The primes less than 100 are as follows:

2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59 61 67 71 73 79 83 89 97

There doesn’t appear to be a pattern in the distribution of primes.

How about the "gap" (spacing) between the primes? Is there a pattern in that?

1 2 2 4 2 4 2 4 6 2 6 4 2 4 6 6 2 6 4 2 6 4 6 8

There doesn’t appear to be a pattern in the gaps, either.

Spiraling

Stanislaw Ulam was a Polish-American mathematician who was involved in the Manhattan Project during World War II.

One day he was bored in a meeting and began to write numbers in a spiral. He started like this, moving in a clockwise direction.

The next round continued the "spiraling" pattern, as follows.

He kept going (it must have been a long meeting), then highlighted the prime numbers and found something interesting.

Many of the primes appear to line up when arranged in such a sprial.

Let’s go much bigger and see what happens. We observe there are many places where the primes form line segments, mostly at 45°, but sometimes horizontal and vertical.

You can see an even bigger display, courtesy of Prime Curios.

What I found interesting in the large picture is where primes are not – there are distinct blocks and patterns of white space where no primes occur.

This spiral appeared on the cover of Scientific American in March 1964 and continues to generate research interest to this day.

Check out more such prime number information in the book Prime Curios! The Dictionary of Prime Number Trivia  by Chris Caldwell and G. L. Honaker, Jr.

Why do we care about primes?

Apart from many other things, prime numbers are vital in the development of encryption algorithms, used in generating secure Internet transactions.

Related posts:

1. Largest known prime: 2^43,112,609 – 1
2. Friday Math Movie – Math Illuminated
3. Message to G8 Environment Ministers

But even in the traditionally high-interest rate countries (like Australia and New Zealand), banks lend at around 7 to 8% currently, while in much of the world, interest rates are at, or very near, 0%.

So it’s extraordinary (to me) how banks are allowed to charge such whopping amounts on credit card debt.

Here’s an excerpt from Citibank Singapore’s credit card fine print (“RC” stands for “Ready Credit”).

I’ve highlighted some of the charges you’d be up for if you are (heaven forbid) just 1 day late with your payment.

Additional Information Box for Credit Card & RC

The kicker is they are allowed to vary the conditions as they go along.

And did you know if you have loans with other institutions, and you default with them, you can be charged a higher interest rate on this card?

The high interest rates are justified by the banks on the grounds they are lending to people who pose some risk. This is certainly the case in the US where it is still easy to get a credit card. In the Singapore case, there are income limits (normally S$30,000 per year fro locals and $60000 per year for foreigners) before you can apply for a card.

There are thousands of people in the US who are now owing more than they borrowed (using their credit cards) and the interest (and fees) are growing so fast, it’s sending them bankrupt, especially as the Global Financial Crisis results in job losses.

Oh, and what was the main cause of the financial crisis? Lending money to people who couldn’t afford it, and didn’t understand the risks, of course.

It used to be simple. Banks could only lend to people who had the means to pay it off and could secure the loan via some mortgage (this means if you stopped paying, the banks could sell your house.) It reduced excessive and risky lending. And I doubt we’ll see that situation again anytime soon.

To be fair, it’s not just Citibank. All credit card providers do this because they can, giving us a US economy with around 1 trillion in credit card debt.

See Frontline’s The Card Game, from PBS.

Be careful what you sign.

Related posts:

1. Online banking usability
2. Another misleading credit card advertisement
3. Financial literacy for students

The Gini Coefficient is one way to measure how evenly the income (or wealth) is distributed throughout a country.

The Gini Coefficient is calculated as follows. We find out the income of all the people in a country and then express this information as a cumulative percentage of people against the cumulative share of income earned. This gives us a Lorenz Curve which typically looks something like the following.

Image Credit: Wikipedia

In plain English, the graph above indicates the proportion of the income going to the poorest people, middle-income people and richest people.

There will always be rich and poor, but we are interested in how evenly wealth is distributed and most governments put effort into keeping this coefficient as low as possible.

The Gini Coefficient ranges between 0 and 1 (or it can also be expressed as a number from 0 to 100) and is given by the ratio of the areas:

If A = 0, it means the Lorenz Curve is actually the Line of Equality. In this case, the Gini Coefficient is 0 and it means there is "perfect" distribution of income (everyone earns the same amount).

