# The Maths They Never Taught Us - Part One

## The Basics

Let us begin this Article by an explanation of the the Title. When I speak of the Maths they never taught us, I do not mean that teachers or the education system in any country is particularly lax, but rather that there are things in Mathematics that either were once taught, and no longer are, depending on which Country You are in, or some things that have been left out for various reasons, and I am especially looking at my own experiences in learning, which may or may not be the same as those of others. Suffice to say, that the following are certainly things we do need to know.

We thus commence our exploration into the World of Mathematics by a look at what the British television character Basil Fawlty** ®** once said would be his wife

**Sybil's**specialist subject on Mastermind (TV series)

**®****® :**" the bleeding obvious ". By this I mean certain mathematical and linguistic traits we take for granted or do not even understand, where we may all get confused as to what is meant when something is said. I say the obvious, as it may even seem so to some, but these explanations are primarily for those who may not as easily grasp such concepts, with no judgment on their intelligence or ability. Perhaps we could all do with such explanations, as some things are in fact better not left unsaid.

A prominent example of what I am trying to point out is the reason why for example, the 1900s is referred to as the Twentieth Century, although it begins with the number **19**, and not **20**. I shall explore this kind of thing and certain others, in order to clarify these meanings. Then in the Hubs to follow I shall get into basic mathematical concepts as well as some which are often overlooked, or not fully appreciated.

So why indeed do we refer to the time we are in now as the 21st Century when we are still only in the Twenty Hundreds ? Yes, we are told that this is the time we are in, but are not always informed as to why it is referred to as such. Let us think back to the beginning of our **A.D.** ( Anno Domini ) era, from which we take our current dates. If we believe such, we understand that **Christ** was supposed to have been born on December 25th, 1 A.D., and so in countries with a Christian tradition, we take our dates from there, with those Before Christ being referred to as **B.C.**

As an aside which I shall look at in other Hub, if one looks through the King James Bible properly, and understands that Dionysius Exiguus, who gave us such dates, was out by a few years, it may be closer to the truth to say that the Baby Jesus was born in about September, 4 B.C., although other sources may differ. But given the dating system we do have, we then say that the years from 1 A.D. to 100 A.D. inclusive, covering a whole hundred years, make up the *First Century A.D*** .**, since this is the first lot of one hundred years from the official time of

**Christ's Birth**( see Nativity of Jesus ).

Now this will only work if one counts *all of* 1 A.D. and goes through to the end of 100 A.D. This then gives another of the obvious things I am here to point out, that although the length of time from the 25 December 1 A.D. to the 25 December 100 **A.D.** will make **99 **years, we can make up the whole century by ( arbitrarily ) deciding simply to say that the *First Century A.D.* will begin at the very start of the 1st January 1 A.D., just as midnight had past of the 31 December of the previous year, and this same century will last up until midnight on the 31 December 100 A.D.

So if we take each single year as it is from the very beginning of its 1st of January until midnight at the end of its 31st of December, this is *one whole year*, and since we are going from 1 A.D. to 100 A.D. inclusive, this is *100 separate years in total*, and indeed, as noted, *one whole century*. Accepting this, we take the *Second Century A.D. *to be from the very start of the First of January, 101 A.D. to the very end of the 31st of December of 200 A.D. So it is called *The Second Century A.D.*,* *even though most of its years are in the 100's.

## Y not 2K, OK ?

This pattern is therefore repeated, so that the years from the beginning of the First of January, 1901, until midnight of the 31st of December, 2000, were referred to as the Twentieth Century, yet most years are in the 1900s. This brings to light an interesting issue from, if some of You can remember, say, 14 years ago, ( at the time of this writing, November, 2013 ), as 1999 drew to a close, and the term **Y2K** ( with reference to the Year 2000 problem ) came more into prominence. People assumed the *21st Century* and therefore the *New Millennium* it ushered in began at the stroke of the 1st of January, 2000, a Saturday, as I recall, but Horn and Abbot's *Trivial Pursuit**™ *even had as a kind of trick question, what the first day of the *21st Century* would be. This was where people might have been expected to give 1st of January, 2000 as the wrong answer, but if we take our whole A.D. System as being from the very start of 1 A.D., then all subsequent years ending in a 1 will begin centuries and even decades.

