LECTURE 3: The Whole Numbers
Last day we defined two sets to have the same cardinality (size) if they are in 1-1 correspondence.
A finite set is one that is in 1-1 correspondence with a set in the sequence
0 =
1= {0} = {}
2 = {0,1} = {,{}}
3 = {0,1,2} = {,{},{{}}}
4 = {0,1,2,3}= {,{},{{}},{,{},{{}}}}
Each set contains as its elements all the previously-constructed sets! This can be written just in terms of the empty set, but
it get horrible quite fast.
Suggested reading: “Surreal Numbers”, Donald Knuth.
This book deals with John Conway’s use of the same “feeding back in” idea to develop a much more interesting number system (with applications to the theory of games). It’s written in a very readable way – as a short novel about a couple on a desert island!
“Winning Ways” (Berlekamp, Conway, Guy) is another accessible book on the same topic, from the game side. Vol. 1 is the most directly relevant.
Conway’s book “On Numbers and Games” is also good, but is written specifically for mathematicians.
The use of previously-constructed numbers is just a convenience; what we are avoiding is having the same element each time as a placeholder!
The elements of
3 = {0,1,2}
can be paired
with any of
So all these sets have “cardinality 3”. The textbook uses n(A) for the cardinality; other symbols are |A| and #(A).
No two of the sets 0,1,2,3,4,5... are in 1-1 correspondence. (This is another of those things that is hard to prove rigorously.)
We call the set {0,1,2,3,4,5...} the whole numbers.
Every finite set can be counted by one, and only one, whole number.
It’s clear from the construction that the one of any pair that was “constructed first” is a proper subset of the other. For instance,
3 = {0,1,2} Ì{0,1,2,3,4} = 5
If we try to pair a set that is paired with 3 and a set that is paired with 5, there will always be elements left over in the second set. This depends only on cardinalities;
so it makes sense to write
3 < 5.
If m and n are whole numbers, then we always have exactly one of m n, m=n, m n
This property is called trichotomy (“cutting in three”)
INDUCTION: the basic property of the whole numbers
“Every whole number can be counted to, starting from 1”
This is the basis of a famous method of proof for statements about the whole numbers.
(1) Prove what you want to prove for 1.
(2) ASSUMING it’s true for n, show it’s true for (n+1)
You’re done!
True for 0
=> true for 1
=> true for 2
=> ... =>true for n.
EXAMPLE: Show that the sum of the first n odd numbers is n2.
(1) The sum of the first odd number is 12.
(2) Assume that the sum of the first n odd numbers is n2:
(1+3+5+...+(2n-1)) = n2
Add the (n+1)stodd number, which is 2n+1, to both sides
(1+3+5+...+(2n-1)) + (2n+1) = n2 + (2n + 1)
sum of the first n+1 = (n+1)2
odd numbers
completing the proof.
Probably the earliest way to represent numbers was the tally. One stroke is recorded for each element of the set.
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This was probably done to record and display data (sheep, cattle, loans, etc) rather than for arithmetic purposes; so it can be argued that this was the beginning of statistics, not of mathematics – and that statistics is therefore the older of the two subjects!
Later people started to group the strokes, probably first in fives (based on counting on fingers!):
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because a large number in “simple tally notation” is very hard to read.
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At some point, people would have got the idea of copying a tally that had been made up before. At this point a group of five would be drawn all at once, and might be stylized as a hand. In many early languages the word for “five” may have developed from the word for “hand”.
“six hands of arrows and an arrow”
from the shopping list of Fred Flintstone (from Fredrockton)
Once the idea of a symbol for a group of things caught on, it was generalized to numbers too big to make good tally groups.
EGYPT (ca 3400 BCE):
1 10 100 1000 10,000 100,000 1,000,000
12,345 =
BASE TEN PIECES: (ca 2000)
1 10 100
1000
ROME (500BCE – 100):
I V X L C D M V
1 5 10 50 100 500 1000 5000 (etc)
This is a decimal system with a sub-base of 5.
