Math: proof that 1 ≠ 0

Tutoring math, you often hear the comment that math involves itself in silly pursuits.  The tutor gets the point, and apologizes.

Sometimes, non-math people will point out that math harbours absurdities. The infinite string of numbers with a finite sum (see my post here) is one example. Irrational numbers might also qualify (see my post here), if due to their name alone.

I’ve heard people joking that there’s a proof for the fact that 1 is not equal to 0. Who would need to prove something so obvious? Surely, those math people are just amusing themselves, while everyone else has to put up with the output.

Those sentiments are definitely understandable; I’ve even heard my own math professors say so. However, there are two mollifying considerations:

  1. The ideas that seem absurd are often taken out of context.
  2. Often, the most obvious ideas are the hardest to prove.

The fact that 1 is not equal to 0 is obviously true in our common number system. We all count on its truth. Yet, to discover further ideas about our numbers, we often have to prove what we already know. That’s why math people so often embark on apparently ridiculous errands.

For those who still want to see proof that 1 is not equal to 0, here’s my version:

I’ll use an indirect proof (see my post here about that): I’ll first suppose the opposite of what I want to prove.

Suppose 1 is equal to 0.

In our number system, a number is itself times 1. For example, 10=10(1).

In general terms, the number n is equal to n(1).

Now, if 1=0, then n(1)=n(0). Therefore, n=0.

It follows that if 1=0, then every number is equal to 0. However, we know that not every number is equal to 0. Therefore, 1 cannot equal 0.

No doubt, there are other ways to prove 1 is not equal to 0. Is it worth proving? In some circles, yes – but probably not for most people:)

Jack of Oracle Tutoring by Jack and Diane, Campbell River, BC.

Statistics: the meaning of mean and standard deviation

Tutoring math, you get asked about statistics.  The tutor offers an interpretation of mean and standard deviation.

Back when I was in university, there was a 2nd-year statistics course. It was the introductory stats course and was required for many degrees. Of course, being in math, I had to take it. However, I recall that business students, as well as social science students, commonly took it as well. I rarely had those people in my classes; that stats course was the exception.

I was told that the class had a 50 percent failure/dropout rate – the highest of all the courses at the university. It was even higher than calculus, which had a 40 percent non-completion rate. However, the stats course did have the enrollment of many students who weren’t math-oriented.

A common sentiment expressed by statistics students is that it’s hard to apply the ideas to everyday life. In a way, the feeling is ironic, since the first couple of courses in statistics – much more than most math courses – focus on everyday applications.

Introductory statistics focuses closely on the mean and the standard deviation of a population. The mean is another name for the average; most people understand it as the “expected” value. Consider mean height, for instance. If you imagine a person you can’t see, but want to guess their height, your best guess is the mean height of the population.

Standard deviation is harder to understand for most people. It’s the measure of how far apart the population’s values are – how spread out they are. Thinking of heights again: in a population with low standard deviation, people’s heights would mainly be close to the same. In a population with high standard deviation, the heights would likely be quite different from person to person.

I’ll be discussing more technical aspects of mean and standard deviation in coming posts. This, I hope, might be a good first step:)

Jack of Oracle Tutoring by Jack and Diane, Campbell River, BC.

Auto batteries, part two: cold cranking amps and internal resistance

The tutor continues to explore auto batteries (aka car batteries).  While this topic is rare during tutoring, it’s probably relevant to virtually everyone at some point….

 
I recall, decades ago, a friend of mine bought a new battery for his car. “It’s got 600 cold cranking amps,” he smiled. “No more doubts about starting.”

I soon learned the meaning of 600 cold cranking amps: his car started perfectly from then on. At the same time, he was impressed by the number because of its technical meaning; being a mechanic, it spoke to him in a more precise way.

Years went by: I finished my degree, then went back for (of all things) some mechanics training. Today, when events from the past pop in my head, I try to answer questions that I let go at the time.

So it is with the issue of “cold cranking amps”. The other day I suddenly realized: if a 12-volt battery is pushing 600 amps, its internal resistance must be less than 12/600=0.02Ω. The obvious question: do car batteries really have internal resistance that low?

