22nd August 2011 - How Infinity Messes Up Stuff

I just love this video. It's caused me much amusement for me in tormenting my friends so I thought maybe I should write about it.

On Minute Physics, Henry Reich 'proves' that

 1+ 2 + 4 + 8 + 16+... = -1.

You can see how in the video so I'll just let you watch it before proceeding.

Now, according to the mathematics you're taught at school, every step is correct yet something seems terribly wrong here. How can you be adding positive numbers and end up with a negative number?

Let's look at what he does. 

1. He multiplies by 1 which doesn't change anything.
2. He rewrites 1 as (2 - 1) which is also fine since 2 - 1 = 1.
3. He applies the distributive law. The distributive law says that (a + b)c = ac + bc and a(b + c) = ab + bc). For example, (3+5)x2 = 3x2 + 5x2. If you check this, you will see that (3 + 5)2 = 8 x 2 =16 and also 3x2 + 5x2 = 6 + 10 = 16. To apply this to the example, he does 2(1+2+4+8+...) - 1(1+2+4+8+...) and then applies it again to multiply each number inside the bracket by the number outside to get 2 + 4 + 8 + 16 + ... - 1 - 2 - 4 - 8 - 16 - ...
4. He cancels all the terms and is left with -1.

So where is the flaw? It's in step 4 because when we deal with infinity strange things happen, and the distributive law doesn't work. The method in itself is a useful mathematical tool, as we'll see below, but unfortunately it doesn't work when we are dealing with infinite series.

Let's look at how it should work. What we will do is use another mathematical trick, and by trick I don't mean something bad or deceitful, but something clever. We will look at a finite series which has N+1 terms, and then we will let N get as big as we want, infinite in fact, and see what happens. This will show us how infinity is confusing us in the video above. We're going to follow the Minute Physics method 

Now, as we're doubling each time, so we have

1st Term: 1 = 2⁰

2nd Term: 2 = 2¹

3rd Term: 4 = 2x2=2²

4th Term: 8 = 2x2x2=2³

5th Term: 16 =2x2x2x2=2⁴

Notice that the power of 2 for each term is one less than its position in the series. This means for the N th term, where N is some number (we don't care what number it is at the moment, just some large positive whole number.)

N th Term: 2x2x2x..x2=2^N-1.

Right, back to the method given in the video, but this time for a finite series.

1 + 2 + 4 + 8 + ...+2^N-1 = 1(1 + 2 + 4 + 8 + ...+2^N-1)

= (2-1)(1 + 2 + 4 + 8 + ...+2^N-1)

= 2(1 + 2 + 4 + 8 + ...+2^N-1)-1(1 + 2 + 4 + 8 + ...+2^N-1)

= 2 + 4 + 8 + ...+ 2^N-1+2^N - 1 - 2 - 4 - 8 - ...- 2^N-1

= 2^N - 1.

Well, that means that 1 + 2 + 4 + 8 + ...+2^N-1= 2^N - 1, which is quite a nice formula but how does it help us with the infinite series in the video?

What we'll do now we've got this nicely written down is make N get bigger and bigger so it tends to infinity. That means the left hand side becomes 1 + 2 + 4+ 8 + ... where the dots mean carry on forever. What happens to the right hand side? Since N tends to infinite, then 2^N tends to infinity even more quickly, so the right hand side tends to infinity. This means that the sum 1+2+ 4+ 8 + ... tends to infinity too, and most definitely not -1.

19th August 2011 Fact or Fiction? The Importance of Looking More Closely

Today I came across the following article being shared on Google+. The headline is "Science Reveals Women Who Wear Less Clothing Live Longer". The article claims the following.

A prominent British anthropologist has completed a ground-breaking, 10-year study proving that women who wear less clothing live up to 20 years longer. And the fewer clothes they wear, the longer they live. Writing in the Royal Journal of Social Anthropology, Sir Edwin Burkhart presents the results of over 5,000 interviews with women ages 70 to 120. It's clear from the interviews and graphs shown that those women at the older end of the spectrum consistently wore either fewer clothes throughout their lifetime -- or no clothing at all!

It goes on to further discuss the topic, giving examples and possible explanations.

Copies of this article have spread all over the place. Do people believe the content? Should they? Is it a case of confirmation biased in that people want it to be true?

There are no links to the source, nor is it referenced properly. Sadly, this is often the case in journalism and in itself doesn't mean the information is incorrect. In this case, however, I can find no evidence for the existence of the Royal Journal of Social Anthropology, nor for the 'prominent' anthropologist Sir Edwin Burkhart.

The design of the study seems strange. A 10 year study which doesn't appear to have any follow-up? Why would it take 10 years and how could it be extrapolated to mean 20 extra years' life? Where is the proof of causation and not just correlation (not that any evidence for that is presented here)?

