# Exploring 0.99999…. = 1 using fractions and binary

Last week my younger son read chapter 2 of Jordan Ellenberg’s How not to be Wrong. In that chapter Ellenberg discusses the the nunber 0.999999…. and whether or not it equals 1.

We discussed his thoughts on that chapter here:

https://mikesmathpage.wordpress.com/2021/01/09/talking-through-chapter-2-of-jordan-ellenbergs-how-not-to-be-wrong-with-my-younger-son/

Today I thought it would be fun to approach the idea from the (slightly) different perspective of using fractions and binary.

We started with a review / refresher of how to write integers in binary since we haven’t talked about that in a while:

Then we talked about how you write fractions in binary including fun problem of writing 1/3 in binary:

Now I posed the question of how could we write 1 in binary – this part turned out to be the rare discussion that was as fun as I’d hoped it would be ðŸ™‚

Finally, having found an interesting way to write 1 in binary, we moved on to the question of how to write 1 in base 10:

This was a enjoyable project. The discussion of infinite series in How not to be Wrong is fascinating and accessible to a wide audience. Talking through the ideas in that chapter with my younger son has been really fun!

# Sharing Grant Sanderson’s Hamming Code video with my younger son

Yesterday Grant Sanderson published a fantastic set of videos on Hamming codes. I watched the first one with my younger son last night:

Today we talked about some of the ideas in the video – starting with some of the things he thought were interesting:

Next I had him work through one of the examples in Grant’s video – I didn’t realize it was an example of an error since I just pulled it off of a screen shot, but we discovered the error talking through the example:

Finally, we went back to the same example. This might seem like a strange thing to do, but Grant’s example had an error in the parity bit and I wanted to make sure my son understood that the error correcting codes could also detect that kind of error.

I love Grant’s work – it makes for such a fun and easy way to explore ideas that kids wouldn’t normally see in their school math.

# Writing fractions in binary

I saw an interesting tweet last wee that got me thinking about fractions:

Today as sort of a unusual way to play around with fractions I thought it would be fun to try to write some fractions in binary. It has been a while since we talked about binary, though, so I had my son tell me what he knew about binary first:

Next we moved on to writing fractions in binary – we started with some simple cases:

Finally, we tried to write 1/3 in binary. This video shows what a kid thinking through a math problem can look like, and also shows why I thought this exercise would be a nice fraction review:

# Playing around with the PCMI books

After seeing a plug for them on twitter I bought the PCMI books. They arrived yesterday:

The first book I picked up was Moving Things Around since the shape on the cover of the book is (incredibly) the same shape we studied in a recent project.

One more look at the Hypercube

I found a neat problem in the beginning of the book that by another amazing coincidence was similar to a (totally different!) problem we looked at recently:

Revisiting Writing 1/3 in binary

We started by talking about the books and the fun shape on the cover:

Now we moved on to the problem. It goes something like this:

Consider the number 0.002002002…. in base 3. What is this number? How about in base 4,5,7, and n?

We started in base 3 and the boys had two pretty different ways to solve the problem!

Next we moved on to base 4:

Now we moved to the remaining questions of base 5, 7 and N. Unfortunately I got a phone call I had to take in the middle of this video, so I had to walk away while the solution to the “N” part was happening.

We finished up with the challenge problem -> What is 0.002002002…. in base 2?

This is a pretty neat challenge problem ðŸ™‚

Definitely a fun start to playing around with the PCMI books. Can’t wait to try out a few more problems with the boys!

# Sharing Stephen Wolfram’s MoMath talk with kids

I saw an amazing tweet from Stephen Wolfram today:

Based on the blog post, his talk at MoMath must have been incredible!

I decided to try out one of his explorations with the boys tonight. We did the first few parts by hand and the last part using Mathematica and the code from Wolfram’s blog post.

The process we studied works as follows:

(1) Pick an integer to start with and pick a number $n$ to multiply by in step (3),

(2) Cycle the digits of the number to the left. A few examples will make the process clear:

123 goes to 231
402 goes to 024, or simply 24
111 would stay 111

(3) Multiply the new number by $n$ and then add 1.

The video below shows how our exploration began. Our initial integer was 12 and we multiplied by 1 at each step (so, starting easy, though I picked 12 at random so I really didn’t know what was going to happen):

Now we moved to a slightly more complicated example -> the same process as in the first part but we’ll be working in binary rather than in base 10.

We started with the number 6 (110 in binary) and multiplied by 2 at each step. Once again we found a fun surprise:

To get one more round of practice in before moving upstairs to the computer we looked at the same situation as in part 2, but this time starting with 1 and looking at several cases – multiplying by 1, by 2, and by 3:

Finally, we went to the computer to explore the process in many different situations. We used code from Wolfram’s blog post to recreate the work from the MoMath talk:

What I *love* about this project is that the exploration works really well with kids on the whiteboard and on the computer. The whiteboard exploration gave us a great opportunity for a little practice with arithmetic, with binary, and with algorithms. We also saw some really fun surprises!

The computer exploration is obviously fantastic, too. I’m so grateful that Stephen Wolfram shared the ideas from his talk!

# Writing 1/5 in binary

I’ve spent the last couple of days talking about binary with my younger son. We were inspired a bit by Kelsey Houston-Edwards’s latest PBS Infinite Series video on binary. It has been a fun little review.

Tonight we talked about how to write 1/5 in binary. I didn’t really know how the conversation would go, but it ended up being a nice little arithmetic review.

