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Number 6 Fair View,
The Garth Chambers,
Dexter University Chapel,
Dexter.
NORREX
DX1 1AY
My Dearest niece Josephina,
thank you for your recent letter, which I enjoyed reading enormously. In answer to your question, yes I agree with you. Without doubt the most fascinating number of them all is zero which was discovered independently in India, the Middle East and China, but the concept came late to Europe.
Late in the twelfth century, a wealthy entrepreneur, Joseph King of Oxenford, who was a friend of many learned men of the day, in conversation asked where he should educate his son and heir at the highest of levels. The reply he recorded in his journal was interesting.
“Should it suffice that he needst learn addition and subtraction then any of the great western centres of learning are adequate. But should he needst learn multiplication and division he must go east. There is talk of Pisa but nothing is certain there. To be certain of mastery of the arts he must go to Persia at least.”
This may seem surprising today, but the explanation was the European number system in common usage was Roman and had no zero, so multiplication and division were nearly impossible whereas the number system used in the east was Arabic and used a positional system i.e. it used a zero.
For example 5 could be 5, 50, 500, 0.5, 0.005 &c. So only ten symbols were used to represent all possible numbers, but the tenth symbol zero was crucial to place the 5, in the correct place which made multiplication and division easy.
The Roman system needed an unlimited number of symbols to do that and hence it couldn’t.
I have been so fascinated by this concept that years ago I wrote a small book called ‘This Book is About Nothing’, of which I enclose a copy.
As to you question, ‘Is zero odd, even, both or neither?’ You do ask interesting questions, my dear.
The question has no validity if one considers ℕ the set of natural numbers which start at one and are {1, 2, 3, 4...} the counting numbers all children learn, for zero is not a member of ℕ.
However, if one considers ℤ the set of integers we discussed last month which zero is defined as a member of and which includes all positive numbers, zero and all negative numbers your question indeed has validity.
If one starts at say seven and counts down in twos one gets the double open ended sequence {...7,5,3,1,-1,- 3,-5,-7...} which is of course the odd numbers, all of which if divided by two leave a remainder of one.
With a positive number, say 7 one gets 7 = 2 x 3 + 1 the 1 is the remainder when 7 is divided by 2.
With a negative number, say -7 one gets -7 = 2 x -4 + 1 the 1 is the remainder when -7 is divided by 2.
Similarly if one started at six one gets {...6,4,2,0,-2,-4,-6...} which gives you the even numbers and zero is amongst them, all of which if divided by two leave a remainder of zero.
With a positive number, say 6 one gets 6 = 2 x 3 + 0 the 0 is the remainder when 6 is divided by 2.
With a negative number, say -6 one gets -6 = 2 x -3 + 0 the 0 is the remainder when -6 is divided by 2.
And finally with zero 0 one gets 0 = 2 x 0 + 0 the last 0 is the remainder when 0 is divided by 2.
Have your mother drop you off for dinner and a chat, my love, at my chambers any evening other than a Tuesday when I have a tutorial with a student. I’d be delighted to discuss such matters with any five year old who asks such delightful questions, and I’ll see if I can beat you at chess this time.
We shall certainly discuss ℚ, and ℝ and possibly even ℂ! Which I sure will fascinate you.
Here’s a puzzle for you. Why does none sometimes take a plural verb and sometimes a singular one? I know mathematics and not grammar is what you are interested in, but unfortunately they do overlap sometimes.
And finally something for you to investigate before we meet. Hexadecimal counting uses the digits, 0,1,2,3,4,5,6,7,8,9,A,B,C,D,E,F. How many numbers can one represent using just four digits? Don’t forget 0000.
My very best to my dear sister, your father and yourself,
Your loving Uncle Peter
P Halthorp
Professor P. Halthorp
P.S. I was reminded of a very old joke the other day. I think you’ll like it.
Only one in a thousand people understand binary, the other seven don’t.
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Comments
65536
There are 10 types of people, those who know binary, and those who don't.
65536
Indeed as in (FF+1)^2. And that's why the Commodore 64 was so called that being 64K or more properly 64* 2^10. Somewhere I've got a collection of the jokes, but I remember being told yours. We all tend to be a bit smug and elitist about it, but...like I said before if you become a preacher tell me. I want a place in your choir.
Regards,
Eolwaen
Eolwaen
Beyond Zero
there is the
Square root of -1 and the wonderful world of complex numbers without which we could not properly describe (in a mathematical sense) the world in which we live.
