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.)
6 YEAR OLD:
It’s not as big as a million though.
9 YEAR OLD:
Or Graham.
3 YEAR OLD:
6 YEAR OLD:
No, a million's bigger than ten. A hundred is bigger than ten. A million is really big.
3 YEAR OLD:
Who's Graham?
6 YEAR OLD:
Infinity is bigger than Graham.
3 YEAR OLD:
Nifity?
6 YEAR OLD:
No, infinity. In-fin-ity.
3 YEAR OLD:
(Straining to speak; arms outstretched as wide as he can make them go.)
9 YEAR OLD:
That’s not even a number.
3 YEAR OLD:
Nifity?
9 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...
6 YEAR OLD:
Infinity plus a hundred then.
9 YEAR OLD:
(Sarcastically.)
Infinity plus infinity.
6 YEAR OLD:
Two times infinity.
9 YEAR OLD:
3 YEAR OLD:
(Quietly, in the ear of his 6 year old sibling.)
Is Graham friendly?
6 YEAR OLD:
Infinity times infinity then.
9 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.)
3 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.
- This article by Robert Crowston in NRICH maths is a thoughtful introduction to the countably infinite via (David) Hilbert's hotel — to the idea that adding a finite number to an infinite set is countably infinite (moving guests from room \(n \to n + 1\)), as is adding an infinite number to an infinite set (moving guests from room \(n \to 2n\)).
- For more on infinity paradoxes, including Hilbert's Hotel, watch this video from the always inspiring Numberphile.
- Introduce students to the idea of \({\aleph _0}\), ("Aleph-Null" or "Aleph-Nought").
- Segue then into the uncountably infinite, why they are bigger infinities than the countably infinite, intuitively through comparing the cardinality of real numbers \(\left| \mathbb{R} \right|\) on the number line to the cardinality of the naturals \({\aleph _0}\), i.e. \(\left| \mathbb{R} \right| > {\aleph _0}\), and then perhaps via Cantor's diagonal argument, with this article from Plus Magazine.
- For more on these different infinities, watch these superb videos:
- How to Count Infinity, by Minutephysics.
- Infinity is Bigger Than You Think, by Numberphile with James Grime.
- How to Count Past Infinity, by VSauce.
- In addition, this article from Katherine Körner in NRICH maths and this from Peter Macgregor on 'Cantor's paradise' in Plus Magazine, both provide more depth, and this, on raising infinity to the power of infinity, from Reginald Braithwaite via GitHub.
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):
\[\begin{array}{c}\begin{align}\sum\limits_{k = 0}^\infty {\left( {\frac{1}{{{2^k}}}} \right)} &= \frac{1}{{{2^0}}} + \frac{1}{{{2^1}}} + \frac{1}{{{2^2}}} + \frac{1}{{{2^3}}} + ...\\ &= \frac{1}{1} + \frac{1}{2} + \frac{1}{4} + \frac{1}{8} + ...\end{align}\end{array}\]
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:
\[\sum\limits_{k = 0}^\infty {\left( {\frac{1}{{{2^k}}}} \right)} = 1 + \frac{1}{2} + \frac{1}{4} + \frac{1}{8} + ... = 2\]
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:
\[\sum\limits_{k = 0}^\infty {\left( {\frac{1}{{{2^k}}}} \right)} = 1 + \frac{1}{2} + \frac{1}{4} + \frac{1}{8} + ... = 2\]