Let's look at some of the information given: "... 5,000 interviews with women ages 70 to 120." Where did the researcher find this 120 year old? According to the Guinness Book of Records,

[t]he oldest person living is Maria Gomes Valentim (Brazil), of Carangola, Brazil, who was 114 years 313 days old as of 18 May 2011. [1]

The quotations from the supposed research subjects are somewhat ridiculous.

Philomena Bushfield, 120, is a life-long nudist who hasn't "worn more than a pair of socks since I was 7." She is rarely sick, still runs in marathons, and has the energy and enthusiasm of a woman half her age. "It also made it much easier to meet men," says Bushfield, who currently has three different boyfriends. 

Could someone who was born in 1891 really have lived her life from 1898 without clothes? Wouldn't we all know about her is she had, especially if she were running marathons while wearing nothing more than a pair of socks!

Now, apart from looking at the content, and there is more that could be said about it,  if we look at the author of the article it is Mark C. Miller, humorist. Surely that is sufficient evidence to take this all with a pinch of salt?

Much as I can see that this is meant as a humorist piece, I find it worrying that people seem to believe it, or to take it at face value. The subject concerns me in that I wonder if it could be used as a weapon against women and their right to dress as they wish. Maybe that is taking it too seriously, but seeing how subtle pressure can be applied to people 'for their own good', I don't really like to see something that could potentially be used like this.

Although this doesn't seem to have been picked up by any newspapers, a recent incident with an article by the BBC on the IQ of Internet Explorer users highlights the danger of not checking our facts and believing everything we read without a critical thought.

17th August 2011 Overcoming a Problem by Making it a Game

Today I read about the actions of a father whose daughter was struggling to cope with the breathing exercises she had to do to clear her lungs because she has cystic fibrosis. Instead of just battling on with the problem and forcing her to go through the necessary steps every night he turned it into fun. He made use of his contacts to develop a game which is controlled by the puffs on the breathing machine.

What I like about this story is that someone looked beyond the problem and traditional way of doing things to find a new and innovative solution. It's a story of one person making a difference with an idea and then pushing it through to development.

If something as onerous and painful can be made fun by making it into a game, what other things can be made more fun in some way? We might not all be able to come up with something that will have an effect on many other people, but how can we use our own creativity to make life easier and more enjoyable for ourselves and those around us?

15 August 2011 - 1 Multiplying Numbers When the First Digits are the Same & the Last Sum to Ten

Continuing the theme of multiplying various numbers, we come to some which satisfy a particular condition as explained in the examples below.

Example 1

We'll multiply 26 by 24. The method we'll use requires the first digits to be the same, here 2, and the last digit from each number sums to ten, here 6 + 4 = 10.

1. Take the first digit	2
2. Add one	2 + 1 = 3
3. Multiply the numbers from step 1 and 2.	2 x 3 = 6
4. Multiply the last 2 digits of the original numbers	6 x 4 = 24
5. Write the answer to step 4 after step 3.	624

Hence 26 x 24 = 624. The method also works for larger numbers.

Example 2

We'll multiply 113 by 117. We'll check the numbers satisfy the conditions for the method. The first digits are the same, both 11, and the last two digits sum to ten: 3 + 7 = 10.

1. Take the first digit	11
2. Add one	11 + 1 = 12
3. Multiply the numbers from step 1 and 2.	11 x 12 = 132
4. Multiply the last 2 digits of the original numbers	3x7=21
5. Write the answer to step 4 after step 3.	13221

Hence 113 x 117 = 13221.

Proof

Again, let's check that the examples above weren't just flukes. Let's multiply two numbers a and b which both start with c which could be a single digit or several digits, and end in d and (10 - d) respectively. Hence

a = 10c + d and b = 10c + (10-d).

Multiplying a and b, we get

ab = (10c + d)(10c + (10-d)) 
= 100c²+ 100c -10cd + 10cd + 10d - d²

= 100c²+ 100c + 10d - d².

We'll now check to see if the method gives the same result.

1. Take the first digit	c
2. Add one	c + 1
3. Multiply the numbers from step 1 and 2.	c(c + 1) = c2 + c
4. Multiply the last 2 digits of the original numbers	d(d-10) = d2 - 10d
5. Write the answer to step 4 after step 3. This is the equivalent to multiplying the answer to step 3 by 10 and then adding the answer to step 4.	10(c2 + c)+d2 - 10d = 10c2 + 10c + d2 - 10d

Hence the method gives 10c² + 10c + d² - 10d which is ab so it works for numbers of the prescribed form.

