Mathematical Grammar

Given the original question:       6 ÷ 2(1+2) = x

People younger then I were handed an acronym to help them process equations such as the one above, this acronym is PEMBAS and the alternate which is BODMAS. The expansion of the acronyms are provided below:

 

·         PEMDAS: Parentheses, Exponents, Multiplication and Division (from left to right), Addition and Subtraction (from left to right).

·         BODMAS stands for Brackets, Order, Division, Multiplication, Addition, and Subtraction.

 BODMAS is common in the UK while PEMDAS is used in the US. The terms Exponents and Order are interchangeable, as they both refer to raising a number to a power or finding a root of a number; these operations are usually presented as x^y and √x. It must be noted that numbers within brackets are to be considered as a single number, please keep this idea in mind. It is at this time I’d like to thank my father for teaching me math and not an acronym.

 

It must also be noted that MATH is not a natural language. In a natural language one can still get their point across if the sentence is grammatically wrong, where as with math when you get the grammar wrong you get the wrong answer.

 

For the question above some people get to the answer 1, while others get to the answer 9. The answer is 1, and not 9, below are examples of how people get to their answer. Things will be made a bit tricky later on to prove the point.

 

Following the explanations we can return to the original equation:

Right:             6 ÷ 2(1+2) = 6 ÷ 2(3) = 6 ÷ (2×3) = 6 ÷ 6 = 1

Also right:       6 ÷ 2(1+2) = 6 ÷ (2+4) = 6 ÷ 6 = 1

Wrong-1:         6 ÷ 2(1+2) = 6 ÷ 2 × (1+2) = 6 ÷ 2 × 3 = 3 × 3 = 9

Wrong-2:         6 ÷ 2(1+2) = 6 ÷ 2 + 4 = 3 + 4 = 7

 

Recalling the idea that equations within brackets, as part of a larger equation are to be resolved first, brackets can be nested. In some cultures there was an emphasis put on the brackets e.g. {5 + [5 × (1+2)]} = 20 or in the more modern form 5 + 5(1+2) = 20.

 

Now for the tricky part, in the original equation we are going to declare that (1+2) equals ‘x’ and the answer will be provided, giving us the two possible equations 6÷2x=1 and 6÷2x=9. These equations can be expressed as shown below.

 

Right

6÷2x=1 is        6   = 1 multiply both sides by x thus arriving at         6 = x OR x= 3

(2x)                                                                              2

 

Wrong

While 6÷2x=9 can be expressed as 6÷2×x=9 when ignoring the influence of the brackets and then moving from left to right it can be seen that the equation gets altered. This new equation would result in 3×x=9 and while dividing both sides by three still results in x =3 there is a serious problem.

 

The problem is that both equations can’t be right seeing as 1 ≠ 9. For those of you who do not know or maybe forgot, ≠ means ‘not equal to’.

 

The Proper Perspective

As with most situations, laziness causes confusion and mistakes. The most accurate way of writing the original equation would be “{6 ÷ [2 × (1+2)]} =?”. Know that when it comes to brackets, or parentheses, there is also a hierarchal order of operations where the mathematician starts with the round brackets, and then on to the square brackets and finally the curly or brace brackets.

 

Most people who enjoy maths, or those unfortunate souls who hate math yet have to learn it, known that (a+b)²= x results in a² + 2ab + b² = x. While this shortcut formula can provide the answer for x, does anyone remember why this comes about? Just to finish off the page it will be shown below using the bracket types provided above.

 

(a+b)²= x                    

 

Expanding on the above this is what we get

[(a+b) × (a+b)] = x     

 

Multiplying each element in the first brackets with each element in the second bracket we get

[(a×a) + (a×b) + (b×a) + (b×b)] = x

 

Because a×b is equal to b×a we get

{(a×a) + [(a×b) + (b×a)] + (b×b)} = x

 

All of which results in

a² + 2ab + b² = x

 

Of course, if a=3 and b=5 it is simply easier to add them together to get 8 and then return the answer of 64. Mathematics is not just about playing with numbers, while that can be fun; mathematics also provides a methodology and framework towards successful problem solving. 

Of Mice and Men

 I was recently asked to review a documentary on some of the experiments that were done from the late 1950s to the early 1970s by John B. Calhoun an American ethologist and behavioural researcher. The video I was provided, ‘The Mouse Utopia Experiments | Down the Rabbit Hole’ is available at https://youtu.be/NgGLFozNM2o.

 While it is recommended that the reader views the video first, this article will try to explain some of the observations and conclusions I made. In a response to the video, I made the following statement: ‘People are more complex than mice and rats, how much more complex must be left to time because while the rodent-like behaviour might be observable; the cause may be multifaceted. The rodent experiments placed all of the subjects on a non tilted field, lacking affirmative action and welfare for the single mom rats.' I should have included mice, why didn't I include the mice, at the end?