Therefore, for *practical purposes *the Y2K Bug was assumed to attack computer systems as we clicked round to 2000, since we thought they would assume it was the same as 1900, and this made people think it was the start of the *Millennium*. In reality, this *Millennium** * did not begin until the 1st of January, 2001, on this occasion, a Monday. ( A study of dates and how they work, re perpetual calendars and the ways in which the days of the week alter with the dates as years go by, will follow in another Hub ).

Related to this idea is that of decades beginning with the year ending in a 1. Now what this strictly means is that the *nineteen seventies*, say, did not really begin until the 1st of January, 1971, and sure, that is logical and correct. But naturally, when any talks of the *seventies* I am sure they mean, as I do, the time from the very start of the First of January, 1970, until Auld Lang Syne on 31 December, 1979, and does not exclude 1970, nor include 1980. Sure, in reality, 1980 was in fact the *last year of the 1970s*, but perhaps we can allow our minds a little laxitude if only on that matter, and say instead that the *seventies* were the inclusive time I gave, with years like 1970 and 1980 as those transitional periods into the next era, but remember that the other option is more technically valid.

## Include me in

Another aspect of numbers about which we get confused, and has been mentioned here, is the idea of what I call *Inclusiveness*. If I mention a range of dates, say from the Fifth of January up to the Fifteenth, we might say that this covers **ten **days, but we would only be right if we are taking our measure from a given time on the Fifth until the *very same time of day* on the **Fifteenth**, and so the total time within that range would be **ten **days. But if we instead take *each day in its entirety*, we are dealing with a total of *eleven **different days*. Were we rather to consider our range of time to be *just past midnight* as the Fifth of January begins, thereby including *all of this date*, right up to *midnight at the very end of the Fifteenth of January*, then this covers *eleven **days inclusive*, these being the 5th, 6th, 7th, 8th, 9th, 10th, 11th, 12th, 13th, 14th and 15th.

Another matter we shall look more at later is the confusion over winning targets in Cricket. Depending on the format, basically one teams sets a total, the other tries to chase it down. Say for example that in a match where each team gets one innings, the first team scores **307 **runs, so the winning target for the second team is now **308**. This is the score needed *to win. *But to tie, they would only need this **308**. Sometimes people confuse the target with the amount of runs the first team scored, and forget that the target has to be **one **run above the first score in order to ensure victory by scoring more runs. Sometimes one might see that with say one ball left the chasing teams needs **two **runs to win, and when they get only **one **run, they think they have lost and forget that the scores are in fact tied.

Now sometimes in cricket though, there will be the situation, in Test Matches especially, that one team has batted first, then the other. Now say the second team had a lead of **78** runs, then the first begins their second innings. Here they are in slight danger of an innings defeat, but that is less likely than if the lead was **378 **runs.

Now in this case as yet there is no target as it is only the third innings, and the graphics supplied by TV will say that the second team leads by so many runs, so then that is there actual lead. The target in a sense is for the first team to wipe out the deficit and force the other team to bat again. In this case a minimum of **78** runs is required.

Say they made exactly that, then after three innings, the second team needs only one run to win, and if their batsmen go out and get one with out loss, they will win by ten wickets. If instead they bowled the first team out for a paltry **53**, they would win by an innings and **25 **runs, because on this occasion, **78 **was the overall lead, not the target. But the first team could set a target by say, having trailed by these **78** runs, they are all out for **406**, giving them an overall lead of **406 **- **78** runs, which is **328 **runs, and therefore a target of **329** for the second team to bat to in order to win. We in New Zealand do not assume even **26 **runs is enough to avoid an innings defeat, as that was what our team was once bowled out by England for in 1955. My late father was a teenager at the time and he once told me he remembered the controversy over that.

Another matter that might cause confusion is the idea of placements in say Olympic style events. Imagine a** 110 yard **dash, and two runners win the race - not even the microscopic evidence in the finish photo can separate them. So what about the person who comes after ? Some could argue they get a silver, but if one thinks about it, this should be wrong. Two people crossed the line before them, and we assumed all else is above board if you know what I mean, so even though they share the gold and get one gold medal each, third place is still third place, because after all, if two finish ahead of you, you would expect that, and a runner should not be rewarded more than they deserve just because two better runners could not be separated. Each of them gets the gold because they earned it, and since the third placed runner could not conceivably have done ought to make them do so, they have come in third place behind two others, so third place is what they get, and if they wanted more they shoulda run fasta.

