# 4. INFINITY

To see a World in a Grain of Sand And a Heaven in a Wild Flower, Hold Infinity in the palm of your hand And Eternity in an hour.

— William Blake

#### IN THE CLASSROOM

###### Ontario Curriculum
• Gr. 3-8
• C1. – identify, describe, extend, create, and make predictions about a variety of patterns, including those found in real-life contexts
• C3.1 – solve problems and create computational representations of mathematical situations
• C1.2 – create and translate repeating, growing & shrinking patterns using various representations
• Gr. 4/5
• B1.4 – use drawings and models to represent fractions
• C.3.2 – read and alter existing code; sequential, concurrent, and repeating & nested events
• Gr. 9
• B1.1 – research a number concept to tell a story
• B1.3 – use patterns and number relationships to explain density, infinity, and limit as they relate to number sets
• C2 – apply coding skills to represent mathematical concepts and relationships dynamically, and to solve problems, in algebra and across the other strands
• Gr. 12
• infinity & limit in Calculus
###### Implementation
• Gr. 3-9
• Using 8×8 square grids, student shade to represent the fractions 1/2, 1/4, 1/8, 1/16 & 1/32.
• They use scissors to cut out the shaded parts and combine them to form a new shape.
• They are surprised to discover that all the fraction (to infinity) pieces fit in a single square, and that they can hold infinity in their hands.
• Gr. 7-9
• Using Scratch code that models the relative density of number sets, they edit the code to solve density puzzles.
• Gr. 12
• Students investigate infinity and limit in the context of Calculus

#### 4A. INFINITY IN YOUR HAND

Let’s shade squares to represent an infinite number of fractions.

For example, the image below shows how to represent the fraction 1/2.

And the image below shows how to represent the fraction 1/4.

Q1. Now you do this, using identical squares, like the ones shown below.

• Sketch these squares on a piece of paper
• Or, print this PDF handout of the squares
• Shade the first square to represent the fraction 1/2
• Shade the second square to represent half of 1/2, or 1/4
• Repeat for fractions 1/8, 1/16 and 1/32

Q2. Now you need a pair of scissors.

• Use the scissors to cut out the shaded parts (as shown on the right)
• Then join all the shaded parts to form a new shape
• Imagine doing this forever, shading, cutting out, joining
• How big would the new shape be?
• Would it fit in your room? In your house? In your town or city?

The animation below summarizes the above activity.

#### 4B. Infinity in a walk

Q3. Imagine walking to an open door this way:

• Walk half way to the door
• Then, walk half of the remaining distance to the door
• Then, walk half of the remaining distance to the door
• Keep doing this forever
• Will you ever get to the door?
• Will you ever walk past the door?

Q4. Stuck? Try it another way:

• Don’t think about the fractions
• Just walk to the door
• Then stop and look back
• Use your imagination to see the infinite number of fractions you walked to get to the door

#### 4c. WALK FRACTIONS WITH SCRATCH

Q5. The code below walks fractions 1/2, 1/4, 18 and so on. You can run the code at https://scratch.mit.edu/projects/1056011059/editor

• Alter the code to find sums of these fractions of 300:
• 1/2, 1/8, 1/32, 128 and so on
• 1/4, 1/16, 1/64, 1/256 and so on
• 9/10 + 9/100 + 9/1000 + …
• What did you learn?

#### 4D. SUM OF FRACTIONS WITH PYTHON

Q6. The code below finds the sum of the first 10 fractions 1/2, 1/4, 1/8 …

• Enter the code at https://cscircles.cemc.uwaterloo.ca/console
• Run the code.
• Edit the code to model the sums of each of these fractions:
• 1/4 + 1/16 + 1/64 + …
• 1/2 + 1/8 + 1/32 + …
• 9/10 + 9/100 + 9/1000 + …
• What did you learn?

#### 4E. MODELLING INFINITY WITH PYTHON

Follow this link to investigate infinity + limit by editing snippets of code: https://colab.research.google.com/drive/1D7Z7Uxsw-qgwG5odFxzWZ6oDbL3SH4u7?usp=sharing

Q7. Edit the code to model the sums of each of these fractions:

1. 1/4 + 1/16 + 1/64 + …
2. 1/2 + 1/8 + 1/32 + …
3. 9/10 + 9/100 + 9/1000 + …

Investigate sums of other fraction patterns.

#### 4F. What is natural density?

###### Natural density of even numbers
• Natural numbers = {1, 2, 3, 4, 5, 6, 7, 8 …}
• Even numbers = {2, 4, 6, 8 …}
• The natural density of the even numbers = the chance of randomly selecting a natural number that is even = 0.5
• We may also write this as d(even numbers) = 0.5

Q8. Does d(even numbers) = 0.5 makes sense?

• Every other natural number is even.
• Natural numbers = {1, 2, 3, 4, 5, 6, 7, 8 …}
• Therefore, the natural density of the set of even numbers is 1/2 or 0.5.
• How might you also explain this using a dot density diagram, like the one shown  on the right?

###### Natural density of other multiples

Q9. What is the natural density of the each of the following subsets of multiples?

• {100, 200, 300, 400, 500 …}
• {3, 6, 9, 12, 15, 18 …}
• {5, 10, 15, 20, 25, 30 …}

Q10. More

• The natural density of the number 1 = 0.
• How might this make sense?
• The natural density of square numbers (1, 4, 9, 16 …) = 0.
• How might this make sense?

#### 4G. Natural density with Scratch

###### Natural density of {1} using Scratch

The code below uses experimental probability to approximate the natural density of even numbers. The code is available at https://scratch.mit.edu/projects/550227059/editor/

Q11. Alter code to estimate the natural density of {1}.

1. Execute the code to see the output.
2. Alter the code to randomly pick numbers from 1 to 1000.
3. What do you notice?

NATURAL DENSITY OF SQUARE NUMBERS USING SCRATCH

The Scratch code below models calculating the density of square numbers in the interval 1-10 of natural numbers. The Scratch code is available at https://scratch.mit.edu/projects/565845359/editor

Q12. Change the interval size and notice the effect.

#### 4H. Mathematician interview

We interviewed mathematician Graham Denham (Western University) and he completed, extended and discussed some of the infinity activities above.

###### 1/2 + 1/4 + 1/8 + … = 1
• How can the sum of a never-ending sequence of fractions have a finite sum?
• 1/2 + 1/4 + 1/8 + 1/16 + 1/32 + 1/64
= 63/64
= 1 – 1/64
• “It looks like there’s some suspicious pattern going on here.”
• “It turns out that you can make some real mathematical sense out of saying that an infinite sum is actually equal to 1.”

###### 0.9999… = 1?
• “I have to confess something that bothered me when I was in about grade 5.”
• “How is it possible that 0.99999… = 1?”

Such infinity paradoxes puzzled mathematicians and philosophers of the past.

by George Gadanidis, PhD