 The email statement above brought on a conversation where some interesting points were raised. I restated my position and expanded upon it by adding in that mice are not susceptible to propaganda. My opponent, and friend, brought up the portion of the experiment where some of the mice were given food harder to get, while mice in a different area were given food that was easier to get. Following that statement, is where I made my mistake by conceding that the difference in food access was a form of propaganda.

 My friend made a good point, indirectly though, that if mice/rats didn’t behave close enough to men then scientists wouldn’t use the rodents for behavioural experiments. Of course the tin-hat wearing individuals will insist that the rodent experiments were done so that the anti-social and the meek can justify their actions and lack of responsibility, more easily.

 The title of this writing was taken from the John Steinbeck classic published in 1937. The Steinbeck novella tells the story of two men, one physically weak yet smart and another man who is mentally weak yet physically strong, who have a cooperative dream of one day owning their own farm. Lennie, the large one, persistently gets into trouble due to his lack of emotional control not tempering the use of his strength. Spoiler alert: George the smarter one, murders Lennie at the end of the story after he realizes that the dream will never come to fruition so long as Lennie is part of that dream.

 While the Steinbeck story may be the boilerplate template for many TV shows, as it shows each person through a single characteristic, it is somewhat related to the Calhoun experiments because the experiments stressed the rodents to a point where all they had left was their base characteristics. This is where I feel my friend and I interpreted the validity of the experiments differently; a difference I attempted to convey in my emailed response and obviously failed.

Calhoun’s experiments were taken up by a plethora of people across all levels of society as a doomsday call for urban centers, back in the day. Today, many cities seemingly demonstrate that Calhoun was correct in his conclusions and that a given area can only support a given population and that the inevitable constraint of population density will eventually cause a self culling of that society. This line of thinking, for me has a number of problems when the comparison is applied to present day man.

 As mentioned in the email response I provided to my friend, the human urban condition is a multi-faceted regarding to the physical and social inputs, when compared to the simplistic model of an urban rodent life. The mouse environment provided personal/familial shelter, food, water, and a lack of natural predators. During the course of the experiment the food and water were maintained while the personal/familial shelter was limited due to the growth in population until the mice became predatory towards each other. The one thing that was missing in the source experiments that is of interest to me is that the experiment did not provide for the ability for the mice to leave and never return.

 Before returning to the realm of men, I’d be remiss if I did not ask this simple question, why not use cats in a similar experiment. Cats seem to be more finicky, a TV commercial once told me so, and cats seem to have more personalities within smaller groups. I feel that cats would be more representative of how men would behave in a similar situation because cats also take ‘pride’ in their hierarchal culture. The reason I’m introducing this point, is that mankind has assigned itself the title of apex predator, though this is only a truism after humans invented tools and language. I’m pretty sure that in the early years of our species development, we were both the feeder and feed.

 With your indulgence I’m going to toss in a wooden shoe, a sabot, into the works. The behavioural scientist decided to study the gatherer over the hunter and present that as the stereotypical behaviour of mankind. Understanding the bifurcation housed within the capabilities of men, should the mouse experiment be provided such weight when only one of the stereotypical roles be poked?

 Moving back to the realm of men now, there are many inputs in comparison to the world of those constrained mice. Within the world of men, the urbanites are subjected to a proverbial/literal carrot attached to the proverbial/literal stick via laws and social pressures. These pressures are applied by Governments, the Main Stream Media, Non-Government-Organizations (NGOs), activist types, corporations and religions. Please consider the following pressure points: commuter congestion, affirmative action mandates, family support laws, the ‘mostly peaceful protests’ and how they were covered during the COVID lock-downs, ad hominem attacks, cancel culture, social media, and being alone in crowd.

 Conversely, the rural folk are obviously exempt from some of the pressure points listed above and in fact some of the pressure points are completely reversed, commuter congestion and being alone in crowd are the two obvious pressure points.

 Going off topic momentarily, many people take situational pressures differently based on personal characteristics and which inputs, this means that any impacts are obviously going to be relative between peoples. As an example, one person may feel that not having the latest "smart (it’ll make you stupid) phone" makes them poor, meanwhile another person may be in awe as to why that first person is willing to shit in drinkable water.

 Adding in the element of ‘some way out’ as a key variable would for me, make a closer resemblance to the human condition. The mice were intentionally constrained, in a way similar to mankind until the means of travel matured. The mouse city experiment with the three constants and only one variable; is for me the main reason why the experiment is a flawed model for modern cities; better comparisons for ‘The Mouse Utopia Experiments’ would be Europe well into the late 19^th century one considers all of the territorial wars, or incarcerated felons.