## Maybe You Can Count Your Thumb

Thinking on all these concerns, having watched TV and Movies over the years I have noted the odd arithmetical error from time to time. Now these are not just where one is confused about a range of dates, but are actually statements that simply do not add up. Right now I cannot remember a specific example, but I recall something along the lines of someone mentioning a character born a certain year, but getting their age off by two years or so, and this not due to any mistakes about *Inclusiveness* or anything like that - in these cases it was all quite simply bad adding.

If anyone can help me, from decades ago I recall that one of the last scenes in the original TV version of *The Hitchhiker's Guide to the Galaxy*,* *had a man spelling out a question. The answer to this was supposed to be *the Meaning of Life, the Universe and Everything, *which the computer **Deep Thought **said is **42**. But in this scene the person seems to spell out *" What is 9 times 7 ? "*, but the answer to this is **63**, ( which may even be the answer to Life, the Universe and Everything plus **21** or something ). Perhaps he had put the tile with the number **6 **upside down, and made it a **9**, although I think I remember he spelled out the numbers. If anyone can confirm this, this would be good. I have not seen that scene in say over **25 **years. ( Yes, I am that old, and I do feel even older ).

## Oh, Heck, No ! ( after all, children might be logging on )

Let us now address the more understandable confusion we can get about ages and dates. Imagine someone has September 25th as a birthday. Indeed they do. One of my nephews, as well as Will Smith, Catherine Zeta-Jones and her husband Michael Douglas. For a good deal of the year, if one asked them what year they were born in and they gave it, then without knowing the date, one might make a false assumption about their age. Look at Smithy. He was born the same year as myself, 1968, and, as I said, on September 25, while I was brought in on January the Second. Now back in 1988, with its people wearing bad hairstyles and all, for all but one day of the year, the First of January, I was 20 years old. But Mr. Smith was **19**. Yet someone may have assumed, that since he was born in 1968 and it was then 1988, that he too was **twenty**. So when I hear the year someone is born in, I have to consider what time of the year it is now and work out the probability they are the age they may be assumed to be. In fact, the only date we can be sure of anyone's age, based on knowing only their year of birth, is the 31st of December. This is the worst date to be born on, only in the sense of therefore not being the right age people would guess at. The best date for guessing then, is January the First.

To do with this is some confusion as to which year of life one is in. This was looked at previously with the idea of the 1900s being the *Twentieth Century*. In 1988, up to perhaps the end of September 24, and in fact, all the way back to September 25, 1987, Willard Carroll Smith, Jr., was in his *twentieth year*, which is what we are in from as soon as we turn **19** right up to before we become **twenty**, even though his age is given in **19**. In fact, if we say a joker is **19**, which we date from the day of his birth, we mean he is *at least 19*, which is why some young kids, in an effort to make themselves seem older, will say they are **seven and a half**, when called **seven** years of age, where they are in their **eighth year**. Interesting that girls will continue this trend of trying to seem older until they get there, and once they hit say **35**, every successive birthday will be their **21st**. Some will even seem convincing.

## Dinner with Agrippina and Nero - Mushrooms, Anyone ?

Having looked at and thought about dates, and doing so also from an interest in History, I realized something that had never occurred to me. We acknowledge that the Anno Domini Years begin at what we now call the First of January, 1 A.D**.** ( when Jesus** **was perhaps in reality about **three **or **four **), but the year before that is denoted 1 B.C. ( I understand others will differ on what they call these or even on their agreeing to our way of reckoning dates, since the Jews and Moslems ( see *Muslim*) each have different systems, but I am using here the dating system relevant to me and what I am comfortable with ).

