Math: Numbers in other bases

Tutoring math, you rarely hear of this now.  For computer science students, the math tutor gives a five-minute read on it.

In everyday life, the number system we use is base ten.

The number 4062 means

4×1000 + 0x100 + 6×10 + 2×1

More specifically, it means

4×103+0x102+6×101+2×100

Recall, of course, that x0=1, x being any real number.

So, what is the number 57 in base 2?

Well, the powers of 2 are as as follows:

20=1

21=2

22=2×2=4

23=2x2x2=8

24=2x2x2x2=16

25=32

26=64

and so on.

57, written as a combination of powers of 2, can be thought of as

1×32 + 1×16 + 1×8 + 0x4 + 0x2 + 1×1

or, in other words,

1×25 + 1×24+1×23+0x22+0x21+1×20

Therefore, we have

57=111001 in base 2.

Exercise: Verify that 129, when converted into base 5, is 1004.

Solution: 129=1×125 + 0x25 + 0x5 + 4×1. Of course, 125=53, 25=52, and so on.

In computer science, the hexadecimal system – aka, base 16 – is often used. We’ll take a look at that in a future post:)

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

Area: size of a pizza

As a math tutor, you notice these little surprises about everyday things.

We all know that, with a circle,

Area=Πr2.

Years ago, I worked in a pizza place. A small was 10″ (diameter), while a large was 14″. Let’s compare the sizes.

First, we’ll agree that from an eating point of view, the size of a pizza is really its area, rather than its diameter.

For the small, the diameter being 10″, its radius is of course 5″. Its area is

A=Πr2=Π(5)2=Π(25)≈78.5 square inches

For the large, the diameter being 14″, its radius is 7″, so its area is

A=Πr2=Π(49)≈154 square inches

The ratio of the areas:

(1)   \begin{equation*}\frac{large}{small}=\frac{154}{78.5}=1.96\end{equation*}

I think we can agree that 1.96≈2.

Therefore, the 14 inch pizza is really twice the size of the 10 inch. You can check what kind of deal you’re getting on the large:)

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

Math: interest vs inflation

Tutoring math, interest and inflation do come up, though not often enough.  To all my financial readers:  hang on to your hats!

When I earlier talked about inflation and interest here, I mentioned the rule of thumb that, generally, interest hovers around 3% above inflation. A saver might believe they are protected by that three percent spread.

Assuming interest stays three percent above inflation, you are (more or less) protected while both inflation and interest are low. However, witness the outcome of the following scenario:

Example 1: You have $1000 in a savings account. Inflation is high: 16%. Interest, following the 3 percent rule, hangs at 19%. Calculate your balance at the end of the year, adjusted for inflation.

First, the interest:

Converting 19% to a decimal, you get 0.19, which you then multiply by the principal of $1000:

1000*0.19=190

Adding the interest at the end of the year, your $1000 becomes $1190.

Now, consider inflation:

To adjust for 16% inflation, your balance of $1190 is worth 84% of its nominal value. (100 – 16=84)

Convert 84% to its decimal of 0.84, then multiply it by your year end balance:

0.84*1190=999.60

Adjusted for inflation, your $1000 has become $999.60: you’ve lost value.

Where did the lost value go? After all, you didn’t withdraw it.

The answer is that the borrowers got it. In an environment of high inflation, value is gradually transferred from lenders to borrowers. The reason is that the borrower pays back the lender with money that is worth less and less.

Through the 70s, many people’s mortgage payments remained basically constant while prices ballooned. Specifically, for the people who didn’t need to refinance, the payment got smaller and smaller relative to the paycheque. For those lucky people, inflation picked up a lot of the tab.

Of course, such high interest and inflation rates are very hard to imagine today. People likely felt the same in 1965….

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

English: What is a homonym?

As an English tutor, I’ll share a definition I learned in elementary school.  I had no idea it was controversial.

I remember my spelling text from grade three – yes, it was a textbook.  Each week had a new list of words.  However, the book went further:  it covered a new theme each week as well.

One magic chapter talked about homonyms.  Homonyms, it explained, are words that sound the same but are spelled differently; dear and deer are homonyms, for example.

What was my surprise to discover, 35 years later, that the Yanks don’t necessarily agree with that definition!  Leafing through Websters yesterday, I read that homonyms are spelled the same, but have different meanings.  An example is screen, in the following two contexts:

1)  We’ll screen the applicants carefully.
2)  At the airport, you watch the screen for the arrivals.

That’s homonyms from a Yank point of view.