Source: http://www.vedicmaths.org/Introduction/Tutorial/Tutorial.asp

14 August 2011 - 1 Sea Pen Video

A couple of days ago, I wrote about sea pens. Today, youtube suggested the following video which shows sea pens contracting to bury themselves under the sand over a four hour period. It's a clip taken from the BBC's Ocean series, so I hope it's visible outside the UK.

13th August 2011 -1 Squaring Two Digit Numbers Ending in 5

I saw this method in a GCSE foundation paper, as well as here. Squaring a number is simply multiplying it by itself, so 3 squared, written as 3² means 3 x 3 = 9. That is 3²=9.

Method

This method works for numbers ending in 5. As an example, we'll square 35.

Example	Method
3	1) Write down the tens figure.
3 + 1 = 4	2) Add 1 to it.
3 x 4 = 12	3) Multiply the numbers from steps 1 and 2.
1225	4) Write 25 after the number obtained in step 3. This is the original number squared.

Does this method really work?

Proof

Let the number to be squared be a where a is a number ending in 5. Let the tens digit be b. Then we can write

a = 10b + 5. 

Squaring a gives

a² = (10b + 5)²

= 100b² + 100b + 25. (1)

Now we'll apply the method to a and see if we get the same result. First we write down the tens figure b and then add one to it, giving b + 1. These are then multiplied to give

b(b + 1) = b² + b.

Since, we write 25 immediately after this number, we are actually multiplying by 100, which shifts the 12 in the example above to 1200,  and then adding 25.

100(b² + b) + 25 = 100b² + 100b + 25,

which is what we obtained when we multiplied a by itself in (1). Hence the method gives us a², that is, it works.

Note that this proof can be adapted for any number ending in 5 as we never use that b is a single digit. For instance 115² is found as follows:

11 x 12 = 132     (since 12 = 11+1)
so 115² = 13225 (putting 25 after the number above).

12th August 2011 - 1 Can Leaf Veins Be Used To Predict Climate?

Benjamin Blonder, an ecology graduate student, working with his advisor Brian Enquist at the University of Arizona in Tucson is examining the connection between the veins in leaves and climate. Preliminary results from his models predict temperature and precipitation. I'll be very interested to see if these preliminary results hold up when tested more widely and whether they can be extrapolated to accurately determine climate from fossil records, as he suggests.

His website has some useful resources including details on making leaf skeleton, and his blog has fascinating articles about leaves from the cost of producing veins and their function to colour change to how plants survive with only the occasional flash of sunlight through the dense canopy above.

Source

1. http://blogs.nature.com/news/2011/08/using_fossil_leaf_veins_to_rec.html

11th August 2011 - 2 Jigsaw Pieces Everywhere

I read a lot of articles and blogs every day, along with my Google+ stream, books and conversations with friends. I'm very hungry-for-knowledge but then some days it feels like there is too much, and that nothing fits together. It's as if each piece of knowledge is a jigsaw piece and they are strewn all over the place.

How can I ever make sense of all these pieces? Nothing ties up. Such small snippets but surely they go together somehow? I jump from one thing to another, wondering what is going on and then suddenly I get a feeling that something is familiar. Just a moment, didn't I hear something about that a few days ago? Then it dawns on me. I did! I love that AHA moment when things link up.

I had that feeling today when I was reading up on sea pens. A predator of the sea pen is the nudibranch. Nudibranch, nudibranch, oooh, wasn't that something someone posted a picture of on Google+?  I checked back and sure enough, +Kjetil Greger Pedersen had posted a photograph of one which eats anemones and uses their poison to protect itself. Great feeling.

11th August 2011 - 1 First Pencils, Now Pens - Pen Urchins

Back in April, I wrote about the beautiful pencil urchins. Today, I came across this lovely collection of photographs of sea pens, so I thought I'd carry on with the stationary theme.

Name

The sea pens are types of soft coral of the order Pennatulacea, sub order Subselliflorae. The name 'sea pen' arises from their feather-like appearance which resembles a quill pen.

Photo by fiveinchpixie

The visible part of the sea pen can be up to 2 m in some species, such as the tall sea pen which can be found off the west coast of Scotland.

Life Cycle

A sea pen is a colonial animal, which means it is made up of individual polyps which together function as a whole animal. It begins life as a larva which roots itself and then develops into a stalk. This stalk is called a rachis. It has a root-like structure at its base which anchors it to the sea floor. Both this and this stalk are strengthened by calcium carbonate. The feather-like protrusions are made up of two new types of polyps formed through asexual reproduction. These polyps are responsible for feeding (gastrozoids) and respiration (siphonozoids). The gastrozoids capture food while the siphonozoids move water around to allow for gas exchange.