We started talking about the problem and he settled on the idea that we needed to find a number that would equal to 1 when we multiplied by 5. That got us going on the arithmetic review since that idea works in any base.

Now we had to figure out now to divide 1.000000000…. by 101 in binary. This long division problem gave us an opportunity to talk about subtraction (and borrowing) in binary:

The last step was multiplying the number we thought was 1/5 by 101. Once again this was a great opportunity to review some basic ideas about arithmetic and multiplication.

So, an unexpectedly fun project! We learned what 1/5 was in binary and had a nice review of subtraction, division, and multiplication along the way ðŸ™‚

# Sharing Kelsey Houston-Edwards’s binary video with kids

Kelsey Houston-Edwards released a new math video last week:

So far we’ve been able to use all of her videos for great weekend projects. This video had a fun little surprise because we’d seen the problem she talks about in a (seemingly) totally different context – an old magic set! Once we dug out that out magic set from under my younger son’s bed we started the project ðŸ™‚

Before jumping to the challenge problems, we looked at the old magic trick:

Finally, we tried to answer the two challenge problems from Houston-Edwards’s video. I’m sorry this got a little rushed at the end – I’d not noticed that we were out of batteries! We finished with about 10 seconds to spare!

The two challenges are great problems for kids to think through – the boys found a few interesting patterns even though the relationship with powers of 2 was a little hard for them to see.

# Po-Shen Loh’s coin problem

Last year I saw this amazing presentation from Po-Shen Loh at the Museum of Math:

In the presentation Loh does something that seems close to impossible. He takes a difficult problem from the International Mathematics Olympiad (problem #5 from 2010) – a problem on which roughly 2/3 of the 2010 IMO participants received 0/7 points! – and turns it into a fantastic public lecture.

Tomorrow I’m going to give a talk at the East Coast Idea Math camp and I’m using problem in Loh’s lecture for one part of my talk. For our Family Math project today I used the problem in the first part of Loh’s talk with my kids.

Here’s how it went.

(1) First I introduced the problem and the boys gave their initial reaction:

(2) Having found a value of 63 cents in the last video next we tried to see if we could find some other values by moving the coins in different ways. We also had a nice discussion about how you could determine if 63 was the maximum value.

(3) Before talking more about why 63 is the maximum value, I wanted to have my younger son try an new approach to the game just to see if we’d ever find a value other than 63.

(4) Finally, we wrapped up by talking about one way you can see that the value of this game is 63. One lucky coincidence is that my younger son was learning about different bases this week. That coincidence helped him see the connection between this game and binary counting fairly quickly.

So, I love Loh’s presentation. It is so cool (and inspiring) to see him take a super challenging IMO problem and turn it into a public lecture. I won’t walk the kids all the way through the solution of the original problem tomorrow, but I will use Loh’s approach to create a big discussion about the 2nd part of the problem. Can’t wait to see what values the kids find for the 2nd game.

# 120-sided dice!!

Our d120s arrived!!

Here’s the unboxing:

We were pretty lucky to have gotten our order in before all of the publicity caused the number of orders to explode. Before the boys ran off to show their friends the new dice, we did a few projects. I asked the boys to think of a question that they thought would be interesting to study with the dice. They didn’t have a lot of time to think about it, but I just wanted their gut reactions anyway.

My older son thought it would be fun to see how far they rolled. We have several different types of dice around the house, plus a few 3d printed shapes, so we saw how far the different shapes rolled:

The “winner” wasn’t actually one of the d120s, but rather a pentagonal hexecontahedron that we’d printed from Laura Taalman’s blog:

Day 194 of Laura Taalman’s Makerhome blog – the Pentagonal Hexecontahedron

The Dice Lab actually makes a d60 in the shape of a deltoidal hexecontahedron, so – no surprise, really – they are way ahead of us!

My younger son wanted to use them to make binary codes. I didn’t quite understand what he meant, but we eventually decided that the odd numbers would represent a 1 and the even numbers would represent a 0. We rolled the dice to create some 5-digit binary numbers. Strangely, we rolled lots of even numbers:

Finally, we did a project that my wife suggested – how many rolls to you think it will take until we see a number that we’ve already rolled? Fun! We had a great time exploring this question.

So, some fun little projects with the dice. Now the boys are off showing their friends and using them for some advantage playing Magic: The Gathering ðŸ™‚

Oh, and just in case you’ve not seen the video about these new dice, here it is:

# Revisiting 1/3 in binary

Last night we talked about writing $pi$ in base 3.Â Â  A long long long time ago we talked about writing 1/3 in binary.Â  Here are those two projects:

Pi in base 3

Writing 1/3 in Binary

I suspected that the boys wouldn’t remember the project about writing 1/3 in binary, so I thought it would make a good follow up to last night’s project.

I started by just posing the question and seeing where things went. They boys had lots of ideas and we eventually got most of the way there:

At the end of the last video they boys figured out that if our number was indeed 1/3, if we multiplied it by 3 we should get 1. That reminded them of the proof that 0.9999…. (repeating forever) = 1.

We reviewed that proof and applied it to the situation we had now.

Just one little problem . . . what if we apply the idea in this proof to a different series, say 1 + 2 + 4 + 8 + 16 + . . . . ?

We’ve looked at the idea in this video before:

Jordan Ellenberg’s “Algebraic Intimidation”

We felt pretty comfortable believing that 0.9999…. = 1 and that we’d found the correct series for 1/3 in binary, but do we believe the results when we apply the exact same ideas to a new series?

I love projects like this one ðŸ™‚