It is additive and also subtractive. This means that under some circumstances one numeral’s value is subtracted from another in evaluating. (The subtractive method was actually rarely used in ancient Rome; it was more used in the Middle Ages.)
23 = XXIII
29 = XXIX (numeral out of order is subtracted)
1989 = MCMLXXXIX (movie title screens!)
Compare in English
“five minutes to nine”, “quarter to twelve”
The overbar multiplies the value by 1000; a system with such a feature is called multiplicative.
Most spoken languages (and sign languages) are multiplicative as well as additive, because expressions have to be simpler or people lose count.
“One thousand, two hundred and three”
(1x1000 + 2x100 + 3)
Commonly used numbers often break the pattern, often due to gradual change.
“eleven”, “twelve” are not additive.
“vingt” in French is not multiplicative
ASL uses some irregular forms too:
There are two signs meaning 20, one used on its own and one in combination for 21,23,24,...,29.
22,33,...,99 are signed as repeated digits, a place value method.
The Babylonians (BCE 3000-2000) introduced the concept of place value. In their system, as in ours, a numeral can have different values depending on its position. They also used base 60 (surviving in our “minutes” and “seconds”) with a sub-base of 10. Thus the “basic numbers” corresponded to
1, 10, 60, 600, 3600, 36000, ...
Our written system is additive with place value and is a pure decimal system.
NOTE:
English 1,234,567.8
French(incl. Quebec) 1.234.567,8
Systeme
Internationale 1 234 567.8
SI usage is common among European scientists and some North American scientists.
IMPORTANT FACT: Every whole number has a unique base 10 positional representation.
To explain the meaning of our notation, and the algorithms we use for arithmetic, schools sometimes use expanded notation:
3,248 = 3x1000 + 2x100 + 4x 10 + 8
or for more advanced students
3x103 + 2x102 + 4x10 + 8
(this makes a valuable connection with polynomials such as 3x3 + 2x2 + 4x + 8 .)
We can also use bases other than 10. Again, representations are unique.
Binary: 27 = 16 + 8 + 2 + 1
= 24 + 23 + 2+ 1
= 1x24 + 1x23 + 0x22 + 1x2 + 1
= 11011 2
Base 7:
101ten = 2x72 + 3 = 2037
Some educators consider this gives a valuable perspective on decimal arithmetic.
Our spoken system uses a large number of multiplicative units:
ten 10
absorbed into “twenty”, “ thirty”, etc.
hundred 100 102
thousand 1000 103
(Germanic in origin)
myriad 10,000 (obsolete)
lakh 100,000
(in India, even among English speakers)
million 1,000,000 106
(from the Italian for “big thousand”;
in use before 1400 in English [used by Langland, Piers Plowman])
billion 109
trillion 1012
......
nonillion 1030
(from French, ca 1500)
Originally meant powers of a million: 1012, 1018,...,1054 but this changed around 1650 (except in Britain!))
Argument for the change: Nesting six-digit numbers inside each other gets too tricky:
1,234,567,890
one billion, two hundred thirty-four million, five hundred and sixty-seven thousand, eight hundred and ninety
OR
one thousand two hundred and thirty-four million, five hundred and sixty-seven thousand, eight hundred and ninety?
Counterargument: most of us never have to give all the digits of a number over a billion anyway. “One point two billion” does nicely for most purposes; so does “one point two thousand million” in the British system.
We have further units, very rarely used:
decillion 1033
undecillion 1036
...
novemdecillion 1060
vingetillion 1063
unvigentillion 1066
...
quattuortrigintaducentillion 10705
(from Latin for 234; 705 = (234+1)x3 )
...
novenonagintanongentillion 103000
(from Latin for 999)
A typical “random” number of this length would require about 18 syllables for every group of three digits, so would require over two hours to read aloud!