2 + 4 x 3 = ?

There are innumerable Facebook posts posing the above question (or similar). It usually results in more or less acrimonious discussions. There are two main camps: those who believe that the answer is fourteen (14), and those who believe that the answer is eighteen (16).

So, which camp is correct? Well, the answer is that the question is not complete. It doesn’t specify the way that the answer should be evaluated. If you do it one way, you get one answer, and if you do it another way, you get a different answer.

How can this be? The answer lies in how we break down the question in order to evaluate it. A mathematician or a computer programmer would perhaps ask how we intend to ‘parse’ the question.

When we learn arithmetic, we learn how to add and subtract, and how multiply and divide. For instance, we learn that:

7 + 6 = 13
4 x 8 = 32

Simple! A crucial point here is that it doesn’t appear to matter which way round the numbers appear.

6 + 7 = 13
4 x 8 = 32

If we want to add three numbers together or multiple three numbers, it doesn’t matter what order we perform the operation:

6 + 7 + 12 = 12 + 6 + 7 = 25
4 x 8 x 3 = 3 x 8 x 4 = 96

It also doesn’t matter if we work from left to right, or from right to left. If we start from the right we have the number 12, we add the number 7, giving the number 19, then finally we add the number 6, giving the final result. So far, so simple.

But if we had an arithmetic problem which involves subtraction or division, then things start to become complex. The order of symbols used and the direction in which the problem is processed does matter.

8 - 3 = 5          (Left to right)
8 - 3 = -5         (Right to left)
2 / 4 = 0.5        (Left to right)
2 / 4 = 2          (Right to left)

OK, a convention is called for. If I pose you an arithmetic question, I don’t want to have to tell you how I want it to be processed. So the convention, at least in languages which are written from left to right, is that arithmetic problems are also processed in the direction that the language is written. So from now on, I will assume that any arithmetic problem is processed from left to right.

I’d like to add that, though I’ve chosen the convention that the expression is processed from left to right, the issue can be resolved in other ways. For instance, I could suggest a convention that a bare number is always associated with the operator that precedes it. In other words:

8 - 3 = 8 + (-3) = 5 (LTR or RTL)

But this could cause other issues for more complex expressions.

If we mix addition and subtraction with multiplication and division, we get something like our original problem:

2 + 4 x 3 = ?

We can, some people may be surprised to learn, get more than one answer to this problem, depending on how we process the expression.

2 + 4 x 3 = 6 x 3 = 18 (proceeding strictly from left to right)
2 + 4 x 3 = 2 + 12 = 14 (using the BODMAS convention, see below)

Many people would argue that the first answer is correct. Why jump to the multiplication first in the second example? The answer is that it is simply a convention among mathematicians and computer scientists and programmers use. It’s the answer that you would get if you put those numbers and symbols into most calculators.

Some calculators (eg Microsoft’s Windows Calculator) can give either answer depending on what mode the calculator is set up to use. There’s a simple explanation in the linked article on why that is. Maybe too simple.

The convention that mathematicians and computer scientists use is not a law of arithmetic or mathematics, as some people believe. So, why is a convention necessary? The real answer is so that you can pass a random piece of mathematics to someone else and they will understand how to process it unambiguously, if there is a commonly used convention for processing such expressions.

In particular, in algebra and computer science, using the common BODMAS conventions actually reduces the complexity of the strings of symbols necessary to express a mathematical idea. Einstein’s famous equation would be more complex without the convention – there would need to be a multiplication symbol between the ‘m’ and the ‘c’, if the equation was to be understood strictly left to right.

E = mc2 is more explicitly E = m*(c2)

The convention that I’m using here is that if two non-operator symbols are adjacent to one another, there is an implied multiplication operator between them. e.g.

2ab is equivalent to 2 x a x b
2a + b is equivalent to 2 x a + b

This convention is, strictly speaking, not part of BODMAS.

