Friday 27 April 2018

On Overheard Conversations About Maths #1





S
SCENE:     
The floor.  Just before bedtime.
Three siblings are islands in a sea of Lego.
The eldest is trying to make a Lego Penrose triangle.


3 YEAR OLD:

(Silently counting the studs on a red 2 × 4 Lego brick,
mouthing each number until, to no-one in particular...)

Eight!

(He opens up his thumb and fingers successively
from the fist of one hand as he counts, this time out loud.) 
                       
One, two, three, four, five...

(Now to the other hand.)

...six, seven, eight.

(Some time passes as he collects all the 2 × 4 bricks
he can find, piles them together, and pretends to count them all.)

A hundred!  A hundred is bigger than four.  Or five.  Or eight.


6 YEAR OLD:

It’s not as big as a million though.


YEAR OLD:

(With a nod to his Numberphile education...)

Or Graham.


YEAR OLD:

Or ten.  Ten's bigger.


YEAR OLD:

No, a million's bigger than ten.  A hundred is bigger than ten.  A million is really big.


YEAR OLD:

Who's Graham?


YEAR OLD:

Infinity is bigger than Graham.  


YEAR OLD:

Nifity?


YEAR OLD:

No, infinity.  In-fin-ity.


YEAR OLD:

(Straining to speak; arms outstretched as wide as he can make them go.)  

Is Nifity big like this?


YEAR OLD:

That’s not even a number.


YEAR OLD:

Nifity?


YEAR OLD:

Infinity.  In-fi-nity.  In-finity.  No, it’s not a number it just means the biggest thing.  People say it's a number but it's not because you could just say infinity plus one but that's still just infinity because...


YEAR OLD:

Infinity plus a hundred then.


YEAR OLD:

(Sarcastically.)

Infinity plus infinity.


YEAR OLD:

Two times infinity.


YEAR OLD:

Duh, that's the same.  And it's still just infinity.


YEAR OLD:

(Quietly, in the ear of his 6 year old sibling.)

Is Graham friendly?


YEAR OLD:

Infinity times infinity then.


YEAR OLD:

Do you even know what that means?  OK then, infinity times infinity infinity times.  It's still just infinity!  It's still...

(Establishing the correct perspective
from which to see the Lego Penrose triangle
doesn't seem to matter quite as much right now.)

Isn't it...?  Erm...


YEAR OLD:

Nifity?


CURTAIN.



Suggested explorations, diversions and links:

What is infinity?
  • Share this typically accessible introduction to the idea of infinity from MathsIsFun.com.  (Play, for example, with the provocation 'An infinite series of A's followed by a B will NEVER have a B'.)  The page gently introduces infinity as 'the idea that something has no end', that doesn't grow, and subsequently moves into arithmetic properties and the exciting idea that there are different sizes of infinity.  

    Why does \(\infty  + 1 = \infty  + 100 = \infty  + \infty  = \infty  \times \infty  = \infty \)?

    We can (sometimes) add an infinity of numbers.

    Play around with infinite sums, starting perhaps with this article by Luciano Rila from Plus Magazine.   And consider, for example:
    \[\underbrace {1 + \frac{1}{2} + \frac{1}{4} + \frac{1}{8} + ...}_\infty \]Or, to put it another way:
    \[\begin{array}{c}\begin{align}\sum\limits_{k = 0}^\infty  {\left( {\frac{1}{{{2^k}}}} \right)}  &= \frac{1}{{{2^0}}} + \frac{1}{{{2^1}}} + \frac{1}{{{2^2}}} + \frac{1}{{{2^3}}} + ...\\ &= \frac{1}{1} + \frac{1}{2} + \frac{1}{4} + \frac{1}{8} + ...\end{align}\end{array}\]
    If we add up the first few terms (i.e. find the first few partial sums):
    \[\begin{array}{c}\begin{align}\frac{1}{1} + \frac{1}{2} &= 1\frac{1}{2}\\\frac{1}{1} + \frac{1}{2} + \frac{1}{4} &= 1\frac{3}{4}\\\frac{1}{1} + \frac{1}{2} + \frac{1}{4} + \frac{1}{8} &= 1\frac{7}{8}\\\frac{1}{1} + \frac{1}{2} + \frac{1}{4} + \frac{1}{8} + \frac{1}{{16}} &= 1\frac{{15}}{{16}}\\etc.\end{align}\end{array}\]
    We can see that each iteration produces a number that gets closer and closer to 2.  We produce another infinite sequence in others words with a limit of 2, thus:
    \[\left\{ {1,\;1\frac{1}{2},\;1\frac{3}{4},\;1\frac{7}{8},\;1\frac{5}{{16}},\;...} \right\}\]
    And we can therefore show that the sum of infinite series is 2.  Say:
    \[\sum\limits_{k = 0}^\infty  {\left( {\frac{1}{{{2^k}}}} \right)}  = s = 1 + \frac{1}{2} + \frac{1}{4} + \frac{1}{8} + ...\]
    Then:
    \[\begin{array}{c}\begin{align}\ 2s &= 2\left( 1 \right) + 2\left( {\frac{1}{2}} \right) + 2\left( {\frac{1}{4}} \right) + 2\left( {\frac{1}{8}} \right) + ...\\2s &= 2 + 1 + \frac{1}{2} + \frac{1}{4} + \frac{1}{8} + ...\end{align}\end{array}\]
    And so:
    \[\begin{array}{c}\begin{align}\ 2s - s &= \left( {2 + 1 + \frac{1}{2} + \frac{1}{4} + \frac{1}{8} + ...} \right) - \left( {1 + \frac{1}{2} + \frac{1}{4} + \frac{1}{8} + ...} \right)\\s &= 2\end{align}\end{array}\]
    Thus:
    \[\sum\limits_{k = 0}^\infty  {\left( {\frac{1}{{{2^k}}}} \right)}  = 1 + \frac{1}{2} + \frac{1}{4} + \frac{1}{8} + ... = 2\]

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