Some species reproduce by releasing eggs and sperm into the water, whereas in others the female retains the eggs and fertilisation is internal. The developing embryos are brooded until they have reached an advanced larval stage when they are released into the water, where they root to form new pens.

Habitat

Sea pens prefer deeper water, from 10 m to 2 km deep, where they are less likely to be uprooted. They tend to stay in one place but can re-establish themselves if necessary. They position themselves so currents ensure a steady flow of plankton, their main source of food.

Threats and Defences

Unfortunately, sea pens are often destroyed by prawn trawlers and dredgers as well as being prey for sea stars and nudibranchs. In defence, a sea pen may luminesce or deflate and retreat completely underground.

Sources

1. http://en.wikipedia.org/wiki/Sea_pen
2. http://www.bbc.co.uk/nature/blueplanet/factfiles/jellies/seapen_bg.shtml
3. http://www.naturalengland.org.uk/ourwork/marine/protectandmanage/mpa/mcz/features/habitats/seapen.aspx
4. http://www.marlin.ac.uk/speciesfullreview.php?speciesID=3353#
5. http://www.advancedaquarist.com/2003/10/inverts
6. http://www.ehow.com/about_6418563_sea-pens.html

10th August 2011 - 1 Multiplying Numbers just over 100

Today, we're going to look at extending the methods discussed earlier this week to numbers just above 100. Again, we'll look first at the method, then its feasibility and finally a proof.

Method

We'll multiply 105 by 107. How does the method differ from earlier methods? Instead of subtracting the numbers from 100, we subtract 100 from then, and instead of subtracting diagonally, we add. (See table below.)

Numbers to Multiply	Left hand column minus 100
105	5
107	7
ADD the number diagonally opposite. It doesn't matter which pair we use as we get the same answer. The pairs are shown in red and blue.	Multiply the numbers in this column
105 + 7 or 107  + 5 = 112	5 x 7 = 35
Multiply this by 100
112 x 100 = 11200	35
Finally, we add the two numbers.
11200 + 35 = 11235

When is it useful?

If you can multiply the two digits after 100 by each other, then this method works.

Example 1
For instance, 102 x 145 since 2 x 45 = 90 is easy if you're comfortable with doubling.  Once you've multiplied those, all that is left to do is add 102 and 45 which is 147, put a couple of zeroes on the end and add the 90 you got earlier. That is,

147 x 100 = 14700,

14700 + 90 = 14790,

and so

102 x 145 = 14790.

Example 2
Similarly, 120 x 145 is as follows. Do 20 x 45, which is just 2 x 45 = 90 with a zero on the end, that is

20 x 45 = 900.

Then

120 + 45 = 165,

165 x 100 = 16500,

16500 + 900 = 17400,

and so

120 x 145 = 17400.

Proof

The proof is similar to those I've given for the last three posts on this topic.

 We will multiply a and b, which are numbers greater than 100. Next we subtract 100 from each of them to give a - 100 and b - 100, and then multiply them together.

 (a - 100)(b - 100) = ab - 100a - 100b + 10000. (1) 

 Moving on to the other part of the calculation, we add a - 100 to b to get a + b - 100. Next we multiply by 100 which gives

100(a + b - 100) = 100a + 100b - 10000. (2) 

 Adding (1) and (2), we obtain

ab - 100a - 100b + 10000 + 100a + 100b - 10000 = ab, 

again showing that our method does indeed multiply a and b together.

 Source: http://www.vedicmaths.org/Introduction/Tutorial/Tutorial.asp

9th August 2011 - 1 Moeraki Boulders in New Zealand

Inspiration

A photograph that caught my eye this morning on Google+ was a picture by . My initial reaction was "Whoa, that's amazing!" It's a lovely image in soft pastels of something I've never seen before. Fortunately Mr Sojka gives some background to his shots and explains that this is a broken Moeraki boulder on the Otago coast of New Zealand. I don't know the scale of the boulder from the shot, but it looks quite large to me. Quite how such large boulders could be just sitting there on the beach puzzled me so to google I went.

Description

The picture below shows unbroken boulders. Moeraki boulders are roughly spherical and up to 3m in diameter. They are covered in cracks called septaria.

Moeraki Boulders at Sunrise by Karsten Sperling via  Wikipedia Commons

Formation

Not to scale!

The Moeraki boulders started to form  in mud on the sea floor 60 million years ago. They are thought to have formed from the inside outward by the depositing of layers of mud, fine silt and clay which was them cemented together with calcite. Over 4.5-5 million years, they continued to grow while marine mud was deposited over them.

The cracks which taper off from the centre to the outer layer are taken as evidence that the surface was more rigid than the softer inside and so could not shrink as much. This is attributed to the differing amounts of calcite in the different layers.