Notice the brackets around the exponentiation. Brackets are the ‘B’ of BODMAS, and are always evaluated first. The ‘O’ stands for ‘orders’ or powers, so an expression with multiplication and powers is interpreted as follows:

3 * 24 = 3 * 16 = 48

It is not interpreted as follows:

3 * 24 = 64 = 1,296 (Wrong!)

The D and M of BODMAS stand for division and multiplication. If there are both multiplications and divisions in an expression, division is not always done first. The multiplications and divisions are processed, by convention, from left to right, and the same holds for addition and subtraction, but multiplications/divisions are done before additions/subtractions. I’ve seen explanations of BODMAS that say that divisions should precede multiplications and subtractions should be carried out before additions, but this is not so, and gives wrong answers. Or rather answers that don’t really comply with the BODMAS convention, as understood by most people.

Fine, that’s all sorted. Except that it isn’t. There are cases where the simple BODMAS, left to right, convention is insufficient. One such case is the case of exponentiation on exponentiation:

2 ^ 3 ^ 4 interpreted as 2 ^ (3 ^ 4) = 2 ^ 81 = 2.417x10^24
2 ^ 3 ^ 4 interpreted as (2 ^ 3) ^ 4 = 8 ^ 4 = 4096
Note: The '^' is used here for the exponentiation process, because it is difficult to apply superscripting twice. It also makes things a little clearer.

This case is usually interpreted by the first method, above. Such cases aside, the BODMAS convention clearly describes how to evaluate any arithmetic or mathematical expression. If you are not sure of the correct methods to use to create a complex expression, you should use brackets to clarify matters, whichever convention is used. If you are trying to evaluate a dubious one, you are out of luck, unless you can contact the author of the expression!

So, given that mathematicians and computer scientists (and many others) use the BODMAS conventions, what does that say about the expression ‘2 + 3 x 4’? Is the correct answer fourteen (14)?

The problem is posed ambiguously on purpose. The original setter was not really requesting an answer. He/she was inciting debate. Therefore the ‘solution’ doesn’t really matter. For what it is worth, I understand and use BODMAS, so I favour the answer of 14, but if the poser of the conundrum really wanted a unique answer, then they would have included brackets. Either:

2 + (4 x 3) = 14 or (2 + 4) x 3 = 18

Unfortunately the debate often quickly becomes acrimonious, with one side or the other hurling insults. But that’s the Internet for you.

What Happens When Your Body Encounters a Virus?

Image by Zeathiel. Downloaded from Freeimages

When your body encounters a virus, any virus, the virus enters your body and starts to multiply. It multiplies fast. It does this by taking over the genetic systems of the cells, and so the cells can’t maintain themselves and die. When this happens the cells burst open and release many copies of the virus from each cell into your body. Each copy of the virus finds another cell to invade. This applies to all viral infections, not just Covid-19.

Let’s say that each cell releases 100 copies of the virus, a number which is far smaller than the real number. The real number is much higher. So in the first step one virus becomes 100 viruses. Each copy infects another cell, and each produces 100 viruses. So, at the second step, 10,000 virus copies are released. At the third step, 1,000,000 copies are released. At the fourth step, 100,000,000 copies of the virus are released. By the time that the replication has gone through 10 steps, an astronomical number of copies of the virus are floating around your system. After 20 steps… Remember that replication factor is enormously more than 100, too. This is termed exponential growth.

Of course, the virus doesn’t have things all its own way. Your body has an immune system. It detects the virus and starts to fight it. If the virus has not yet managed to reduce your lungs to a bag of slime, or dismantled your nervous system, or whatever, your immune system starts to fight the virus. It starts killing infected cells thus preventing them from creating copies of the virus.

It recognises infected cells by proteins on the surface of the cell. These proteins are different for different viruses, and the immune system ‘remembers’ the signature of an infected cell, and eventually, if you are lucky, the immune system succeeds in killing off all virus infected cells and the infection is over.

