Forming Equations from Integer Sequences

Recently I've been using Twitter to create a daily tweet that records my "day count" (number of days I've been alive) plus its factors (if not prime) and some interesting facts about the number itself or one of its factors. Sometimes there's little to say about the number and in such cases I've found that I can usually form an equation by inserting mathematical operators between one or more of the digits. 

For example, yesterday the count was 23518 and 23-5=18. Today the count is 23519 and 2+3+5-1=9. I was wondering if it's always possible to create an equation from five digits using the standard mathematical operators (addition, subtraction, multiplication, division and exponentiation in combination with brackets). Obviously with just two digits, it's only possible when the digits are repeated e.g. 99 becomes 9=9. With three digits, it's sometimes possible e.g. 819 becomes 8+1=9 but generally it isn't e.g. 219. With four digits, it's more possible e.g. 2119 becomes -2+11=9 but I'm doubtful whether this is always so. There must come a point however, where the number of digits is sufficient to ensure that it's always so. Maybe five digits is that point.

From now on, I'll try each day to form an equation to test out this theory. For example, tomorrow the count is 23520 which becomes 2+3-5=2x0 and it works for tomorrow but beyond that let's see.

Learning English on the Beach

In an earlier post, I wrote about learning English via rhyming monosyllabic words and I used the words that rhyme with bake as an example. Here I'm using beach as another example, hence the title of the post. The most common rhyming words are:

• beech
• breach and breech
• bleach
• each
• leach and leech
• peach
• preach
• reach
• speech
• screech
• teach

There are of course Internet resources that can be used to check whether you've gotten all possible words. For this example, I used Rhymer. It's particularly useful for my purposes because it starts out by explicitly listing all the rhyming words of one syllable (followed by those of two and three syllables). In my reference system, the word that comes first in the alphabet is used to identify the list of rhyming words. This is different to the approach proposed in my initial post where I started with a root sound ake. 

Learning would take place using a suitable software program or combination of Internet resources that would allow the learner to hear the word, see its meaning explained in words and pictures, and be given examples of its contextual usage. For example, in the case of teach, the content might be:

• pronunciation (a site like howjsay.com will provide this service as well as various TTS (Text-to-Speech) apps)
• simple definition (give lessons to students) and graphic

• contextual usage like Mrs Jones will teach us Mathematics this year.

Lastly, why does the word monosyllable have five syllables?

23505 and the Visualisation of Triprimes

Yesterday I was 23505 days old. This number is triprime with factors of 2, 3 and 1567. I'm not sure how widely the term "triprime" is used in the mathematical community but it follows on logically enough from the term "biprime". One way of visualising a triprime number is to associate it with a rectangular prism whose length, breadth and height correspond to the three factors. Suppose we want to create a rectangular prism with a volume of 23505 cubic metres using only square plates with sides of one metre. There is only one way to do this and that is with a prism whose sides measure 2, 3 and 1567 metres (or 1.567 kilometres).

Viewed in one way, this prism has a very narrow cross-section of 2 metres by 3 metres and it is very inefficient in terms of the surface areas required to enclose the volume. The most efficient shape would be a cube with a side of \(23505^{1/3}\) metres. This cube would have a surface area of \(6 \times 23505^{2/3} \) square metres as opposed to our prism with a surface area of$$(2 \times 3 + 3 \times 1567 + 2 \times 1567) \times 2 \text{ m}^2 $$As was done with biprimes earlier, we could then determine the percentage "surface area efficiency" of our prism according to the formula:$$ \text{efficiency}=\frac{\text{surface area of cube}}{ \text{surface area of rectangular prism}} \times 100$$In the case of 23505, the efficiency turns out to be a little over 31.39% by my calculations.

So biprimes can be visualised as unique rectangles (that in some cases can be squares) and triprimes can be visualised as unique rectangular prisms (that in some cases can be cubes). Tetraprimes and beyond of course can have no visualisation in 3-dimensional space.

23501 and the Visualisation of Biprimes

Today I'm 23501 days old. This number is biprime, meaning that it has two prime factors (71 and 331). One way to get a handle on understanding the significance of biprimes is to imagine that you have an area of 23501 to enclose with a rectangular fence made up of modular 1 metre sections. This means of course that both sides of the rectangle must be whole numbers. There is only one way to do this and that is by creating a rectangle with a length of 331 metres and a width of 71 metres.

With other numbers that are neither prime nor biprime (such as 23502), there is more than one way. For example, because 23502 = 2 x 3 x 3917, there are three possible rectangles:

• 2 metres by 11751 metres representing a perimeter of 23.506 kilometres
• 3 metres by 7834 metres representing a perimeter of 15.674 kilometres
• 6 metres by 3917 metres representing a perimeter of 7.846 kilometres

These are admittedly very narrow rectangles but rectangles none the less. Of course, they are very inefficient in terms of the perimeter length required to cover the required area. Without the restraint of whole numbers, the most efficient four sided figure would be a square with a side of a little more than 153.3 metres and perimeter of about 613.2 metres or 0.6132 kilometres.

You could measure, in percentage terms, the efficiency of the perimeter required to enclose the area by using the formula:$$ \text{efficiency}=\frac{ \text{perimeter of square}}{ \text{perimeter of rectangle}} \times 100$$In this case, the efficiencies become:

• 2.6%
• 3.9%
• 7.8%

Getting back to 23501 however, the efficiency of the rectangle turns out to be 76.3%. Of course, with more factors the number of possible rectangles increase and rectangles of greater "perimeter efficiency" can be defined. Of course, biprimes with equal factors will convert to squares and thus have 100% efficiency.

