NUMBER AND ALGEBRA
We have glanced at the development of numbers and geometry up to the 19th century. Let’s go back now for a look at the development of numbers, arithmetic, and algebra.
A much too simple definition of mathematics says it is the study of quantity and space. Mathematics began with the abstraction of number from the elements it described - as four being a separate concept from four birds or four fish. Four is even separate from space - such as four dots, four lines, or four miles. This was not always so as mentioned earlier.
We now believe that infants do not have a clear concept of what we consider to be a separation of the infant and everything else. In other words, they may perceive things they look at as things that are a part of themselves. (Philosophically, this separateness – standing back and observing – or even indifference that “I” have could be an illusion; the infant may be right!) Early man felt a part of nature and this is exhibited in words used for the numbers “1” and “2.” These were usually adjectives or used as adjectives, often being a part of the nouns they described.
Sometimes pairs were considered the whole – such as in Chinese, a single word was developed for eyes (meaning two eyes). In Sanskrit, ahani means “the day,” which is actually a duel for day and night. Also, duels were used in Greek and Semitic languages using a singular form, a double form, and then a plural form – such as in Arabic (radjulun for “man,” radjulani for “two men,” and ridjahun for “men”).
The separation, the abstraction of numbers, is evidenced by the changing of adjective forms for number words to noun forms. Counting stopped for a long time before it resumed with three, four, etc. The Arabic number words, ancestors of our present-day number systems, begin with the numbers one and two as adjectives, and continues with three, four, five, etc., as nouns. This is exciting evidence of man’s historical abstraction of numbers.
It is interesting that the next counting number – three – was also a milestone. Almost universally we find forms of counting “one, two, many.” In old language forms, there is the singular, duel, then plural. The meaning of 1, 2, 3, in Sumerian counting is man, woman, then many.
Now, consider the sets of marks below:
Instead of nicks on sticks or dots or marks like above to represent counting, words became more useful. A different word was used for a different number. Of course, it would be difficult to count with words if you had to keep making up different words. Counting up to a million would mean you would have to memorize a million words!
There is another way that people today write the above set.
The above is actually a natural ordering for these marks since we have digits on each hand. A primitive numbering may use a special symbol for . Spanish expeditions to the Yucatan in the early 1500s uncovered the Mayan numeral system which began by using dots (pebbles) for a unit and a horizontal stroke (stick) for five. The Mayan system actually used 20 for their number ‘base,’ perhaps because we have 20 digits on our hands and feet.
We will use part of their system to illustrate a number ‘base’ of five. The previous set of marks can now be replaced by:
A more modern system would handle this by using five symbols and five names; this is what we mean by base 5. Where we position the symbols will enable us to handle large numbers. Let us use the five symbols and names below:
0 = zero 1 = one 2 = two 3 = three 4 = four
From these, we can make up the following number system in base five noting that the only symbols we are allowed to use are 0, 1, 2, 3, and 4.
If you caught fish, it could be written as 32 (threety-two for 3 fives and 2 ones)! Also note that is represented by 10 and is represented by 100.
We can write any counting number we desire using this made up scheme. However, unless we make up names for 10,000, 100,000, etc., we would have to read 312,114 as ‘three one two one one four.’
Let’s push this system one step further by making up an addition table and adding.
Hopefully, you have followed this system in base five and can push it further yourself with a multiplication table and other operations.
Another number base, base two, is highly useful with computers since base two only uses the two symbols 0 and 1. 0 and 1 can represent electrical current that is on or off, or they can represent a spot that is magnetized or not magnetized; or any other type of on-off situation that is recorded and read by digital computers. Below, we compare how we count with these symbols in base two, base five, and our base ten system that we learned in school.
The early cities of Summer (which became the Babylonian Empire) were the first to formally develop counting using a base of sixty. They probably used sixty since the shekel was a unit of weight of grain, and it took sixty shekels to equal one mina, and sixty minas to equal one talent. Sixty was probably determined by the size of the container used. Shekels, minas, and talents ultimately became the names for their minted coins. They did move to a ‘positional’ number system, a predecessor of our system, where the position of a digit represents a multiple of the base. For example, in our base ten, 325 represents 5 units, 2 tens, and 3 hundreds. The Hindu-Arabic system was developed from the Hindu-Arabic system that used base ten.
The concept of addition seems to be inherent in the concept of counting. After all, one, two, three, four is merely taking one, then adding one to one, again adding one to one to one, and again adding one to one to one to one to get four. We first streamlined it by letting 1 + 1 = 2, 1 + 2 = 3, 1 + 3 = 4, etc.
By experience, we learn that adding is independent from the order of the numbers we add. In other words, 3 + 2 = 2 + 3, 1 + 5 = 5 + 1, 8 + 12 = 12 + 8, etc. Therefore, one of the first concepts learned by children is:
This is called the commutative law for addition of natural numbers. Of course, children don’t begin by using the symbols a and b to represent natural numbers, nor do they understand the words “commutative law . . .”. They usually first see this rule by numerous examples similar to the illustration below:
Counting the left side yields six fish, whether you have 2 + 4 or 4 + 2 fishes.
When we do learn to state this commutative property for addition of natural nubmers, using symbols such as a and b, then we have progressed from arithmetic to algebra! Arithmetic deals with known constants such as the problems below:
Algebra incorporates symbols that either represent unknown quantities such as the x in the three problems below
521 + 98 = x |
2x – 1 = 5 | 3x + x = 2 , |
or more general statements such as
let a and b be natural numbers, | then a + b = b + a. |
In the three problems above, x can only equal 619 in the first, x only equals 3 in the second, and x equals 1/2 in the third problem. For a + b = b + a, we are stating an infinite set of possibilities such as 1 + 1 = 1 + 1, 1 + 2 = 2 + 1, 1 + 3 = 3 + 1, 2 + 3 = 3 + 2, 2 + 2 = 2 + 2, 3 + 3 = 3 + 3, 1 + 4 = 4 + 1, and 2 + 4 = 4 + 2, and so on. The x in the three problems above simply represents a particular unknown value which we ‘solve for.’ The a + b = b + a represents a generalized way of stating an infinite set of equations and is given the name commutative property for addition.