I wish more young people appreciated this instead of saying 'Maths is hard so I don't do it'.
That's the engineer in me speaking sorry.
Samantha
Mathematics
Life is hard too, and it seems to me many don't do that either for the same reason it requires effort. The sad thing is many have no interest in anything and know nothing about anything either. They have no idea how boring they are as a result. I make no apology for describing experts on soaps and sport as knowing nothing, after all I too am entitled to a view. e^(iπ) + 1 = 0 may indeed have little application at the breakfast table, but in order to have achieved understanding, what one will have learnt on the way will stand in good stead for after dinner conversation. I agree with the Victorian belief that studying botany was not of any particular intrinsic value but it trained the mind.
Regards,
Eolwaen.
Eolwaen
e^(i*pi)
http://imgs.xkcd.com/comics/e_to_the_pi_times_i.png
http://imgs.xkcd.com/comics/e_to_the_pi_times_i.png
I've had that reaction too many times to count giving lectures and tutorials.
Regards,
Eolwaen
Eolwaen
Gotta love the math whingers...
"Algebra is useless!!!"
"I hate story problems!!!"
I like to tell people that engineering school consists mostly of story problems.
On the whingers
I like to tell them how much entertainment the whingers provivide for those us us that at least tried at mathematics even if we didn't get very far. They don't like the idea that we laugh at them. I can live with that. I'm not a snob about Mathematics and I'm more than happy to provide explanation to any who want it. I'd far rather they asked, so they too can enjoy it, whatever it happens to be, than they wistfully wished they understood.
You've done it again, Ray. Provided me with ideas for stories. How does "The Virgin Maths Whinger and the Fields Medal Winner," or "I've Never Tried at Mathematics Because I Don't Like It," or "I was born a Geometer but I Cross Dress as an Algebraist," sound as titles?
Truly my response, after copying those ideas into the Ideas folder of course, is simply 'You don't have to read them, but if you complain about their existence you can expect those who do like them to give you a hard time.'
Regards,
Eolwaen.
Eolwaen
lol
The math whinger's guide to wisdom?
And let's not forget all those Facebook questions like, "What is 2+3*5?" You get to choose between 17, 25, and a few others. I always comment that anyone who made it through grade school math ought to know about the order of operations.
2*3-3*2
2*3-3*2 has a similar effect, though somehow more get that one correct. As to order of operations, if you want a real laugh, have a look at a few YouTube videos on the subject. I don't do social media, but probably similar material is there too. I'm not sure what level 'making it through grade school implies' (the internet suggests 16 or 17 year olds, but I couldn't find a minimum school leaving age for any state) but it's similar over here. Qualifications issued with coupons cut off boxes of cereals with the powers that be insisting that standards are not being degraded.
Regards,
Eolwaen
Eolwaen
Grade school
It's a bit of an exaggeration. I believe they introduce order of operations, but just barely.
Grade school is first through sixth grade -- about ages 6-12.
Grade school
I wrongly thought it was the equivalent of compulsary education in UK which ceases at 16. English state primary schools are 5-11, secondary schools are 11-16, with optional 2 years after that. Private schools here start at 13 not 11 (pre 13 is referred to as a prep(aratory) school. In Scotland they switch to secondary school usually at 12.
Thank you for the explanation Ray.
Regards,
Eolwaen.
Eolwaen
Further education ... muddying the waters
But after school, when you continue to university, you learn that this "order of operations" is just a set of arbitrary axioms, nothing more.
And in computer science you get to learn that there are different languages using this infix notation with different sets of operator precedence and/or order of expression evaluation (left to right vs. right to left). That mess was one of the reasons why prefix and postfix notations were invented (and who doesn't love HP calculators with RPN or Forth?).
So, both 17 and 25 can be correct answers, depending on the context.
Conventions
All you have written is true, however most folk have never heard of anything other than the default, BODMAS or BIDMAS or whatever acronym they were taught. If one is to use other than the default it needs to be stated, whereas by definition the default does not need to be expressly stated. The point being made is that there is a poor understanding, if any, of the default, not that the default is the only convention available. If you ask most people who have a pass grade or better at GCSE (in the UK taken at 16) indicating they should understand, what does two x squared, 2x^2, evaluate as when x is 3 they will tell you 36. That is the issue, not whether there are other legitimate ways of evaluating the expression used by those with specialist knowledge. Interestingly they will evaluate 4x^0.5 as 6 but 4√x as 12 when x is 9. I have known first year mathematics undergrads do that too. The conventions may be arbitrary, but it is a universally agreed upon arbitraryness, which is just anotherway of saying it is a convention.