I started searching the net. The answer was harder to find than I’d expected, but here are some numbers:

car battery resistance link
0.003Ω chiefdelphi.com
0.01Ω tap.iop.org
0.001Ω furryelephant.com

So I guess it’s true: 600 cold cranking amps – or even more – is possible, based on internal resistance alone.

I have yet to define “cold cranking amps”; I will do so in a coming post. Cheers:)

Jack of Oracle Tutoring by Jack and Diane, Campbell River, BC.

Perl: repeated text

Tutoring Perl programming, you are aware of so many little tricks Perl contains for doing specific tasks.  The tutor shares a fun one.

 
I recall, when I was a kid, a common trick to show off was to fill up the computer screen with your name. A kid with a little programming instruction could easily do it; a kid with no knowledge of programming, had no hope of pulling it off.

At the time, I had no knowledge of programming. However, I saw a kid do the trick on a Commodore Vic-20 (I’m really showing my age here). For years I wondered how he did it.

The kid likely did it in BASIC; Perl wasn’t even invented then. However, I’ll show how it’s done in Perl:

#!/usr/bin/perl
$name=”Johnny Smith! ” x100;
# the x100 means repeat the name 100 times
print “$name”;

Of course, you could replace x100 with x500 or whatever. Don’t overdo it; you just want to fill the screen, right?:)

I might start coverage of BASIC; it’s a fun language. If I recall correctly, I think it uses dollar signs in front of variables the way Perl does. It’s been awhile.

Source: McGrath, Mike. Perl in easy steps. Warwickshire, UK: Computer Step, 2004.

Jack of Oracle Tutoring by Jack and Diane, Campbell River, BC.

Auto batteries: seasonal reflections

Tutoring high school chemistry includes electrochemistry.  The tutor shares some reflections about auto batteries as winter approaches.

 
I recall more than one mechanic talking about how a device might work fine for a long time but have a hidden weakness. When that device is, for whatever reason, put under abnormal stress, it likely fails. The failure is surprising: hasn’t it worked for months (or years) with no problems? Why, then, does it suddenly fail, at a time you really need it to work?

From my experience, auto batteries can give that “false sense of security.” It’s not the battery’s fault, of course; it’s just the life cycle in most of North America (ie, summer to winter).

From purchase, an auto battery of good quality is likely strong and reliable for a few years anyway. Depending on driving habits, it may maintain its vigor much longer than that. However, time is working against the battery: potentially, the chemical process of sulphation, among other factors.

So, the battery likely weakens over time, yet continues starting the car just as expected. Let’s imagine the battery becomes considerably weaker in late May. The driver likely won’t even notice: from late spring through early fall, the weather is warm. In the heat, the car’s oil might be more agreeable to letting the engine turn over. The days are bright, warm, and dry: the driver doesn’t use the headlights, heater, or wipers as much. Life is easy for the battery. Yet, during those carefree months, the battery may already be too weak to start the car in the cold.

When the inevitable “first winter storm” comes, the car may not start. The driver is surprised. (I’ve been there.) Really, though, the driver’s been on borrowed time for weeks or months already. The battery just wouldn’t reveal its weakness until put under stress.

I’ve been reading up on auto batteries lately (just for the pictures, of course:). In coming posts I’ll discuss some of my findings about this fascinating topic we all depend on.

Jack of Oracle Tutoring by Jack and Diane, Campbell River, BC.

Math: simplifying fractions with radicals

Tutoring high school math, you work with radicals.  The tutor discusses simplifying fractions with radicals.

I’ve written several articles about working with radicals; you’ll find them by keying “radicals” in the search box. Ones that might be helpful towards this one are here, here, and here.

Example: Simplify √(28)/√(32)

Solution: First, we use the rule

√(a)/√(b)=√(a/b)

which leads, in our case to

√(28/32)

Now, the fraction can be reduced on the inside:

√(28/32)=√(7/8)

Now we can back out again:

√(7/8)=√(7)/√(8)

Now, from my article Simplifying Radicals…part 1, we know that

√(8)=√(4)√(2)=2√(2)

So we have

√(7)/√(8)=√(7)/(2√(2))

Next, the denominator needs to be rationalized (see my article here):

√(7)/(2√(2))*√(2)/√(2)

=(√(7)*√(2))/(2√(2)√(2))

=√(14)/4

Apparently, √(28)/√(32) simplifies to √(14)/4

Radicals are posed in seemingly endless combinations during high school math. They are part of the daily diet in calculus as well. Therefore, I’ll be exploring further examples with radicals in future posts:)

Jack of Oracle Tutoring by Jack and Diane, Campbell River, BC.