If A is a very large area (making B very small), then the Gini Coefficient is large (almost 1) and it means there is very uneven distribution of income. Countries with a high Gini Coefficient are more likely to become unstable, since there is a large mass of poor people who are jealous of the small number of rich people.

What Does it Mean?

Let’s try to understand the above graph.

For example, say we have 10 people in a village and the income for the village is $100 per day. If every person shares this income evenly, they get $10 each per day.

So the income distribution would be as follows. ("Cumulative" just means add up the number you have so far for each step.)

So for this society with perfectly-distributed income, we could draw a graph of the cumulative proportuion of population (on the horizontal axis) against the cumulative percentage of income (on the vertical axis) as follows.

In the above case, A = 0 so the Gini Coefficient is 0.

Now, people being people, some of the villagers decide they should be paid more because they work harder, or because they are older, or because they have more children, or whatever. So three of them (persons H, I and J) decide to keep 15% of the income each, and distribute the rest evenly among the others. However, that doesn’t work out evenly, so they decide the laziest 3 people in the village (persons A, B and C) should only get 5% of the income. Our table now looks like this:

Let’s graph it and see what it looks like.

In summary, the bottom 30% of the population earns 15% of the income, while the top 30% earns 45% of the income.

I’ve shaded 2 regions in the above graph, region A (with light magenta shading) and region B (with light green shading).

Recall the Gini Coefficient is the ratio of the areas:

Area A = 0.095 (from calculating area B – one triangle and 2 trapezoids – and subtracting it from 0.5)

Area (A + B) = 0.5 (this is half of the rectangle)

So the Gini Coefficient in this case is:

Let’s take it another step. The three richer guys (H, I and J) have a fight and J wins. He demands 50% of the income and leaves it to H and I to distribute the rest.

Then H and I have a fight and I wins. He wants 33% and gives 10% to H and they decide to give what’s left (1% or $1 a day) to each of the rest of the village.

(Millions of people live on less than $1 per day.)

Now we have a very uneven income distribution. The bottom 70% of the population earn only 7% of the income, while the top 30% earn 93% of the income.

Here’s the graph.

The Gini Coefficient for this situation is very high:

Finally, let’s take the extreme case, where “Person J” becomes a dictator and decides all the income should go to him and everyone else gets nothing.

The cumulative income is 0% for Persons A to I, then it jumps up to 100% for Person J. Here’s the graph.

This time area A is very large and the Gini Coefficient is:

Why isn’t it equal to 1?

The highest possible Gini Coefficient is 1 and this implies 1 person gets all the income.

In our story, we only have 10 people in our example population. If there were, say, 100 million people in the country, and one person had all the income, then the Gini Coefficient would be 0.999999, or very close to 1.

Using Calculus to find the Gini Coefficient

The above story is simplified and with a large data set, the Lorenz Curve will appear to be a curve, not a series of straight lines.

This time I have modeled the Lorenz curve using:

Cumulative share of income = (cumulative share of people)⁵

If we use I (for income) and P (for people), this would be written

I = P⁵

We find the area A using the following:

This gives:

So the Gini Coefficient in this case is very high, at:

Gini Coefficients in Various Countries

These are sorted highest to lowest.

China’s coefficient is quite high and this is causing a lot of concern. The Eastern provinces are now well-developed and responsible for most of the income growth, whereas the rural west is still quite poor.

You can see the full list here: Gini Coefficient by Country.

Singapore’s Case

Here’s the coefficient for Singapore over the last decade. The rapid rise from 2002 and spike in 2007 weres due to several factors, including rapid population increases (through immigration) of higher-income people, and a subsequent boost in the overall economy.

The drop in 2008 and 2009 is due to the Global Financial Crisis, where many high-paying jobs either disappeared, or bonuses were slashed.

Information source for graph: Straits Times

Further Reading

There are plenty of interesting links on this topic in Wikipedia.

You may also be interested in: Database of Happiness.

Related posts:

1. IntMath Newsletter: Functions, Gini Coefficient, math anxiety
2. Database of Happiness
3. Singapore’s population bubble

Yousuf, one of my regular correspondents, got stuck on the following problem recently.

What is the area of the rectangle ADEB shown in the diagram?

The curve is the graph of y = 1/x² (for positive x), and r is some arbitrary value of x.

We’ll come back to this question a little later. I suspect his problem with this question was due to a rusty conceptual understanding of functions.

Functions Overview

A function is simply an expression involving variable(s).