So we have no Year Zero in Gregorian calendar System. ( Actually the calendar or dates as we know them were supposedly not worked out until the time of, as mentioned, Dionysius Exiguus, about 525 A.D., but as also noted he may have gotten the Birth of Christ wrong. At 1 B.C. to 1 A.D. the dates are noted as they appeared in the Julian calendar and are out by some days, until the change over to the Gregorian which we shall look at in another place, with certain explanations, as to why the Julian Calendar got out of sync with the Universe, not just being due to inaccurate setting of Christ's Birth. ) Now because there is no zero, when we work out the age of a person who was born in the B.C. era and lived into the A.D. one, we need to be careful. Since there is only one year in difference between 1 B.C. and 1 A.D., where this 1 A.D. follows right after 1 B.C., someone born in 1 B.C. would only be *one year old *on the same date in 1 A.D.

Extending this, if we look for example at Tiberius Claudius Caesar Augustus Germanicus,** **( 1 August 10 B.C. – 13 October 54 A.D. ) who was Roman Emperor from 41 to 54, we can work out that since he was born on what we now deem to be the First of August ( birthday for all horses ), 10 B.C., then on the First of August 1 B.C**., **he turned **nine**. One year later, the First of August, 1 A.D., he would then have been** ten**, so that **53** years later, on the First of August 54 A.D., Claudius turned **63**, which *is* the age given as to what he lived to. We take our age to this up to the day before our next birthday, so Willard Smith Junior was **19** up to and including the 24th of September, 1988, even though in effect he was simply a day short of **twenty**, and sure, one could argue he was effectively **twenty** years of age, but not officially, until the 25th of September, 1988, and if one wanted to be even more pedantic, not until his exact time of birth, although these days we don't normally consider that necessary, otherwise imagine me at **five** having to wait up to just before midnight on the 2nd of January, 1973, to blow out the candles and get my presents. Again we say when one is say **19** we mean he ( or she ) is *at least nineteen. *So Claudius was **63**, and yet to be honest, before I was aware of how to work out these dates properly, if I had seen he was born 1 August 10 B.C., and died ( poisoned fungus, I understand, by Nero and his mother Agrippina the Younger ), 13 October 5.4. A.D., I once would have added **10 **to **54 **to get **64**, and that would have been wrong.

Looking now at a slightly different example, let us note the man first known as **Octavian**, and more familiarly, Augustus ( 23 September 63 B.C. – 19 August 14 A.D.).** **He founded the Roman Empire, and became its first Emperor, ruling from 27 B.C. until his death in 14 A.D.. On the 23 September 1 B.C. he turned **62**, so that on the 23 September 1 A.D., Augustus then made it to **63**, such that if he had lived until the 23 September 14 A.D., he would be **76**, but died *five weeks* short, and as such is recorded as having lived to be **75**, by which we mean *at least *75, and the rest is History. So all one need do to work out ages for a person who live from B.C. into A.D. is add the years together, but remember to subtract **one** as there is no Year Zero, but also be aware of what date the person was born on, and the date they then died, then one shall get the age right.

## Work this one out, Sherlock !

Let us look at another aspect of how our impressions of time can tend to give us the wrong idea. Imagine two babies born about the same time in different parts of the World. First of all, **Hamish McDavish**, born in Invercargill on Thursday 7th of November, 2013 at 7:15 pm,** New Zealand **Daylight saving time, and **Dudley Wright**, brought into this World at 11pm** Pacific Standard Time**,( PST ) Wednesday the 6th of November, 2013, in Vancouver, British Columbia. Now don't go all goo goo, ladies, I made them both up. What I am trying to illustrate here is that, although young Dudley appears to be *twenty and a quarter hours older*, he was in fact born *45 minutes after* Hamish, due simply to the difference in Time Zones between the two Dominions. This shows us that sometimes things are not always as they seem, and so when dealing with anything we need to be sure of all the details so we do not come to any wrong conclusion.

Let us look at one more way in which years of birth could give us the wrong impression. Imagine two men who lived as near neighbours in the County of Devon a very long time ago, the first, **Master Jedediah Trumthwake**, from 1638 – 1719, the other, **Kingdom** **Murrycroft**,** ** from 1639** **– 1718. At *face value*, it would appear that the older man lived about *two **years longer*, since he was born the previous year to the other, and died in the year after he did. But did he *necessarily* live so much longer ? Let us look at their respective birthdays :

**Jedediah : December 31 ^{st}, 1638 – January 1^{st}, 1719**¸ having lived

**Eighty Years and One Day**.

**Kingdom : January 1 ^{st}, 1639 – December 31^{st}, 1718**¸ having lived

**One Day short of Eighty Years**.

So in this case, at least, the older one actually live ** two more days** rather than

**.**

*two more years*But then, if we had :

** Jedediah : January 1^{st}, 1638 – December 31^{st}, 1719**¸ having lived

**One Day short of Eighty Two Years**.

** Kingdom : December 31^{st}, 1639 – January 1^{st}, 1718**¸ having lived

**Seventy Eight Years and One Day**.

In effect we have added nearly *two years* to the older man’s time, and taken just about *two years * off the younger one, to give a difference in age at death of almost *four **years*.

