The Continuum Hypothesis:
A Look at the 20th Century History of the Real
Numbers from Cantor to Cohen/Scott/Solovay.
Initial Questions (pre-test)
What is a counting number? a natural number? an integer?
Ans:
-
counting numbers: 1,2,3,...
-
natural numbers: 0,1,2,3,...
-
integer: ...-3,-2,-1,0,1,2,3,
...
What is the likelihood (probability)
of choosing a counting number at random from all the integers between -10
and 10?
What is a rational number? an algebraic number? a real
number?
Ans:
-
rational number: k/n where k
is an integer and n is a counting number.
examples: 3/5, -2/7, 36/9
-
algebraic number: a solution to an
equation of the form a0 + a1x
+ a2x2 + ... + anxn
= 0 where a0 , a1 , a2
, ... , an.are integers, not all 0.
examples: 3/5 [(-3)
+5x = 0] ; sqrt(2) [ (-2) + x2
= 0] ; sqrt(-1) [ 1 + x2 = 0]
-
real number (?) : can be expressed
as a decimal.
examples: 3/5 = .6;
3/7 = 0.42857142857142857142857142857143... ;
sqrt(2) = 1.4142135623730950488016887242097...;
ln(2) = 0.69314718055994530941723212145818...
;
pi = 3.1415926535897932384626433832795...
What is the likelihood (probability)
of choosing a counting number at random from all the rational numbers between
-10 and 10?
Ans.: 0
What is the likelihood (probability)
of choosing a rational number from all the real numbers between -10 and
10?
Ans: 0
Arrange the following sets of numbers from smallest to
largest:
| natural numbers |
algebraic real numbers |
integers |
| real numbers |
transcendental real numbers |
rational numbers |
-
natural numbers
-
integers
-
rational numbers
-
algebraic real numbers
-
transcendental real numbers
-
real numbers