We usually write a function of the variable x using the notation: f(x). A function has at most 1 value for each value of x.

For example, if f(x) = 5x² + 3, we can find the value of the function if we choose x = 0 as follows.

f(0) = 5(0)² + 3 = 5 × 0 + 3 = 3

Now, this is a good example of the notation problem I was talking about at the beginning. We write "f(0)" (f bracket 0 bracket) to mean "evaluate the function expression by substuting 0 every time we see an x" and we see this on the left hand side of this equation.

But on the right hand side, I have written "5(0)²" (5 bracket 0 bracket squared) and this means "5 × 0²". We need to be careful with this – writing 2 different concepts with what is essentially the same notation.

It is a shame that function notation is so clumsy and causes problems for newbies.

Let’s look at some more examples for our function f(x) = 5x² + 3.

f(2) = 5(2)² + 3 = 5 × 4 + 3 = 23.

f(10) = 5(10)² + 3 = 5 × 100 + 3 = 503.

If we were to substitute many more values of x and plot the dots on a graph, we would get the following:

Note: On the vertical axis, I put f(x), but I could have also put "y", since the convention in math is the vertical axis represents the function value. Often you’ll see it written y = f(x).

OK so far?

Now, let’s make things a bit more interesting. What is f(a)? We just substitute a everywhere there is an x in the original function, like we did before with the numbers:

f(a) = 5(a)² + 3 = 5a² + 3

Let’s do another. In this next case, f(a + 4), we are just replacing each x in the original function expression with a + 4.

f(a + 4) = 5(a + 4)² + 3 = 5(a² + 8a + 16) + 3 = 5a² + 40a + 83

Of course, we need to be careful to expand out the brackets properly!

A Different Function

Let’s change our function to .

This is the curve we met in the question at the the beginning of this article.

If x = 1, we replace every x in our expression with 1 and we have:

What f(1) means on a graph is the distance from the x-axis to the graph is 1 unit. The function value is the height of the graph for that x-value.

Now let’s do f(3a).

The value

represents the height of the graph when x = 3a. We need to be careful with the brackets.

Back to Our Problem

Here’s the graph again:

So how do we find the area of the rectangle BADE? The width of the rectangle is quite straightforward, as the distance from r to r − 1 is just 1 unit. But we need to find the height AD.

AD is just the function value f(r):

So the area of the rectangle is just

Area = width × height =

What if we needed the height BC?

We would just find the function value as follows.

BC =

Functions of 2 Variables

The functions above only have one variable and they describe a curve in 2-D space.

To describe a 3-D surface, we need to use 2 variables.

We write a function of 2 variables using this notation:

z = f(x,y)

The "z" indicates the height of the surface for particular values of x and y.

An example of a 3-dimensional surface is z = x² + 3 sin y.

More Information

See this chapter for a lot more examples of functions: Functions and Graphs. (2 dimensional)

This is an introduction to 3-dimensional Coordinate System.

See also Towards more meaningful math notation where I suggest an alternative to the current confusion.

Related posts:

1. Partial differentiation – what is it about?
2. 3D Grapher with contour plot
3. Towards more meaningful math notation

Today’s math movie is from Population Reference Bureau and features Carl Haub, trying to address 3 myths about population growth.

Myth 1: There was no population growth problem in the 1960s and 1970s.
Myth 2: The world’s population is not growing and today’s problem is low birth rates.
Myth 3: Europe will be predominately Muslim by 2050.

His delivery is somewhat bumbling and a little pompous (and leads me to wonder what his agenda is), but the issues are important.

You may also be interested in: World Population – Live Update.

Related posts:

1. Birth dearth?
2. Singapore’s population bubble
3. Friday Math Movie – PeakWater

Say you are on a desert island and you know there are 3 girls of marriageable age on the island. You will meet them one-by-one and you need to choose the “best” one to marry.

If you propose to the first one you meet, there is a chance she is not the “best” one and you won’t have made the best choice.

Or, you could reject that first girl and see if the second one is better. If so you could propose to her, or perhaps reject her, hoping the last one is the best out of the 3. (Any girl you reject will never talk to you again.)

What is the best approach to maximize your chances of proposing to the “best” girl?

The following article analyzes the probability behind this problem. It’s rather tongue-in-cheek (meaning it is based on reasonable assumptions, but may not be the best way to choose your mate!)

The Marriage Problem: How to Choose?