There are several suggestions as to why the cracks formed:

• dehydration of clay rich, gel rich or organic rich cores,
• shrinkage of the boulder's cenre,
• expansion of gases produced by decaying organic matter,
• brittle fracturing or shrinkage of the interior by earthquake or compaction.

These cracks fill with brown or yellow calcite and, more rarely, quartz and ferrous dolomite when a drop in sea level allowed fresh groundwater to flow through the mud surrounding them.

They are exposed when the softer surrounding mudstone is eroded by the elements.

Sources

1. AA Travel -Moeraki Boulders
2. The Dinosaur Egg Boulders of Moeraki - Photos
3. Wikipedia: Concretions
4. Moeraki Boulders
5. Trifter: Moeraki Boulders, South Island, New Zealand
6. Wikipedia:Moeraki Boulders
7. Encyclopedia of New Zealand - Moreraki Boulders

8th August 2011 - 1 Multiplying Numbers Just Below 100

The underlying method I talked about two days ago can be extended so you can multiply numbers just below 100. What do we need to do to adjust it? Wherever we used 10, we replace it by 100. Let's try 96 x 91.

Method

Numbers to Multiply	100 minus left hand column
96	4
91	9
Subtract the number diagonally opposite. It doesn't matter which pair we use as we get the same answer. The pairs are shown in red and blue.	Multiply the numbers in this column
96 - 9 or 91 - 4 = 87	4 x 9 = 36
Multiply this by 100
87 x 100 = 8700	36
Finally, we add the two numbers.
8700 + 36 = 8736

When is it Useful?

So let's consider when this method might be useful. Personally, I think the method for small numbers is interesting but too time consuming compared with the benefit of learning tables up to at least 10 x 10. When the maths gets more advanced, to have to resort to fingers to do 8 x 7 is slow compared with recalling from memory. Small numbers are multiplied fairly often so the investment of time is worth it, even if you struggle to learn things by rote, like I do.

 If you know your tables up to 10 x 10, then any two numbers in the 90s will be easy to multiply together because the differences will be at most 10.

Multiplying 98 by any number is going to be fairly straightforwards if doubling is something you can do easily. For example 98 x 46.

Numbers to Multiply	100 minus left hand column
98	2
46	54
Subtract the number diagonally opposite. It doesn't matter which pair we use as we get the same answer. The pairs are shown in red and blue.	Multiply the numbers in this column
98-54  or I prefer 46 - 2 = 44	2 x 54 = 108
Multiply this by 100
44 x 100 = 4400	108
Finally, we add the two numbers.
4400 + 108 = 4508

If you're happy to double and double again, you'll find you can multiply by 96 as that is 4 less than 100.  Similarly, 80 and 60 aren't too bad since then you're just multiplying by 20 (double and put a 0 on the end) or 40 (double, double again and put a 0 on the end).

And just to make sure, let's check the proof still works out ok.

Proof

Let's apply this method to a and b which are two whole numbers less than 100.

We subtract them from 100 to get 100-a and 100-b. We multiply these (right hand column in the table) to get

 (100 - a)(100 - b) = 10000 - 100a - 100b + ab. (1)

Next we subtract to get a - (100 - b) = a + b - 100 and then multiply  by 100,

100(a + b - 100) = 100a + 100b -10000. (2)

Finally we add (1) and (2) to get


10000 - 100a - 100b + ab+100a + 100b -10000 = ab.

This means that applying the method does indeed multiply a and b as claimed.

Source: http://www.vedicmaths.org/Introduction/Tutorial/Tutorial.asp

Edit: Corrected two typos where I put 10 instead of 100.

7th August 2011 - 1 Multiplying Single Digits Part 2 - Using Fingers

The Method

Using the principle covered in yesterday's post, here are two videos showing how to use your fingers to apply it. The first touches fingers together

and the second folds some down.

I think the videos explains the methods clearly, but I do have one minor issue with the second and that is the comment "To show it's no fluke, let's try another multiplication." Unfortunately, two examples do not show it is not a fluke. Checking all possibilities would, or a proof such as the one I gave yesterday. The proof needs a slight change from yesterday's to apply to this method.

The Proof

Let's look at what is happening with the numbers first, and then generalise to show how the method works for any two numbers a and b, between 5 and 10 inclusive.

The circled fingers in the first picture correspond to the fingers in both circles in the second picture.

Circled Fingers

On my left hand I have 3 fingers folded down because 8 - 5 = 3. On the right hand it's 7 - 5 = 2. These are added, 3 + 2 = 5. This corresponds to the number of fingers circled in the first picture. This number is multiplied by 10 to give 50. In general, on the left hand there are a - 5 fingers circled, and on the right b - 5. Summing these gives

(a - 5) + (b - 5) = a - 5 + b - 5 = a + b - 10.