This is why it is important to think about how old a person really is, and what the dates of life of especially a historical person really represent, which unfortunately is not always obvious when such dates are given, as most sources only deal in years, and also when dates of historical events are given, sometimes the day of the week they occurred on, is not, and that we shall deal with in a later Hub.

## It Works Either Way

You would of course have already learnt in school that for *certain* operations of numbers, it doesn’t actually matter in which order the numbers are put.

For Example **: 4 **×** 5 **=** 5 **×** 4**, and **4 **+** 5 **=** 5 **+** 4 **But : **4 **÷** 5 **≠** 5 **÷** 4**, while **4 **-** 5 **≠** 5 **-** 4**

Now it is important to make one thing clear. The *first* two statements using multiplication and addition are *always *true, whereas with division and subtraction, the *second *two statements are *almost** *always true. To show this in a general way, I shall introduce the use of letters of the alphabet to symbolize algebraic variables, which we shall also look more into in some of the Hubs to come.

With use of algebraic variables **a **and **b **to symbolize possible values that any number might take, what we mean to say, is that although **a **×** b ***always* equals **b** ×** a**, and **a **+** b ***always *equals **b** +** a**, **a **÷** b **does not *normally** *equal **b **÷** a**, and **a** -** b **does not *normally** *** equal ****b** -** a. **The *only* exception is actually when **a **=** b**. So that for example, if **a **=** 5**, and **b ***also* =** 5**, then as much as **5 **×** 5 **=** 5** ×** 5**, and **5** +** 5 **=** 5 **+** 5**, we see also that **5 **÷** 5 **=** 5** ÷** 5**, and **5** -** 5 = 5** -** 5**.

This, as stated, is the only exception where the value of both numbers on which you are performing these operations is exactly the same. Otherwise, if the two numbers are *different*, then, as noted, only the operations of multiplication and addition can be performed *regardless of the order* in which the numbers are put in the equation.

But why ? You see, it’s all very well in any subject to say that something is true simply because it is, or just because somebody says so. If you try that one on, they’ll slap your fingers with a cold, wet ruler, for in modern mathematics you must *always* be able to *prove* what you say.The best way of doing this is to *show* an example of what you are trying to put across, and in particular, one which is clear and accurate. Well, looking at an example of addition, this at least seems reasonably straightforward :

## Move the Plus Sign one Space to the Left :

Now note that there is a total of **nine **asterisks, wherever the plus sign is placed, since the plus sign itself is *not* actually a part of the sum. That is, it is *not* to be counted as one of the asterisks, but is rather the actual symbol of operation. If we do the sums without it, we then get the following .

This is relatively easy to see, since you can quite simply flip the first collection of asterisks **180°** over, to make it look identical to the second collection. Now, as to the matter of multiplication, we can also have a way of visualising this fact, that the order the factors are in makes *no* difference whatsoever to the final value of the product.

Here we have two examples of multiplication, **5 **×** 4 **and **4 **×** 5**, both of which equal **twenty**. We say **five **times** four**, because we mean to say that we are having the number **five **on **four** occasions, that is, that we are actually *adding** ***four **lots of **five**, so that **5** ×** 4** is in fact the same as saying : **5** +** 5 **+** 5 **+** 5**, which is like saying **five, four times**. In much the same way, therefore, to express **4** ×** 5 **is simply the equivalent of working out : **4 **+** 4 **+** 4 **+** 4 **+** 4**. But since it would be long and sometimes very tedious, especially to add long chains of numbers together, we are taught the shortcut of multiplication.