Regards,
Eolwaen
Eolwaen
"There is no order of operations."
I saw a YouTube video by that title, and it was enlightening.
Express it all in addition, and you don't need it. Cumbersome, though.
Multiplication is repeated addition. 2*3 is 2+2+2 or 3+3. Exponentiation is repeated multiplication. That is why the orders are stated as such. If you don't work it that way, using algebra to solve an equation simply won't work. The numbers won't come out.
The left to right order is just convention. But it really doesn't matter because of the commutative property (2+3=3+2.)
No order of Operations
That's why a purist would tell you subtraction and division don't exist. Eg. You don't subtract 2, you add (-2). You don't divide by 2, you multiply by 0.5 .If you think about it there is a bizarre logic to it. In BODMAS or BIDMAS, or whatever you use, the A&S have the same hierarchical place because they are the same, Likewise the D&M. Taking it one step further roots and indices are the same too, so you don't 4√ you ^0.25 that way roots are redundant though kids tend to evaluate roots better than powers.
Regards,
Eolwaen
Eolwaen
Mathematics? Bring it on!
Would love to see more of these.
As far as zero, it cannot be odd if even and odd are being defined in terms of division by two (as you have). As you showed clearly, it has to be even (and can only be even). However, division in the natural numbers is a rather tedious operation due to the sheer number of pairings that are undefined by necessity. The rational numbers are at least closed under division.
Division
That's why you had to go east! If you like this sort of thing try reading Omar Khayyam. His mathematics not poetry. Wonderful stuff!
Regards,
Eolwaen
Eolwaen
As far as the hexadecimal,
As far as the hexadecimal, there are 16^4 different numbers that can be expressed. 16^4 = (2^4)^4 = 2^16 = 65536.
As above
(FF+ 1)^2
Sorry senior moment. Edit done
Regards,
Eolwaen
Eolwaen
???
FF^2 + 1 != F^4 + 1
Senior Moments
Indeed my Brain was on holiday! Edit done. Surprised it took so long for someone to spot it, but
1! = 1
FF^2 +1! = 65026
F^4 +1 = 50626
but (FF+1)^2 = 65536 W^5 (Which Was What We Wanted) English version of QED
Eolwaen
None of the above!
"None" suffers as do so many singular nouns of being given what this pedant (me, or more correctly "I") hates, a plural verb when that singular noun ("group" , "hundred", etc, etc) is followed by "of" and a different plural noun referring to what members of the group referred to by the first singular noun! Geddit?
Tortuous I know but personally I shudder every time I hear something like "there ARE a hundred reasons", and never feel anything over "there IS a hundred reasons".
It is so frequent these days that I fear that common usage will become correct usage, even when linguistically it fails to male sense.
I added this note instead of saying how much I enjoyed your thoughts on zero, because between you and Ray, you had covered all that I might have added -- had I the knowledge at my immediate back and call!
Best wishes to you both
Dave
Collective nouns Singular or Plural verbs?
Interestingly, Dave, the various style manuals, on both sides of the pond, are in agreement on this one. They all put it differently, but say it depends upon context. That is as to whether one is considering the noun as 'a' noun which takes a singular verb form or as 'a collection' of many separate items which takes a plural verb form. All conclude, though it takes some finding in some of them, that there will always be arguable cases which if nothing else illustrates the limitations of a style manual. None, they say, follows the above so it may take either a singuar or a plural verb form which almost makes one wish one hadn't bothered to look it up, or spend the money on the style manuals.
It is summed up nicely on the following, https://www.quickanddirtytips.com/education/grammar/none-or-...
Regards,
Eolwaen
Eolwaen
About Numbers
I always liked the remark by Lord Kelvin (something like) "I you can't describe something with numbers, you don't know very much about it".
Hugs and Bright Blessings,
Renee
Numbers
Agreed. Providing a wavelength of say 7x10^-7m is considerably more precise than merely saying 'It's red.' which tells you it is somewhere between 6.25 and 7.4 x10^-7m. For many folk saying 'She's a ginger' is adequate, but as a ginger myself it just won't do!