This is multiplied by 10 to give

10(a + b - 10) = 10a + 10b - 100,

which is the same as we had yesterday. We need to work out the value obtained from the non-circled fingers and then add that to this value.

Non-circled Fingers

The number of non-circled fingers is 10 - 8 = 2 on the left hand and 10 - 7 = 3 on the right hand. We multiply these two numbers together to give 6 and add it to the answer from the circled fingers, calculated above.

On the left hand we have 10 - a non-circled fingers, and on the right 10 - b. Multiplying these together gives

(10 - a)(10 - b) = 100 -10a - 10b + ab.

Finally, adding this to the circled fingers answer, and simplifying, gives

10a + 10b - 100 + 100 -10a - 10b + ab = ab,

showing that both methods do correctly multiply a and b together.

6th August 2011 - 1 Multiplying Single Digits Part 1

There are many methods for multiplying numbers together.

The first method I'm going to look at is for single digits which are both 5 or above. If one of the digits is below 5, you gain nothing over just multiplying them together, or at best, very little. If both digits are 5 or above, all you need to know is how to subtract and times tables up to 5 x 5.

The Method

As an example, we'll multiply 8 and 6.

First we subtract both numbers from 10 to get 2 and 4, and write the answer in the corresponding right hand column below.

Numbers to Multiply	10 minus left hand column
8	2
6	4
Subtract the number diagonally opposite. It doesn't matter which pair we use as we get the same answer. The pairs are shown in red and blue.	Multiply the numbers in this column
8-4 or 6-2 = 4	2 x 4 = 8
Multiply this by 10
4 x 10 = 40	8
Finally, we add the two numbers.
40 + 8 = 48

So when we multiply 8 and 6 we get 48.

Sometimes when you multiply the two new numbers together, we get a number which is two digits, as shown in the example below where we multiply 6 by 6. This doesn't make any difference to the method.

Numbers to Multiply	10 minus left hand column
6	4
6	4
6 - 4 or 6 - 4 = 2	4 x 4 = 16
2 x 10 = 20	16
20 + 16 = 36

So 6 x 6 = 36.

How does it work?

With some algebra, we can see that the method above really does give the right answer.

Let the two numbers to be multiplied be a and b. 

Numbers to Multiply	10 minus left hand column
a	10-a
b	10-b
a - (10 - b) = a -10 + b = a + b -10	(10 - a)(10 - b)
Multiply by 10.	Multiply out the brackets.
10(a + b -10)=10a + 10b -100	(10 - a)(10 - b) = 100 - 10a - 10b + ab
We now add the two expression above.
10a + 10b -100 + 100 - 10a - 10b + ab =  (-100 + 100) + (10a - 10a) + (10b - 10b) +ab = ab
As we get ab, this means the method does indeed give us the correct answer when we multiply a and b.

Source: http://www.vedicmaths.org/introduction/tutorial/tutorial.asp#tutorial2

5th August 2011 - 3 Correlation but is it causation?

Ben Goldacre posted an interesting illustration of how data can mislead and needs to be analysed before posting. His example shows a correlation between lung cancer and drinking, that is, if you drink you are more likely to get lung cancer than if you don't drink. Does that really mean that drinking causes lung cancer or is there another factor in play?

When the figures are split into smokers and non-smokers, the smoker who drink and the smokers who don't drink have the same occurrence of lung cancer. Similarly, for non-smokers, drinker and non-drinkers have the same occurrence of lung cancer. This shows that the occurrence of lung cancer is explained by smoking versus not smoking rather than drinking versus not drinking.

5th August 2011 - 2 Bad Maths

I noticed as my twitter feed flew past today the following link which amused me in a strange sort of way. It's an image of a poster claiming to solve the world financial crisis along with some conspiracy claims. In the middle of the poster they make the following claim.

*An estimated 51% of retail prices are a result of interest on the National debt. This means that as of right now, everything in stores is 51% more than it would be if we used a national currency like the United States Note.

What amused me was that someone claiming to solve the monetary crisis doesn't understand basic percentages.

Suppose an item costs $10. From the first sentence, we have that 51% of this price is interest on the National Debt. That would be $10 x 0.51 = $5.10, so without this interest on the item, it would cost $10 - $5.10 = $4.90.

The claim in the second sentence claims that everything in stores is 51% than it would be without the National Debt, as the poster claims that the United States Note will solve this. This 51% is now referring to the 51% of $4.90 (the price without debt) which is $4.90 * 0.51 which is about $2.50. If the second sentence were true, the item would cost $4.90+2.50=$7.40 which it doesn't. So how much more is being paid as the 51% is clearly wrong and not enough.

