NUMBERS It s the numbers that count

Transcription

1 NUMBERS It s the numbers that count Starting from the intuitively obvious this note discusses some of the perhaps not so intuitively obvious aspects of numbers. Henry 11/1/2011

2 NUMBERS COUNT! Introduction Although we have all learned to work with numbers since childhood, and we use them daily from setting the alarm clock to counting money or making telephone calls, the true nature of numbers is more fascinating and intriguing than would appear at a first glance. I am indebted to Peter Atkins book Galileo s Finger who started me on this journey of discovery and the supporting web pages from Wikipedia and MathWorld. Cardinal and Ordinal Numbers From the word first used in the paragraph above we can already see that we use numbers in two different ways and even give them different names one and first, two and second and so on. Numbers are used to count things one sheep, two sheep, three sheep and numbers are used to order things in sequence first, second, third When using numbers to count we actually use the number as a sort of a multiplier for the items that we are counting. 1x something, 2x something,3x something and so on. So you could ask what is the meaning of the actual number if just taken by itself? Natural numbers, Whole numbers or Integers Let s look at the most familiar numbers: 1,2,3,4,5,6,7,8,9,10,11,12,13

3 Although when adding numbers, the result is always a larger number, it soon became apparent that when applying the operation of subtraction to numbers that there is a need for negative numbers. 1, 2, 3 and so on. (The development of the use of the symbol + for addition and for subtraction and = for equal is an interesting topic by itself.) We see immediately that not all these numbers have the same properties. Some can be divided by 2 the even numbers and some cannot the odd numbers. Some can be split as products of lower numbers 8=2x4=2x2x2 (an even even number) and 6=3x2 (an odd even number). Some are only divisible by 1 and themselves the primes i.e. 2,3,5,11,13.. and since 2 is only divisible by 1 and itself it is the only even prime. It s sometimes called the oddest prime. There has been a lot of debate as to whether 1 itself should be regarded as a prime. If the definition of a prime is a number divisible by one and itself then it follows that 1 is not a prime because 1 is itself. Quote from MathWorld The number 1 is a special case which is considered neither prime nor composite (Wells 1986, p. 31). Although the number 1 used to be considered a prime (Goldbach 1742; Lehmer 1909, 1914; Hardy and Wright 1979, p. 11; Gardner 1984, pp ; Sloane and Plouffe 1995, p. 33; Hardy 1999, p. 46), it requires special treatment in so many definitions and applications involving primes greater than or equal to 2 that it is usually placed into a class of its own. A good reason not to call 1 a prime number is that if 1 were prime, then the statement of the fundamental theorem of arithmetic would have to be modified since "in exactly one way" would be false because any. In other words, unique factorization into a product of primes would fail if the primes included 1. Unquote We call the numbers we have talked about so far the integers and when mathematicians started thinking about these numbers, it became apparent that there are actually only very few of them a mere infinity. A bit like the stars with lots of dark spaces in between. Any set that can be put in a one to one relationship with the natural numbers (or integers) is called a countable infinity. The infinite set (or collection ) of these integer numbers is called aleph 0.

4 Base of Numbers The next interesting thing to observe with our series of numbers is that when we write them down something happens between 9 and 10. i.e. from a single presentation to two places. We call the number where this takes place the base (in this case 10) and the weight of the number is given by its position in this sequence units, tens, hundreds, thousands etc. Without having a base arithmetic is very difficult try adding two Roman numerals MCLXII + DXVI =??????. Different bases have been used through history by different cultures. Examples are 5,12,20 and 60 but having 10 fingers, the base 10 has become the winner (except for computers where the base 2 and the hexadecimal numbers 1,2,3,4,5,6,7,8,9,A,B,C,D,E,F are used. FF in Hex, for example, stands for the number 256 in Decimal notation.) The empty position Now what about a number like one hundred and one. How do we write it. There is one unit, no tens and one one hundred. This could be written as 1 1, the space indicating that there are no tens. After much debate (see The history of formulating nothing ) the symbol 0 was finally accepted as a number representing nothing The Greeks asked themselves how can nothing be something but the Indians started doing calculations using 0 as a number in the 9 th century and called it sunya or void. Still 0 is a most peculiar number since it is neither positive nor negative. Subtracting a number from itself also points to there being a 0 or nothing left. 0 is also even since it is divisible by 2, it is not considered prime or composite. The Left Hand and the Right Hand endstop of the series of integers, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1,0,1,2,3,4,5,6,7,8,9,10,11,