Now as an interesting side note, just because **four **times** five **does equal **five **times** four**, this does not mean that they are *always* the same thing. Yes, they *do* give the same number, but imagine you are packing a product on an assembly line, and they want you to pack **four** of a product into a box, then **five** of these smaller boxes, each one containing the **four** products you’ve just packed, into a larger box. It would certainly do no good to do it the other way round, if the product only fits into the boxes a certain way, and there you are trying to ram **five** of goodness knows what into a space made for only **four**. So in some situations the arrangements of the numbers multiplied *might not really matter*, but in others they certainly will.

We continue this look at multiplication in the next Hub, Part Two of this same Title, at The Maths They Never Taught Us - Part Two, and if You are curious, take a look at my other Hubs, such as

The Maths They Never Taught Us - Part Three , The Very Next Step - Squares and the Power of Two , And then there were Three - a Study on Cubes, Moving on to Higher Powers - a First look at Exponents, The Power of Many More - more on the Use of Exponents, Mathematics - the Science of Patterns , More on the Patterns of Maths , Mathematics of Cricket , The Shape of Things to Come , Trigonometry to begin with, Pythagorean Theorem and Triplets, Things to do with Shapes, Pyramids - How to find their Height and Volume, How to find the Area of Regular Polygons, The Wonder and Amusement of Triangles - Part One, The Wonder and Amusement of Triangles - Part Two, the Law of Missing Lengths,

The Wonder and Amusement of Triangles – Part Three : the Sine Rule, and The Wonder and Amusement of Triangles - Part Four : the Cosine Rule.

Also, feel free to check out my non Maths Hubs :

## Disclaimer

Any reference to any Copyright or Registered Trademark is credited as such, for example, mention of *Fawlty Towers*, which is a Trademark of BBC Television and Trivial pursuit , now owned by Parker Brothers / Hasbro . Some discoveries are my own, but may also have been found independently by others as Mathematics is a living language, and it was **Ralph Waldo Emerson** who described the demise of any language that does not keep growing. Some information has been referenced in a number of publications, most in the public domain, as well as on Wikipedia ( copyright 2013 Wikimedia Foundation).

## Comments

Y2K

You Said Quote “Therefore, for practical purposes the Y2K Bug was assumed to attack computer systems as we clicked round to 2000, since we thought they would assume it was the same as 1900, and this made people think it was the start of the Millennium. In reality, this Millennium did not begin until the 1st of January, 2001, on this occasion, a Monday.” Unquote

My Reply

It had nothing to do with a bug, nor did when the Millennium started

The scare was that all of our computer systems around the world would cease to function. Why? Because the programmers didn’t think past the 1900s and dates saved used two digits rather than four for example 1999 was 99. So, when the year turned 00 or 2000 the fear was that most computers would be unable to distinguish between 2000 and 1900. In fact, problems had already begun occurring in situations that involved future dates later than 2000. So, the assumption was that all the worlds computers would crash. It had to do with when the 00 started not the new Millennium

Here the number 42 is explained by the one to ask the question Douglas Adams

You can read about Answer to the Ultimate Question of Life, the Universe, and Everything (42) here

Just because a century begins with an 01 does not mean the seventies began in 1971, it did not. The seventies began in 1970. How we calculate a century has nothing to do with the 1970s, 1980s, and so on.

You said Quote "If I mention a range of dates, say from the Fifth of January up to the Fifteenth, we might say that this covers ten days, but we would only be right if we are taking our measure from a given time on the Fifth until the very same time of day on the Fifteenth, and so the total time within that range would be ten days. But if we instead take each day in its entirety, we are dealing with a total of eleven different days." Unquote

The way you worded it "up to the fifteenth" it is only 10 days because "up to" means you are not counting the fifteenth.

If you wanted to count the fifteenth day you would say "through the fifteenth" and then it would be eleven days.

Also December 25th in the year 1, to the end of December 25, in the year 100 is 100 years, not 99. You count both the 1st and 100th year. You do not subtract them. 100-1 does equal 99 however you do not subtract any year you just count them. 1.2.3...100=100 years

About The Centuries

It’s simple, a century is a hundred years, so 1-100 is the first century. The last year tells what century we are in. For example in 301-400, the 400 is the last year and so it is the 4th century.

There is no great mystery to this.

1-100=1st century

101-200=2nd century

201-300=3rd

301-400=4th

401-500=5

501-600=6

9