle.
Since this constant is an irrational number, it cannot be written as a ratio of integers. If it is written as a decimal number, the digits to the right of the decimal continue forever! They will never reach a position where they repeat forever. Therefore, it is impossible for a person (with finite life expectancy) to even write down this number as a decimal number. An 18th century mathematician named Euler decided to use the symbol , the Greek letter pi, to represent this number. ( is the first letter of periphery, referring to the circumference of the circle.)
There are two concepts to point out about this problem. First, a problem that is difficult to solve may be difficult because it is impossible to solve. A proof that it is impossible to solve can be shown whereby there is no longer any need to attempt solving it.
The other concept is that a number, such as this irrational constant that equals C / D and A / r^{2}, cannot be written as a decimal number so someone thinks of a symbol such as to represent it. Many values have been used to represent approximately . The Bible uses three; many classes in school use 3.14 or 22/7. In 1897, one state legislature, backed by the State Superintendent of Public Instruction, almost established a value by legislation. All of these values are only approximately equal to .
About two centuries after Pythagorus, Plato and his followers believed ideal, perfect, objects existed and that we experience them only imperfectly. Platoism has influenced religion, science, and mathematics. According to Plato, mathematicians discover math rather than create it. What do you think?
Around 300 BC, Euclid attempted to base all geometry on a sound system of self-evident statements called postulates or axioms and the use of deductive proofs to develop theorems (proven statements) in geometry. This system was the most important work existing in geometry for two thousand years. Only in recent centuries have other “newer” geometries been developed. Note that the word theorem in math is not the same as the word theory in science. A theory is a conjecture or proposition about the way something behaves. A theorem in math is a statement or inescapable conclusion proven by a deductive argument from other statements.
We no longer consider axioms as self-evident statements. It became too difficult for everyone to agree on what is self-evident! It is important that the Greeks recognized that if theorems are proven logically from other statements – and if those other statements were proven logically from previous statements – then, eventually, to have a starting point, there must be some unproven statements that we call axioms. In a similar manner for definitions, we must also have a beginning with undefined terms.
Euclid used ten unproven statements to prove 465 theorems. The unproven statement that Euclid seemed reluctant to use was the famous 5th postulate or parallel postulate. He used the 5th only after it was absolutely necessary – probably since it did not seem “self evident.” One of the ways we state this postulate today is: for a given line and a point not on the line, there is exactly one line passing through the given point that is parallel to the given line.
It can be shown that another way to state this parallel postulate is by saying that the sum of the angles of a triangle is l80 degrees. These two ways to state the parallel postulate may not appear to be saying the same thing, but it is not difficult to show that they are equivalent. If you are given either one, the other can be proven from it.
For two thousand years, most mathematicians attempted to prove this parallel postulate from the other nine unproven statements. This was not possible since the parallel postulate is independent from the other nine. Many mathematicians showed its independence finally during the nineteenth century. To show that the parallel postulate was independent, a model or interpretation was exhibited where all of the ten axioms were satisfied except for the parallel postulate. Since the words line and point are really undefined, then they may be interpreted in many different ways.
One of Euclid’s axioms states, in effect, that the shortest distance between two points is a straight-line segment. This may seem reasonable on a “flat” surface, but if the surface is a hemisphere (as navigators will tell you) the shortest distance between two points is a line segment of a Great Circle or geodesic, which passes through the two points. If you pass a plane through the center of a sphere, then the points on the plane that intersect the surface of the sphere will form a Great Circle on the sphere (such as the equator).
Figure 1 below shows some of the Great Circles on a sphere. Figure 2 considers only half of the sphere and indicates that for a given line and a given point not on the line that there would be no parallel line through the point. In other words, every line through the given point will intersect the given line. Finally, figure 3 indicates that in this geometry on the surface of a hemisphere, the sum of the measures of the three angles of a triangle will be more than 180 degrees!
Georg Friedrich Bernhard Riemann developed this geometry, where the line is interpreted as a curve on the surface of a hemisphere. Nicholas Lobachevsky took another approach – that of developing a model where there would be more than one line through a point and parallel to a given line. In Lobachevsky’s geometry, the sums of the angles of a triangle add up to less than 180 degrees! Karl Friedrich Gauss actually tried to prove that the world we live in may not be Euclidean by calculating the sum of the angles of a triangle formed by lanterns placed on three mountain tops. The experiment was inconclusive. If the angles add to 180^{0} there will always be some error, and we can’t prove the sum is 180^{0}. However, if the sum of the angles is not 180^{0}, we only need to increase our accuracy of measurement until the error becomes less than the difference of 180^{0} and the actual sum.
Why didn’t the Greeks or others find new interpretations of lines or points? Actually, Euclid attempted to define these terms. As an example, Euclid called a line length without breadth. If you try to explain what length without breadth means, you will see why it is best to leave the term “line” as an undefined term.
The Greeks thought that these terms, such as line and point, and the axioms were virtually self-evident and represented truth. We now treat axioms as simply unproven statements and some words as undefined terms in order to have a common ground upon which to logically develop a system of other concepts. If we change the axioms, we have a different system. We ignore the attribute of an absolute truth.
George Cantor put forth the law of conservation of ignorance to explain why new interpretations take so long. The Greeks thought their math was based on truth, a self-evident Euclidean geometry. It took 2000 years to dislodge this notion for, if you believe you have truth, why worry about other interpretations resulting in different geometries. Since today we lay no claim of truth to our axioms (unproven statements), even the Flat Earth Society should not object to different systems based on different sets of axioms.
Non-Euclidean geometry is an example of mathematics being developed without applications to the real world. For Einstein to develop the general theory of relativity, one of the greatest scientific advances in the 20th century, he was able to make use of Riemann’s non-Euclidean geometry. Mathematics and the applications of mathematics to real world problems have historically used each other. They have moved together, theoretical math stimulating applications sometimes in the real world, and real world problems sometimes stimulating the math needed to solve them.
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