As we want the percentage to be compared to the price without the debt, we write the extra paid, $5.10, as a fraction of the price without debt and multiply by 100 to turn it into a percentage.

5.10/4.90 * 100 = 104% (to nearest percent)

What this tells us is that instead of paying about half extra as the poster claims, people are paying more than double what the price would be without the debt payments!

This illustrates how important it is to know what any percentage is of so that mistakes like this do not occur.

5th August 2011 - 1 Why I Like Photographs

What I've learnt today has been about myself and how I react to things. On Google+ today, Jessamy Goddard asked what everyone loves about looking at photographs. It made me think and this is what I wrote in response.

For me, it depends on the type of photo and who has taken it. Photos fall into different categories. Some are about what the picture is of, and some are the whole picture, and some are both.

Some photos allow me to see what friends have been doing, how they've been enjoying themselves and to get to know them a little better even though I'm far away. To see a friend who has been through a bad patch with a natural smile on his face in a photo has brought tears to my eyes (and even thinking about it now, I can see the picture in my mind and it brings a smile to my face and another tear to my eye). Then it's the content of the picture that matters and if it's blurry or badly composed, it doesn't distract from what I see.

Other pictures bring the reality of a terrible situation home to me. I don't like looking at them, but I need to to understand. The photos connect me to the event in a way words can not.

I'm not good at remembering names but I pictures give me a sort of anchor to link information to. I will probably forget the name of the little creature which +Kjetil Greger Pedersen took a picture of, a nudibranch, but I will remember the picture and the information he so kindly provided to go with it.
Some pictures fill me with awe at the beauty around us whether it's a landscape, flower or tiny creature. I've always thought of cities as horrible, dirty places and but then I see the photos by +Trey Ratcliff and realise I've not been looking at them in the right way.

Other pictures teach me to look for beauty in the mundane such as the light on a curtain, the pattern on a floor or just a ball of wool such as those +Jordana Wright takes. They teach me how to look at details instead of the overall scene.

I love to see pictures of people who are real so I can wonder about who they are and what their story is. Faces with characters like in the street photography of +Thomas Leuthard.

I love the underwater shots that +Elena Kalis takes which transport me into a dream world. I've never seen anything like her pictures before.

From a creative point of view, they teach me about composition and colour which I can use to improve my own photos and craftwork. The pictures +Robin Mead has posted of her craft work have given me some ideas of things I'd like to try.

In summary, for me pictures inspire, allow me to escape, relate, connect. They teach me, aid my memory, change my perspective and share moments.

I could have gone on a lot longer mentioning many more talented photographers but it was long enough already. I feel so lucky to be able to immerse myself in the beauty that so many people are so freely sharing on Google+.

4th August 2011 - 3 The Horizon

Assuming the Earth is a sphere leads to simple calculation, based on Pythagoras' Theorem, of the distance to the horizon given the height, h,  of the viewer's eye. The details are nicely explained by Phil Plait in his blog Bad Astronomy.

The distance, d (in km),  to the horizon is

d = square root of  h(h+2R)

where h is the height of the eye (in km) and R = 6365 km, an approximation of the radius of the Earth.

For instance, if you eye is at ground level, the horizon is 0 metres away, but if your eye is at 1.65 m (about 5'5"),  then the horizon is at 4.6 km (2.9 miles). At the top of Mount Everest, which is 8,848 m high, it would be 335.75 km (208 miles).

4th August 2011 - 2 Earth's Two Moons

A new proposition suggests that when the moon formed from the impact of an asteroid with the earth 4.5 billion years ago, that it wasn't just one moon formed, but two.

Why would anyone think that? There is a difference between the near and far sides of the moon. The two sides have different terrains, a different crust thickness and different amounts of so called KREEP (potassium (K), rare earth elements (REE), phosporous (P)) elements.

For details see Nature: Early Earth may have had two moons - Richard Lovett

4th August 2011 - 1 Light lost in Reflection

In a conversation with a friend, the loss of light between two mirrors came up. That got me thinking as to why the light was lost. Was it some sort of cool quantum effect? Maybe I've been reading too much quantum stuff that I want something weird and wonderful to be happening. I went on a search for information to see if any of my ideas were in the right direction and this is what I've found so far.

In a metal, and thus mirrors with a metallic backing, light is absorbed by electrons and then re-emitted. I don't know why it's emitted at the angle it is but I think it's to do with waves but that is a question for another day. The following are reasons why the intensity of the beam diminishes.