I wondered if maybe I remembered the definition wrong, so I checked the Oxford Canadian Dictionary.  Its definition comprises both the one I remember and the Websters one.

By the way:  I love the -nym ending:)

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

Math: Expected Value

Tutoring math, you might be asked about statistics, in which expected value is an early topic.

In many everyday situations, the expected value is equal to the mean, aka the average.  The difference is more in definition than in practice.  Expected value involves probability, whereas mean involves fact.

Formally, the expected value is defined as follows:

(1)   \begin{equation*}Expected\ Value=\sum(outcome)(probability\ of\ outcome)\end{equation*}

The Σ symbol means “sum”.  In other words, the expected value takes each possible outcome and multiplies it by its probability.  Then, it adds all those products together.

Example 1: Give the expected value of rolling a fair six-sided die.

Solution:  We know that the probability of getting each result is 1/6.

The expected value, E, of the die roll is

(2)   \begin{equation*}E=\frac{1}{6}(1)+\frac{1}{6}(2)+\frac{1}{6}(3)+\frac{1}{6}(4)+\frac{1}{6}(5)+\frac{1}{6}(6)=\frac{21}{6}=3.5\end{equation*}

Notice that the expected value is not necessarily a possible value. Not surprising, really: if you take the average height of ten people, you’ll likely arrive at a height that none of them is.

Example 2: Find your expected payoff in the following situation: 1000 people pay $1 each for a ticket. Then a number is drawn. The holder of the winning ticket gets $500.

Solution: The probability of winning is 1/1000, in which case you get $500. Really though, the payoff is only $499, since you had to spend $1 to buy the ticket. The probability of not winning is 999/1000, in which case you get nothing. Once again, you still had to pay $1 to lose, so the payoff is -$1. Applying our definition of expected value we get

(3)   \begin{equation*}E=\frac{999}{1000}(-1) +\frac{1}{1000}(499)=-0.999+0.499=-0.50\end{equation*}

So, your expected payoff is -$0.50. It had to be a loss, since not all the money paid in was awarded.

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

Exponential Growth: an interesting application

Tutoring math 12, exponential growth is “always on my mind.”

Years ago, I used to read the Economist.  Eventually I became too busy to keep up with it, but I always enjoyed it when I could.

One of the last articles I remember (this was maybe in ’05 or ’06), China’s economy was being compared with India’s.  At the time, China’s growth was 8%, while India’s was 6%. Either rate would signify wildfire growth in a developed economy; I’d say Canada will be lucky to grow at 2% this year. However, the article said that India’s growth, while a very nice 6%, melted in comparison with China’s 8%.

As a math tutor, I thought about that comment for a moment. “Is 8% really that much more than 6% growth?” I asked myself.

The key is that it’s exponential growth. This year’s growth becomes a part of next year’s economy, which then grows again, so you get growth on growth on growth. That’s exponential growth: anything natural grows that way. My earlier article here talks more about it.

Reading that earlier article, you’ll also encounter the law of 72, which states the following about an economic entity:

(growth rate)x(doubling time)=72.

It’s an approximation, but a very good one.

Let’s compare India’s historic growth at 6% with China’s at 8% using the law of 72. Does 8% really “melt” the 6%? Well, what we can say is that, by the law of 72, India’s economy will double every 12 years, while China’s will double every 9 years. For simplicity, let’s imagine the economies begin at the same size. In 36 years, India’s will double three times (every 12 years), so it will be 8 times its original size. (2x2x2=8). In that same period of 36 years, China’s will double four times (every 9 years), reaching 16 times original size. (2x2x2x2=16). If they were the same size at the beginning, China’s economy, having doubled an extra time, is exactly twice India’s at the end of the 36 years.
 
From that point of view, the difference in the growth rates is impressive.

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

Math: inflation and interest

Recalling the 70s and 80s, the math tutor proposes a nostalgic premise for your coffee break.

Inflation eats away at your money.  Right now, inflation is very low; it has been since the late 90s. However, in the 70s and 80s inflation was significant – and very high some years.

You can tell inflation is at work if prices are going up.  I remember chocolate bars, chips, and pop used to cost a quarter each when I was six.  A year later, they all cost 30 cents.

To calculate the inflation rate in the case above, we do this:

(1)   \begin{equation*}infation\ rate=\frac{change\ in\ price}{former\ price}\end{equation*}

In our case

(2)   \begin{equation*}inflation\ rate=\frac{5}{25}=0.20\end{equation*}

Of course, to change from a decimal to a percent we move the decimal point two jumps right:

(3)   \begin{equation*}0.20=20\%\end{equation*}

Of course, 20% inflation is very high; I don’t imagine inflation is even 2% right now.

