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Secondary Magazine - Issue 48: Focus on


This page has been archived. The content was correct at the time of original publication, but is no longer updated.
Created on 24 November 2009 by ncetm_administrator
Updated on 07 December 2009 by ncetm_administrator

Secondary Magazine Issue 48infinity symbol
 

Focus on...interpretations of infinity

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How have views of infinity changed through time? The Compact Dictionary of the Infinite gives a timeline – just go to the bottom left hand side of the screen and click through to infinity.


 
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It is a common misconception that infinity is a number rather than a concept. Wolfram MathsWorld says that infinity is “an unbounded quantity that is greater than every real number”. This is different from being a number and certainly different from being the ‘biggest number’. Infinity is not a number at all.


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German mathematician Georg Cantor (1845 – 1918) explored and re-imagined the concept of infinity. His theory depends on a simplified idea of counting – a one-to-one correspondence.
Imagine a teacher on a school trip – as each student gets off the bus, she puts a mark on a piece of paper. After the visit, as each student gets back on the bus, she crosses out the mark. Without using numbers she has used the one-to-one correspondence of students and marks to ‘count’ the students.
Cantor described any set of numbers which has this one-to-one correspondence with the natural numbers (1, 2, 3, …) as countably infinite. This leads to the counterintuitive result that there are ‘the same number’ of even numbers as there are counting numbers as there are fractions! Cantor defined this number, the cardinality of these sets, as aleph-null(aleph-null).
 

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Cantor showed that the set of Real numbers (which includes the irrational numbers), are ‘more infinite’ than the set of Natural numbers. In fact, he showed that the set of Real numbers between 0 and 1 is more infinite!

A set is countably infinite if ‘a prescription can be given for identifying its members one at a time’ put simply, they can be written in an ordered list.

Imagine one of your students, in an effort to prove Cantor wrong, tried to write every number between 0 and 1 as a decimal in an ordered list. If they’re not able to do this, then Cantor is correct! They bring you their list to be marked (!) - if you can find a number that’s not on the list, then Cantor is right.
If their list is a set of decimals

0.a1a2a3a4a5
0.b1b2b3b4b5
0.c1c2c3c4c5
and so on

To find a number that’s not on the list, ask the student where the number r is where r = 0.r1r2r3r4r5… made in such a way that r1 is different to a1, r2 is different to b2, r3 is different to c3 and so on. Your number, r, is different to every number on the student’s list in at least one decimal place and, therefore, Cantor is correct! This set of numbers is uncountably infinite.  


 
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German mathematician David Hilbert (1862-1943) devised the idea of the Hilbert Hotel to illustrate different types of infinity. Hilbert’s idea has been turned into a short story, a short film, and even inspired the lyrics to a song!


 
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The  symbol for infinity is sometimes known as the lemniscate (from lemniscus, Latin for ribbon). It is thought to have been introduced as the symbol for infinity by English mathematician John Wallis in his 1655 work De sectionibus conicis.


 
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This video is either going to be my greatest accomplishment or my stupidest video ever. Today, I’m going to count to infinity.” Can you guess what happens? 
 
 
 
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