Math: the CASIO fx-991ES equation solver

During his math degree, the tutor used a minimal TI scientific. Nowadays, during tutoring, he sees so many calculators with amazing features….

The Casio fx-991ES NATURAL DISPLAY is, in my opinion, comparable to a Sharp WriteView. Their screens are deeper than most scientifics, so that they can display fractions up-and-down. They can also simplify radicals symbolically, like you would by hand: √(40) will display as 2√(10) (in the right mode). Moreover, they can display radian measures symbolically, using pi: sin^(-1)(0.5) will display as π/6 (in rad mode). Sharp summarizes these features as WriteView; Casio describes them as NATURAL DISPLAY.

Those features I will explore in other posts; today I’m showing, with an example, how one might use the equation solver built into the Casio fx-991ES. It’s surprisingly powerful, and surprisingly easy to use:

Example: Solve 11xe^(-0.44x)-1.2sinx + 5 =0

This is likely not an equation most people want to tackle by hand. Here’s how to do it with the Casio fx-991ES:

Step 1: I was in COMP mode to use the solver. That’s MODE 1. Also, I was in rads(Shift Mode 4), which is important: sinx is in the equation.

Step 2: You need to enter the equation as Y=11Xe-0.44X-1.2sinX+5. Important: to enter the = in the equation, use alpha calc. Use the alpha switch to enter the Y and the X.

Step 3: After you’ve entered the equation, key in shift calc.

Step 4: Y? should now appear. Key in 0, then regular =.

Step 5: Solve for X should now appear. There may already be a number across the bottom; regardless, press regular =.

Step 6: The screen may go blank for a few seconds while the solver works. Soon, though, you’ll be greeted with the solution (if the equation was, indeed, solvable:)). There will be three lines:

  1. across the top you’ll see your equation
  2. in the middle, you’ll see X=-0.415296339 (for this equation, anyway)
  3. across the bottom you’ll see L-R=0 (which, I believe, means “left” minus “right”; it shows how close the solution matches the two sides of the equation).

To double check, I fed the same equation to the solver on my old Sharp EL-520W; it agrees exactly with the Casio’s solution:)

Have fun experimenting!

Jack of Oracle Tutoring by Jack and Diane, Campbell River, BC.

Mass midpoint: a primer towards centre of gravity

Tutoring math, it’s important to remember the “everyday” side of the subject. With mass midpoint, the tutor begins towards the topic of centre of gravity.

Centre of gravity is a concept you oft hear mentioned. Most athletes know that, with a lower centre of gravity, keeping balance is easier. Similarly with vehicles: one with a higher centre of gravity is more likely to tip over going around a hard corner. Of course, that’s the reason sports cars are built so low to the ground.

For our first example moving towards finding the centre of gravity, we’re not going to find the centre of gravity itself. Rather, we’ll find the “halfway point” of the mass of a square pyramid. Since that point is not, technically, the centre of gravity, I’ve given it the name “mass midpoint”.

A pyramid is not the same top to bottom; therefore, its mass midpoint is not simply at half height. We’ll make the assumption that the pyramid has uniform density. Thence, we find the point from which there is equal volume side to side, front to back, and top to bottom:

Example: Find the mass midpoint of a square-based pyramid with base side length 10m and height 12m.

Solution: By symmetry, the mass midpoint is somewhere directly above the centre point of the square base. Let’s introduce three dimensional coordinates of (forward, across, up); aka, (x, y, z). Then the location of the pyramid’s mass midpoint will be (5, 5, z). We need to find z, as follows:

The formula for volume of a square-based pyramid is

V=w2h/3, w=width, h=height

We seek the specific height, γ, at which the volume below is the same as the volume above. For convenience, γ will be measured from the top down, rather than from the bottom up. (Reason: when measured from top to bottom, the height is always 6/5 the width; conversely, the width is 5/6 the height.)