From what I’ve read, interest rates generally run around 3% above inflation. That’s a historical trend, so it may not be the case at any given time.  However, if interest offered to savers is 6%, inflation is likely around 3%.

Let’s find the change in value of $1000 over a year at 6% interest with 3% inflation (we assume the $1000 is in a savings account):

(4)   \begin{equation*} Interest=principal*rate\end{equation*}

The principal is the amount you put in the account. Of course, the rate must be in decimal form. To go from percent to decimal, you shift the decimal point two places left (or else you can just divide by 100):

(5)   \begin{equation*} 6\%=0.06\end{equation*}

Now we can find our interest earned:

(6)   \begin{equation*}Interest=1000*0.06=60\end{equation*}

Before considering inflation, our $1000 has grown to $1060.

Now let’s witness the action of inflation. The real value of our $1060 at year end is as follows:

(7)   \begin{equation*}Real\ value=(1-inflation\ rate)*dollar\ amount\end{equation*}

Once again, the inflation rate must be in decimal form. We realize that 3% is 0.03 and proceed:

(8)   \begin{equation*}Real\ value=(1-0.03)*1060=0.97*1060=1028.20\end{equation*}

So the proceeds are $28.20 after inflation.

In a future post I’ll mention a surprising twist about the relationship between inflation and interest:)

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

Math: Factoring Easy Trinomials

As a math tutor, you teach and review this method constantly.

Back in May, I began a series of posts about factoring polynomials.  To refresh the topic, you can check here, here, and here.

Factoring polynomials is a make-or-break skill for high school students taking academic math. It encompasses about five techniques, of which easy trinomial factoring is probably the best known. Let’s have a quick look:

Example 1: Factor x2 -3x -28

Solution: Since the coefficient of x2 is 1 (which we know because there is no number written in front of it), we can use the easy trinomial method.

Step 1: Write (x      )(x     )

Step 2: After the x’s, write the numbers that will multiply to give -28, but add to give -3.

You have to do some mental math: 7×4=28, but one of the numbers has to be negative to give -28. The numbers must be -7 and +4, since -7+4=-3.

(x -7)(x +4)

The answer is (x – 7)(x + 4). You can verify using the foil method:

First: x*x=x2

Outer: x*4=4x

Inner: -7*x=-7x

Last: -7*4=-28 (remember: negative times positive gives negative)

Now, line up the four terms we just obtained:

x2 +4x -7x -28

We can combine the like terms: 4x – 7x = -3x

Finally we get

x2 -3x -28.

If you foil out your answer and get back the original trinomial, you know it’s right.

Example 2: Factor x2 + 5x + 4

Solution: The numbers that multiply to give 4 but add to give 5 are 1 and 4: 1*4=4, 1+4=5.

Therefore, the answer is (x + 4)(x + 1)

Example 3: Factor x2 -10x + 16

Solution: The numbers that multiply to give 16 but add to give -10 are -8 and -2 (recall that negative times negative gives positive): -8*-2=16, -8+-2=-10

The answer is (x – 8)(x – 2)

Example 4: Factor x2 +5x – 14

The answer is (x + 7)(x – 2)

Good luck with this method. Most people like it once they get used to it:)

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

French education: the challenge of exogamy

In BC, 75% of francophone families are exogamous.

Exogamy refers to the marriage of someone from a certain culture, to a spouse from outside that culture.  From the francophone perspective, an exogamous family has one parent with French as the mother tongue, while the other parent has a different mother tongue.

In the francophone education system, most students come from exogamous families. As homogeneous French families become increasingly rare in Canada, the survival of francophone education outside Quebec depends on the enrollment of children from exogamous families.

Many people wonder why they would send their child to francophone education when they could just send them to the English system.  The answer is that in Canada, children who are educated in French usually turn out to be better in English as well.  Most people accept without question that knowing a second language is advantageous, and that learning it from a young age – if possible – is the best way.

Surprisingly, a francophone parent will often speak English at home to their children.  At the same time, the exogamous parent (usually English-speaking) may be more serious about their children’s learning French – probably because it’s a great opportunity that the English parent never had themselves.

The challenge for the francophone schools is to devise a way to welcome the non-French parents of exogamous families, while still maintaining a French-speaking environment.  Such a solution will likely ensure the growth of French-English bilingualism outside Quebec.

Sources:

Rodrigue Landry, “The challenges of exogamy”

“English Information,” Conseil scolaire francophone de la Colombie Britannique