Reflections on 23500

Today I'm 23500 days old and I was looking around to see if there was any significance to this number. It's factors are unremarkable (2^2 x 5^3 x 47) but the fact that it's halfway between 23000 and 24000 means that it pops up quite frequently in Internet searches, as would 22500 or 24500 I would imagine. A search reveals that the approximate population of Boston in 1620 was 23500 and there are several towns around the world that are listed as having this population currently e.g. Bishopbriggs in Scotland.

Bishopbriggs grew from a small rural village on the old road from Glasgow to Kirkintilloch and Stirling during the 19th century, eventually growing to incorporate the adjacent villages of Auchinairn, Cadder, Jellyhill and Mavis Valley. It currently has a population of approximately 23,500 people.

Source: http://en.wikipedia.org/wiki/Bishopbriggs

It turns out that Mount Isa has the same population (source):

Mount Isa is located just 200 kilometres from the Northern Territory border and 1,829 kilometres from Queensland’s capital, Brisbane. The nearest major city, Townsville, can be found 883 kilometres from The Isa. Mount Isa covers an area of over 43,310 square kilometres, making it geographically the second largest city in Australia to Kalgoorlie-Boulder, Western Australia ... With a population of approximately 23,500, Mount Isa is a major service centre for north-west Queensland.

Many other examples of towns having populations of about 23500 could be quoted. In addition of populations, the number sometimes comes up as a dollar figure (source):

In its third annual funding cycle, the Black Philanthropy Initiative has pumped $23,500 back into the Winston-Salem area to help African Americans improve their parenting skills.

Interestingly, it turns out that the centre of the Sun is about 23500 times more distant from us than the centre of the Earth (source).

The sun is far enough away (about 23,500 earth radii) that it took a long time before people knew accurately how far away the sun was. Certainly the ancient Greeks had calculated the distance, but they also knew that their results could be off. 

Many countries in the world have five digit postal codes or zip codes as they are sometimes known. These codes identify particular locations within the country e.g. Muang Prachinburi, Prachinburi, Thailand has a postcode of 23500. The United States uses a five digit system but apparently there is no location corresponding to 23500, although there is for 23499 and 23501. 

Learning English - A Syllable at a Time in Rhyme

Sometime ago, it occurred to me that an effective way to teach English to non-native speakers would be to start with words of one syllable that rhyme. I don't have any research to support the idea but it seems like it would be easy and fun, as opposed to some of the dreadful approaches to learning the language adopted by institutions like English First. Maybe later, when I've retired from formal teaching, I can experiment with this approach in Indonesia.

For the time being, I'll use this blog to refine my ideas. I'm going to begin with the root sound ake that we find in words like bake and cake. It's a good place to start because this sound is fairly well-behaved, meaning that it doesn't change its sound within different words. Other root sounds such as own that we find in words like grown and sown can be confusing because the same three letters can appear in words like crown and town and be pronounced quite differently. 

My approach will be to follow the letters of the alphabet sequentially from a to z, noting as many monosyllabic words as I myself know. I'm deliberately avoiding finding all possible words because I want to stick with words that are in common use. I'll refer to proper names as well because these will be probably be encountered by the language student. Any exceptions to the pronounciation will be noted as well as any alternative spellings.

In the case of ake, the very first letter of the alphabet presents as with an exception but only one of spelling. A type of pain or discomfort can be described as an ache and in fact a common expression is aches and pains. After this however, it's all smooth sailing with bake, blake, brake, break, cake, drake, fake, flake, jake, lake, make, quake, rake, sake, shake, snake, stake, steak, take, wake. These words are all widely used and I won't formally define them here (any dictionary can do that) but instead I'll make some comment or illustrate the use of the word in some common expressions.

1. You can bake a cake.
   
   
2. Blake is a proper name used as a first name for males e.g. Blake Carrington or as a last name e.g. William Blake.
   
   
3. You can brake sharply to avoid a collision when you're driving. The car's brakes are what helps you avoid a collision.
   
   
4. You can break a bone if you don't brake quickly enough in the car you're driving and have an accident. In this case, break and brake have the same pronunciation. In the common word breakfast however, the break part is pronounced differently (like eck). If you have a break from work, it means you have a rest.
   
   
5. You can cake your face with make-up or bake a cake but, to quote a common expression, you can't have your cake and eat it too.
   
   
6. A drake is a male duck but it's also a popular surname e.g. Francis Drake.
   
   
7. You can fake an injury or you can spot a fake banknote if you look at it carefully.
   
   
8. Flake: have as much chance as a snow flake in hell, then you have no chance at all really.
   
   
9. Jake is common first name for males e.g. Jake Gyllenhaal.
   
   
10. Lake Baikal is the most voluminous freshwater lake in the world and is the deepest lake in the world at 1,642.
    
    
11. Make: you can make hay while the sun shines. Dirty Harry said go ahead punk, make my day.
    
    
12. You may quake in fear when the werewolves are near.
    
    
13. You can rake the leaves on your lawn, gather them up and use them as compost.
    
    
14. For the sake of your health, you might avoid getting too stressed out about work.
    
    
15. You might shake a bottle of juice to remove the sediment from the bottom.
    
    
16. A snake in the grass is a person who is not be trusted because he or she could turn treacherous.
    
    
17. Your reputation could be at stake if you are found to be dishonest. Vampires are commonly disposed of by means of a stake through the heart.
    
    
18. A steak is part of the flesh of a dead animal that is cooked and eaten by those who enjoy such food.
    
    
19. Take away food is becoming increasingly popular in our fast-paced society.
    
    
20. A wake is often held when somebody dies. You wake up in the morning when your alarm goes off.