If at height γ (measured from the top) the volume above equals the volume below, then

Vtotal – (5γ/6)2γ/3 = (5γ/6)2γ/3

We can add (5γ/6)2γ/3 to both sides:

Vtotal = 2*(5γ/6)2γ/3

Therefore,

102*12/3 = 2*(5γ/6)2γ/3

which leads to

400=50γ3/108

Multiplying both sides by 108, then dividing both sides by 50, gives

864=γ3

so

γ=9.524m

Therefore, the height of the mass midpoint, from the bottom, is

12m-9.524m=2.476m

So it appears that the mass midpoint of this pyramid, using the (forward, across, up) coordinate system, is at (5, 5, 2.476)m.

This post is meant as a first step towards exploring centre of gravity. In the case of this pyramid, its centre of gravity is not the same location as its mass midpoint. Why the two are different, and where the centre of gravity indeed is, are topics for future posts – likewise for the concept of three dimensional coordinates:)

Jack of Oracle Tutoring by Jack and Diane, Campbell River, BC.

Perl programming: logical comparisons: equals and string equals

During tutoring, you often need to point out the unexpected details of a language. The tutor brings up a point that is relevant to Perl, but not only Perl….

Back in my October 4 post, I used a Perl construct that I will explain here now: specifically, the eq operator. There is also potential connection between today’s post and my one from October 2.

In Perl, you can compare values for equality (using the double equals comparison operator, ==). For example:

#!/usr/bin/perl

$joe=$ARGV[0];

$rob=$ARGV[1];

if($rob==$joe){

print “$rob equals $joe\n\n”;
}

else{

print “$rob not equal to $joe\n\n”;
}

Let’s imagine test.txt is the above program’s name. If it is run from the command line thus:

perl test.txt 4 4

you’ll likely get the output

4 equals 4.

If, on the other hand, you run it with this command:

perl test.txt 4 5

you’ll likely receive the response

5 not equal to 4

So far, all seems pretty straightforward. However, let’s run the program again, with this command:

perl test.txt coffee tea

You’ll likely be greeted with the somewhat controversial statement

tea equals coffee

In Perl, variables that are words (as opposed to numbers) are called strings. A string is written in quotes, such as “coffee”. A string definition might look like

$var1=”coffee”;

Strings, if defined, have a numerical value of “not zero”. The == operator is a numerical comparison; from its point of view, “coffee” and “tea” are equal, both being “not zero”.

To compare “coffee” and “tea” as strings, you need to use the operator eq rather than ==, as follows:

if($rob eq $joe)

If we change the program above, replacing

if($rob == $joe)

with

if($rob eq $joe)

it should be able to distinguish between “coffee” and “tea”. If we run it like this:

perl test.txt coffee tea

We should now receive the reassuring response

tea not equal to coffee

Various computer languages treat strings and comparisons in possibly different ways. The important point is that a given language might have its own, unexpected way of handling them. Being aware of that possibility, the programmer is better prepared:)

Source: Robert’s Perl Tutorial

Jack of Oracle Tutoring by Jack and Diane, Campbell River, BC.

Equivalent annual rate, continued: finding it on the TI BA-35

Tutoring high school math, you don’t always see financial calculators; more often, you might see the TI-83 grapher.  The tutor recalls the TI BA-35’s use in accounting courses….

The TI BA-35 is a classic among financial calculators.  It’s got that minimal design Texas Instruments so excels with when they want to.  Frankly, the key sequences are sometimes so concise, it’s surprising. The way you calculate the equivalent annual rate (see my previous post for clarification) on the TI BA-35 might be an example.

Let’s take the premise of my previous post: 3% interest, compounded monthly. Of course, it’s called 3%; that’s the nominal rate. Let’s find the equivalent annual rate, using the TI BA-35:

step 1: Make sure you’re in FIN mode. It should say FIN at the bottom of the screen. If it doesn’t, press 2nd N.

step 2: Enter 12 2nd PV 3 = .

step 3: The screen should say 3.0415952: for 3% compounded monthly, the equivalent annual rate is 3.0415952%.

Notice this answer agrees, to six decimal places, with the one from my previous post.

The BA-35 echoes the efficiency of calculators back in the 80s. They were sometimes less intuitive than what you might find today; the user had to learn how the calculator expected the inputs.

I’ll be talking more about the TI BA-35 in future posts:)

Jack of Oracle Tutoring by Jack and Diane, Campbell River, BC.