5 Since it is always possible to add (or subtract) 1 from the preceding number the end stop so to speak of the series of Integers appear to be INFINITY, an idea/concept we will have a bit more to say about later. The symbol was introduced by John Wallis in 1655 and called the lemniscate, the Latin word for ribbon. The set of numbers we have so far discussed is called Z in number theory. The names for some of the large numbers differ in the English speaking world and Europe as the following Table illustrates: USA+UK short scale EUROPE long scale 10 6 Million Million 10 9 Billion Milljard Trillion Billion Quadrillion Billiard Quintillion Trillion Sextillion Triljard Quadrillion Centillion Centillion googol googol

6 (So we need to take some care when trying to gauge how big the American Budget Deficit really is since the American and European use of the word Trillion are an order of magnitude of 1,000,000 apart!) The Rational Numbers Now we all know that besides the whole numbers there are also fractions ½, 22/7, 32/89, and so on. These are numbers of the form p where p and q are integers and q 0. q This set is given the name Q. The Irrational Numbers There are however also a large number of numbers (excuse the pun) that cannot be written as p/q. These are the so called irrational numbers like 2, and,e, λ It should now be noted however that some of the irrational numbers are solutions/roots/answers to Algebraic equations and some are not. The general form of an algebraic equation is : a x a n 1 n 2 1x an 2x... a1x a0 0 n n n (where all the a s are integers) 2 2, is for example the solution of the algebraic equation 2 0 x. The integers, the rational (from ratio) numbers and the irrational numbers are all solutions of polynomial equations. These are therefore also sometimes called the Algebraic Numbers.

7 The Trancedental Numbers The numbers that are not the solutions/roots of algebraic equations are called trancedental numbers. A typical example of such a number is π and the Euler number e, as well as combinations thereof like e^π and π^e although the last is not proven yet. The Real Numbers The sum total of all the integers, all the rational numbers, all the irrational numbers and all the trancedental numbers are known as the Real Numbers. This set of Real Numbers is called R and they can be represented on a number line stretching from infinity to +infinity. It would now appear, just by logic reasoning, that the set of real numbers is much larger than the set of integers as they cannot be paired against each other on a one to one basis. The set of real numbers is thus considered not countable and appears to belong to a larger order of infinity the so called continuum hypothesis. This could be Aleph 1, or perhaps Aleph 2 or even Aleph n. The great mathematician Cantor ( ) in his attempt to bring infinity into perspective was almost driven mad by this problem. It was only in the year 1934 that Paul Cohen was able to show that the continuum hypothesis is undecidable) 1 Imaginary Numbers Now funnily enough, when we start playing around with equations, we also come across numbers that appear to be algebraic in that they are the root of an equation, but do not seem to fall onto this line of real numbers. 2 The solution to the equation x 1 0 resulting in x 1 is such a case. These numbers are given the (perhaps unfortunate) name imaginary numbers and are represented on a number axis at right angles (i.e. vertical) to the real axis (horizontal).

8 Complex Numbers. Thus some equations have roots that are partly real and partly imaginary. We can express these numbers by a combination of their real part, shown on the horizontal number line and their imaginary part shown on a vertical number line. R ϴ A+jB θ So any point distance R away from the origin, making an angle ϴ with the horizontal on this two dimensional number plane, can be represented as the sum of the two components A + jb where Where and. In vectorial notation we can write: when written in polar form. Multiplying a vector by thus implies a rotation of the vector in an anticlockwise direction by an angle ϴ. Thus and 1 i.e ( 2 = 1 or 1 and thus 1 This leads to one of the most beautiful and simple equations ever combining e,π,1 and 0, and was first formulated by the great German mathematician Gaus in ( ) 1 0 combining 0, 1, e, j and π all in single simple equation.

9 Perfect Numbers Contemplating numbers will discovers all sort of relationships between them. For example 4=2x2 an even number multiplied by and even number and 6=2x3 an even number multiplied by an odd number. So there is a different relationship between an even even number and an even odd number. Another example if we look at the sum of the divisors of a number we find the following. 12 divides by 1,2,3,4 and 6. The sum of the divisors is 16 which is larger than is now called an excessive number. Compare it with 10 which is divisible by 1,2 and 5, the sum of which is 8, so 10 is a defective number. A perfect number is a number where the sum of the divisors equals the number itself i.e. 28 since =28. The next perfect number is 496 but Pythagoras noticed that multiples of 2 are just 1 short of being perfect 4, divisors 1 and 2, sum 3 8, divisors 1,2 and 4, sum 7 16, divisors 1,2,4 and 8, sum 17 32, divisors 1,2,4,8 and 16, sum 31 Two centruries later Euclid discovered that perfect numbers are always the multiple of two numbers, one of which is a power of 2 and the other the next power of 2 minus 1. Although there are many defective numbers that are just one short of being a perfect number there are no excessive numbers that are just one more than a perfect number. There is still no proof today that this is true for all numbers.