I have never used prefixes like nm or mm. I use standard form. To me, prefixes are the domain of scientists, not mathematicians. The only exception for me is the Kg because it is the SI base unit which was derived from the MKS system, which can be a little idiosyncratic because it means a gram to a scientist should be a milli Kilogram! (I don't use gramme because it seems to be a British only word in this context) To me a gram is 10^-3 Kg. My argument , and there are plenty of mathematicians who would take issue with me, is that we spent millennia developing standard form and then the System Idiotique committee took us backwards with prefixes.
Regards,
Eolwaen
Eolwaen
Systems of measurement
We engineers use whatever is handy. But I agree with your assessment about the MKS versus the CGS system. Blame it on the people who defined the gram. In a practical sense, it's too small to go with the Meter. Not because of anything in nature, but because it's so small in human terms. Maybe they should rename the kilogram to something else like rock (as in, a fist sized rock.) (Not to be confused with the stone.)
What I have found is that people try to set up their units so that, in common use, you often see them at around 100, give or take. And people generally don't like working with decimals. For example, blood glucose concentration is measured in milligrams per deciliter. I mean, WTF? Who wants to goof with all those decimal points?
But for a healthy person, 70 mg/dl is kinda light, 100 is OK, 130 is OK for a while, 200 is too much.
Ever hear of a decibel? It's actually a comparison between two (generally power) levels. It was invented because our sight and hearing has a logarithmic response. It was named after Alexander Graham Bell. But the bel was too big and would require that we use those nasty decimal points, so they invented the decibel. 3 db Is a 2:1 ratio. 6 db Is a 4:1 ratio. 9 db Is an 8:1 ratio. (All are approximate.) 10 db Is a 10:1 ratio. (exactly) That's handier than talking about .3 bel, .6 bel, .9 bel, and 1 bel.
So it all comes down to working with what we had before (kilogram instead of fistrock or something like that,) and what the majority of us find handy (I mass 110 Kg, not 110000 g.)
As for prefixes, they definitely are necessary. I mean, who wants to have to count zeros when they are buying a 1649267442000 byte hard drive?
But it didn't used to be so hard. I mean, who though that anyone would ever need more than 655360 bytes of RAM? ;-)
Units
In the main I agree, Ray, but, and it is a big but from a purist mathematician's point of view, the convenience of everday units being sensibly sized has nothing to do with mathematicians. Applied mathematics has an often incestuous relationship with engineering, as it should. Discrete mathematics truly is discrete from the rest, but best not shout that too loudly, you know be discreet about it and statisticians, enough said because I thought we pure matheticians were in the main off our heads.
Decibel, pH, the Richter scale and star magnitudes are logarithmic because as you said so are the responses of human hearing, taste, touch or feeling and sight. As far as I' m aware there is no scale for smell. Not that I'd know, but from what I've been told if there were possibly it could be used in men's locker rooms to determine at what point they should be condemmed as a health hazzard?
Yes. You're talking sense, but I'm talking mathematics which has nothing at all to do with sense. I admit numbers play a very small part in the life of most working mathematicians, and I'm an old purist (pedant), though when I was very young I was using slugs, poundals, Abamps and their like (God bless the System Idiotique). I still stand by what I said about standard form and would remind you about your remark concerning 2 or 3 std devs from the norm because I have no problem working with it.
When young I had to lecture a group of year 1 mixed science undergrads in the mathematics they would need. I was the latest into the department so I got given all the stuff nobody else wanted. When I was teaching what they would need for thermodynamics, Gibbs free energy, enthalpy, entropy &c. it dawned on me one day that the reason why so many of them were making mistakes was most enthalpy changes were quoted in MJ/mol whilst most entropic changes were quoted in KJ/mol for the reasons you mentioned, they were sensibly sized numbers to deal with.
Trouble was the students were just adding or subtracting them without taking the unit prefix into account. I refused to mark anything not in std form. Problem solved, the students got the right answers. Yes I know I'm quoting one silly small example where my way helps, and there are many where it makes no difference, but it's not hard. Top end 12-13 year old have to learn it, and top end 13-14 year olds are expected to have mastered it.
Regards,
Eolwaen
Eolwaen
KJ
A killah jewel is one of those big things you wear in your navel. :) Now that's mega-bling!
Hugs,
Erin
= Give everyone the benefit of the doubt because certainty is a fragile thing that can be shattered by one overlooked fact.
It could be worse.
You could crash a multi million dollar interplanetary probe because you forgot to convert units.