• Some light is scattered by imperfections in the mirror, so called diffuse reflection
• Some light is scattered by the air
• Even if the mirror were perfect, some energy would be lost due to conduction between electrons

I can see I have a lot more to read up about on this topic so I'll look at it a bit more over the next few days and see if I can make any sense of it.

Sources:

Physics Forums - Loss in Mirror Reflection

Physics Forum - Light in a Mirrored Sphere

3rd August 2011 - 1 Drawing Bunnies

Today it's been rather hot and so I've not felt much like concentrating. A friend linked me to something on the Guardian site and I spotted this when I was there. It's about drawing bunnies in a very simple way. I'm thinking about whether I can use them in my craft class in the autumn.

This is my attempt.

2nd August 2011 - 4 Echinacea

Recent research shows no link between the self-reported length of a cold comparing those who took echinacea to those taking a placebo.

 Source: Science Based Medicine: Belief in Echinacea - Harriet Hall

2nd August 2011 - 3 Shield Bug

Yesterday, an insect was flying around the room. It settled on my desk. Its body was about the size of my thumbnail. I didn't know what it was. This is it.

Today, on twitter, Bug Girl linked to an article in the LA Times (1) about a severe insect problem they have there. Stink bugs, as they call them, are gathering in their millions and destroying crops, filling houses and generally making a nuisance of themselves. Squashing them isn't a good option as they, apparently, release an unpleasant smell.

I googled for stink bug and found that in the UK they are called shield bugs. This particular one is Pentatoma rufipes, the forest bug or red-legged shield bug.

Habitat & Distribution

At least throughout the UK and Scandinavia in woods, orchards and gardens.

Life Cycle

August: Eggs are laid.
Winter: Larval form.
July-November: Adult.

Note: Sometimes they are found in early spring suggesting a secondary breeding cycle is possible.

Food

The nymphs feed on tree sap, mainly of oak, but also of other deciduous trees such as alder, hazel, apple and cherry.

The adults also eat caterpillars and other insects as well as sucking the juice from fruit leaving the skin damaged and corky brown regions inside.

Sources

1. LA times: Stink bug invasion puts entomologist in demand - Richard Simon
2. British Bugs: Pentatoma rufipes, forest bug
3. UK Safari: Forest Bugs / Red-legged Shieldbugs
4. Nature Spot: Forest Bug - Pentatoma rufipes

2nd August 2011 - 2 Cigarettes & Vitamin Tablets

Source:  EurekAlert!: Licence to smoke: Taking vitamin pills may undermine motivation to reduce smoking - Jean O'Reilly

In two dummy health food tests, participants, who were all daily smokers, were given a placebo. Some were told it was a vitamin tablet, and some knew it was just a placebo. The smokers then took an unrelated one hour survey where they were allowed to smoke. Those who thought they had taken a vitamin tablet smoked almost twice as much as the other group. The more positive the smokers were to the health giving properties of vitamins, the more they smoked. It may be that the perceived health giving properties leads to a false sense of invulnerability to the diseases associated with smoking.

Some other examples of people using this virtuous choice followed by a poor choice later is, the so-called 'licensing effect', are

• to binge drink at the weekend having not drunk all week.
• to do one 'green' thing and then feel we've done our bit so we can ignore the consequences of our next action.

Another side effect of this 'virtuous' behaviour can be selfishness later on. In a study people could choose to order from an eco-friendly site or elsewhere. They then played a game where they had to split an amount of money with others. Those who had ordered from the eco-friendly site shared less of their money.

Source:  Scientific America: Green and Mean: Eco-shopping Has a Side Effect - Emily Anthes

2nd August 2011 - 1 Placenta absorbed to protect brain in mice foetuses

Source: Naturenews: Placenta to the rescue - Zoë Corbyn

When a pregnant mouse is starved, the placenta breaks itself down to provide a steady supply of food to the foetus. The stage of development studied was crucial for hypothalamus development, and this uninterrupted supply of food protects it from damage. This part of the brain controls primal instincts, including maternal instincts. 

Is this result paralleled in humans? It is unknown and it would be unethical to carry out similar research on people but it is possible that some information may be obtained by studying the placenta after birth.

2nd August 2011 A Change

I've decided that having this public means that I don't want to write because it puts too much responsibility on me to not spread incorrect information through my misunderstanding or poorly chosen words. There are so many amazing bloggers out there, who are specialists in their fields that I can not hope to produce anything that is useful from someone else's point of view.

I found my approach changed completely when it became public. I felt I could not longer use it to jot down notes about different topics I was reading about, but needed to produce a structured document. Instead of being something for me, it became something for other people, and those people quite likely don't exist.

Things are going to change. I am going to use this to note down three things that I've learnt each day, or at least I'm going to try to.