(Please note that specialists in this field would probably find the above hysterically funny, but I don’t think that it is too far off the mark.)

So that’s what happens when a person encounters a virus for the first time. The outwards signs of the battle between the virus and the immune system are what we consider to be the disease. That is, raised temperature, headache, spots, cough, sneezing, and so on. Maybe more life-threatening symptoms. The virus does not directly cause any symptoms itself.

When a person who has already encountered the virus encounters it again, the immune system already knows about the virus and kicks in immediately. It doesn’t have to work out, firstly, that there is a virus and secondly, how to fight it. Your temperature might peak and you might have a headache, but any symptoms will likely be much reduced this time around, and virus will be killed off much faster.

But while your body is fighting the virus, there is a short time interval when the virus is in your body and you may be mildly infectious. Since your body is already fighting the infection, the infectious period will be short, and you won’t be ‘shedding’ as much virus to infect others.

The vaccine, any vaccine, is designed to fool the immune system and provide it with the necessary information to fight the virus without actually inflicting the virus on the body. It does this either by supplying the body with the dead virus or with a very much weakened virus, or with the information necessary to detect virus infected cells.

To do this, it needs to trigger some parts of the immune system into action, but doesn’t need to do more than that, so any side effects will be minor. A sore arm, for example.

A side effect indicates that your body is configuring your immune system to handle the virus. If you had previously encountered the virus in the wild, you would, at that time, possibly have had a severe infection. Side effects of the vaccines are rare, but if there is one, it signifies that your body is doing its job, and preparing a defence against the virus.

Of course, it is theoretically possible that you might have a reaction to some component of the vaccine. It’s theoretically possible that you will be killed by a piece of falling space junk such as a falling space toilet, of course. Or you could win the lotto. All these things would be of the same order of possibility. They are theoretically possible but very unlikely.

Worrying about the contents of the vaccine seems to be silly. If you end up in hospital after an accident, or you are hospitalised for something like pneumonia, or something worse, you would be pumped full of all sorts of things. Full blood. Blood plasma. Antibiotics. Many other things that the doctors and nurses would put into your body to save your life. They might even shoot X-rays through you, or irradiate you to kill cancer cells. It is ludicrous to worry about the contents of a simple vaccine, most ingredients of which are present only in microscopic amounts.

If you tell people that there is the possibility of a side effect on taking a medicine, then a number of people will experience that side effect, even when they are not given the medicine, but are instead given a placebo. A chalk pill or an injection of plain saline. If they are worried or concerned, the likelihood of the side effect will be higher. This is called the nocebo affect, and been blamed for up to two thirds of the ‘adverse reactions’ to the Covid-19 vaccine.

When someone who hasn’t been in contact with virus catches it, it takes a relatively long time for the body to learn how to fight the virus, which means that the body is shedding virus during the period that the body is learning how to fight the virus, and while the body is destroying the virus.

On the other hand, if the person has already had the virus or has been vaccinated against it, the body doesn’t have to learn how to fight the virus. That means that the shedding period is much shorter and the level of the virus in the body is lower, so that the total amount of virus that is shed is very much lower.

In summary, a vaccinated person sheds less of the virus for a shorter period and doesn’t usually get sick. There is less chance, therefore that they will infect someone else.

People often ask if you can catch the Covid-19 virus twice. Of course you can. The protection that the vaccine gives you wanes over time, which is why we need booster shots. This is also true for other viruses, like the flu virus. If you have had the flu, then you are protected for a while by your immune system, but the immunity fades over time. That, and variants of the virus, are why we have to have a flu booster shot every year.

If you don’t come in contact with the virus, then vaccination has no effect. If you do encounter it, it doesn’t stop the virus from entering your body. No vaccine does. It just allows your body to deal with the virus quickly and efficiently. Sometimes, in spite of you catching the virus before or being vaccinated, your body doesn’t properly remember that you have had it or been vaccinated, because the protection has waned.