Since it is impossible to try all possible natural numbers a and b to prove a + b = b + a (there is an infinite set of natural numbers), we have to just assume it is true. In other words, the commutative property cannot be proven so it is an axiom for the natural numbers (axioms are unproved statements that, for practical reasons, are assumed true).
Notice that the commutative property doesn’t hold for everything. For instance, it is not true that put on shirt + put on coat = put on coat + put on shirt, since we are not all punk. Also, the commutative property does not hold for division, 1 / 2 is not equal to 2 / 1.
Let’s examine multiplication for this property and define multiplication. Multiplication can be thought of as a short hand addition. Let ‘*’ represent ‘times’. 3*5 = 5 + 5 + 5 and 5*3 = 3 +3 + 3 + 3 + 3. 2*8 = 8 + 8 and 8*2 = 2 + 2 + 2 + 2 + 2 + 2 + 2 + 2. In general, we can define multiplication of natural numbers a and b to be:
It is not difficult to assume as an axiom the commutative property for multiplication, a*b = b*a.
Let’s examine subtraction of natural numbers. First note that given 7 – 4 = x, then since 4 + 3 = 7, x would be 3. Or if 5 – 3 = x, then since 2 + 3 = 5, we say x is 2. In general, if a – b = x, then this difference a – b is a number x - such that x added to b equals a. We have a definition for subtraction – subtraction, like multiplication, is defined by addition.
a – b = x | if and only if | x + b = a |
It is interesting how the Egyptians handled subtraction. The Rhind papyrus of around 1650 BC, purchased by the English Egyptologist, A. Henry Rhind, and now in the British Museum; shows how the Egyptians represented addition and subtraction. Addition was represented by a pair of legs walking from right to left, the direction of Egyptian hieroglyphics. Subtraction was represented by a pair of legs walking from left to right.
If we try to subtract the natural numbers 7 – 10, we don’t have a natural number that 7 – 10 equals. In other words, there is no natural number x - such that x + l0 - 7. The natural numbers have a property under addition that they don’t have under subtraction. This is called the closure property or, since it is an axiom, closure axiom. We say the natural numbers are closed under the operation of addition since a + b is always a natural number. The natural numbers are not closed under subtraction since a – b is not always a natural number.
In general, a set of numbers is closed under an operation, such as addition or subtraction, if for every pair in the set that the operation is performed on, the result is always a number in the set. (Mathematically, given a set, S, of elements, and an operation, o, then for any elements a and b in S, a o b = c where c is also in S. A finite example of this is the set { - 1 , 0 , l } where the operation, o, is multiplication. -1 o -1 = 1, -1 o 0 = 0, -1 o 1 = -1, 0 o -1 = 0, 0 o 0 = 0, 0 o 1 = 0, 1 o -1 = -1, 1 o 0= 0, and 1 o 1 = 1, all products in S.)
A definition of closure is a relatively modern concept. The ancients did not say “natural numbers are not closed under subtraction, so let’s make up new numbers which will make our set of numbers closed.” We can use our hindsight and the closure axioms to see how the number system developed.
Actually, negative integers, -1, -2, -3, -4, -5, . . . were used in some form for centuries for accounting purposes, to keep track of debts. Debts were recorded on sticks or other objects before marks for numbers were ever invented. It was probably the merchants in the western world who had the most influence in forcing the acceptance of the negative integers.
Finally, during the Middle ages, mathematicians in the western world accepted 0 and the negative integers, -1, -2, -3, -4, -5, etc. (0 was the hardest to accept and was considered mystical since its shape was non-ending. It was thought to mean nothing by itself yet a tremendous number when behind other numbers – such as 1000000000!)
The natural numbers or positive integers, 0, and the negative integers became a closed set of integers . . ., -3, -2, -1, 0, 1, 2, 3, . . . under addition, multiplication, and subtraction (but not division). The sum of any two integers is an integer. The product of any two integers is an integer. Now, the difference of any two integers is an integer.
The integers are not closed under division. What the Pythagoreans actually considered to be the ratio of natural numbers, we now consider to be actual numbers called rational numbers. The rationals, when defined as the ratio of two integers, can be expanded to a new set of numbers that are closed under division (except division by 0). We can indicate this set of rational numbers as in the pattern below (where is read positive or negative).
The pattern above indicates the set of all rational numbers even though many duplicates are in the pattern – such as 1/1, 2/2, 3/3, . . . all representing the number l.
If you divide the numerator by the denominator of each of the rational numbers above, you will get a decimal number which either stops somewhere after the decimal point, or it will start repeating the same block of digits somewhere after the decimal point. Examine the examples below. They either stop or repeat. Where they stop, we could continue to add 0s and consider them repeating – such as 3.0000 . . . equals 3. Instead of using . . . to mean continue in the same fashion, we will draw a bar across the top of the repeating part; therefore, where the bar means we continue the 3s forever.
It is possible to define rational numbers as numbers in decimal form that somewhere repeat to the right forever (even though it may be repeating 0s). Then all the rest of the non-ending numbers are in decimal form that do not repeat forever are the irrational numbers. Of course, since an irrational number doesn’t repeat forever and is non-ending, it is virtually impossible to write it down in decimal form. Instead, we use symbols to represent irrational numbers such as and .
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