In which case you will get the infection again, but it is likely that your body will at least partially remember the previous infection, so if you catch the virus twice the second time will likely be less severe.

Can you infect others if you are vaccinated? Yes, you can, if your body is fighting off the infection at the time, but the chances of you doing so are very much reduced. The virus will be in your body for only a short period, and you will not have time to shed much of the virus in that time.

I have tried to explain what happens when you are infected by a virus, such as Covid-19. It is always a good idea to be vaccinated. If you decide not to get vaccinated because of the very small chance of having a usually mild side effect, that is like being in a crashing plane and refusing a parachute because ‘you know, those parachute things have been known to get tangled and people have been killed by tangled parachutes’.

The Creative Process


Photo by Wojtek Piekutoszczak from FreeImages

I’m constantly fascinated by the creative process, at least in so far as it applies to my writing.

I constantly read articles about how to write stories or even novels. The secret, apparently is to plan everything out, usually based on some scheme or template. You have to develop your characters, maybe give them a backstory, a history. Then there must be a problem, an issue, or a difficulty, otherwise there is no story of course.

Only when all this structure is in place can the poor author actually write the story. Avoiding adverbs of course. This works for some people, and there are programs out there to help you if you write in this manner.

There is another way. Draw up a piece of paper or open a word-processing document and write. This is the way that I do it. Oh, I do get an idea in my head, and it does bounce around for a while, attracting characters, plot points, problems and so on, well before I touch a keyboard.

Take the piece that I am writing at the moment. (No, not this post, Johnson!) I’ve already written two stories about a group of people, and I wanted to close out a trilogy. I don’t know why. It just seemed right.

So, there’s a group of characters. They’ve jelled as a group, and are comfortable with each other. I’m comfortable with them, but I can’t just write about their normal lives. No dramatic tension.

So, I just write. I assemble them (it’s the beginning of term), and I have an idea. They’re college age, so boyfriends and girlfriends! They have classmates, and they don’t like one of them, and that character becomes their antagonist. The story is starting to take off.

All the while I am thinking about them when I’m not writing. What if they do this? What if they do that? I guess in a sense that I’m writing the story even when I’m not actually writing the story. I’m considering what they will get up to while I’m wandering the aisles of the supermarket.

Gradually I get an inkling of the main thrust of the story. Obviously, I have my team, I have their antagonist, and maybe their antagonist has a team too. The differences between the two groups escalate, and need to be resolved.

In addition, key scenes occur to me. They may not yet be linked into the story, and not all of them will be in the story in the end, but if I decide that they do belong in the story, the narrative must lead naturally to them.

As regards my current story, I do not have a resolution to the conflict, and only a vague idea of a conclusion, yet, but they will come, inevitably, as I continue to write the story.

The planning approach. It’s not for me, but it may work for you. I’ve tried it and I got nowhere. It may help some people to get their thoughts in order, to steadily work through the plot from introduction to conclusion, but that changes it, for me, from being enjoyable to being, well, tedious. Work, in the negative sense.

But… I can’t emphasise this enough, planning everything out works well for some people. So try it.

But even if you think that you would do best with a well honed plan, I believe that you should try to write a story without a plan. It may be that the method that works for you lies somewhere between the two extremes, and only by experimenting will you work out the way that suits you best.

Rule one of writing is that there are no rules.

Updating my stories

The Mage and the Boffin
The Mage and the Boffin

I’ve been reviewing and updating my earlier stories, and I’ve removed a few spelling and grammar errors, and tidied up the language a bit here and there. I’ve not radically changed them, but I thought that I’d package them as a PDF and make the new package available on my website. So, if you want to have all my early stories in a convenient package, please download it from here.

If you want my stories in a different format, a MOBI file for a Kindle, for example, let me know via my feedback page, and I will see what I can do.

Some of my stories have been published on Amazon (as eBooks and paperbacks) and Kobobooks and Smashwords (as eBooks). Here are the links to my Author Pages on those sites.