[This is from the excellent Parabola magazine, for "secondary schools", by University of New South Wales Mathematics and Statistics department.]

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2. Camera purchase decisions – how math helps
3. Friday Math Movie – Mixed Feelings (Vision through the tongue)

In the SMH article Shania Twain more beautiful than Angelina Jolie? we read the researchers manipulated the photos of certain women using Photoshop so the facial proportions were changed. Then warm-blooded males were asked to choose the photos that were most appealing. It turns out the "ideal" value (the one found to be more beautiful) for the following facial ratio was 36%:

Another key ratio affecting our perception of beauty is the following:

An ideal value for this ratio is 46%.

Apparently, the canadian singer Shania Twain has pretty much the perfect face. Let’s check out her proportions. In this image, the eye to mouth distance is 72px, marking 36% of the facial height (200px).

While the distance from her hairline to the bottom of her chin is quite easy to determine, and the center of the eyes is also quite clear, there is some uncertainty about the "mouth". In the above photo, she is smiling. Even with the mouth closed, it’s not that exact where the "mouth" is.

Next, we look at the ratio of the distance betwen the eyes to facial width. The ratio 72:156 is equivalent to 46%, as claimed in the research.

Once again, there is some ambiguity. What do we take as the "side" of the face? The article says the face width is the distance "between the inner edges of the ears". But it is rather subjective deciding quite where that comes.

A New Golden Ratio?

It is generally believed we are attracted to people whose physical features are symmetrical (a mirror image down the center) and in proportion. That proportion often turns out to be connected with the "Golden Ratio" (Φ = 1.618033…)

The above research claims to be a "new Golden Ratio" for determining beauty.

Comparison with Stephen Marquardt’s Research

Stephen Marquardt has conducted similar stuidies to find the "beautiful" face. He has created a mask which can be placed over different faces to determine whether they fit the classic notions of beauty (at least from a mathematical point of view). The mask is based on the Golden Ratio (or more specifically, the "golden decagon".

Let’s use his mask and measure the same ratios as used by the canadian/US researchers.

For the mouth-eyes to facial height ratio, we get 148:390 = 37.95%. This is greater than 36%, and possibly significant.

Now we look at the other ratio, eye distance to face width.

The ratio 138:279 = 49.5%. Once again, this is larger than the more recent research above, but it’s not clear where the "inner edge of the ear" is for the Marquardt mask.

Conclusion

While the above is interesting mathematically, I wouldn’t take it too seriously…

[Image sources:
Shania Twain
Marquardt mask]

Related posts:

1. Is Phi a Fibonacci furphy?
2. IntMath Newsletter: Functions, Gini Coefficient, math anxiety
3. Laughing scale

(Others include “pi” π = 3.141592653…, which is the circumference of any circle divided by its diameter; and “phi” φ = 1.6180339887…, which is the so-called “beauty ratio“). Both of these numbers are irrational (that is, their decimals go on forever and never repeat).

e is also an irrational number and it has value:

e = 2.718281828459…

The number e was “discovered” by several mathematicians (Oughtred, Huygens, Jacob Bernoulli, Mercator and Leibniz) but they didn’t quite know they had stumbled on it and didn’t know its significance.

There are some curious properties of e, one of which is that it’s the limiting value as n → ∞ of (1 + ¹/_n)ⁿ.

It can also be found by adding the infinite sum:

e = 1 + ¹/_1! + ¹/_2! + ¹/_3! + …

So what is e good for?

It is used extensively in logarithms (which was the only way to do difficult calculations for hundreds of years before calculators came along), exponential growth (of populations, money or drug concentrations over time) and complex numbers (which were used to design the computer or mobile device you are reading this on).

So happy “e” day (February 7th, or 2/7).

[For more information on e, see the MacTutor history.]

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1. My infinity’s bigger than yours
2. Dinosaur Mathematics…
3. Nursing Entrance Test – for mathematicians or nurses?

Let’s have a look at some of the available math graphing tools.

Graphics Calculators

Here are the Texas Instruments TI-83 (left) and Hewlett-Packard HP 40gs (right) calculators.

Graphics calculators are handy, but the screen size is small and there is usually no scale on the axes. Cost is often prohibitive, at around US$100.

Online Graphers

Here’s a few places where you can graph your curve – for free!

The first one is on IntMath.com, and uses Scalable Vector Graphics (you need Firefox browser, or a plugin for IE).

• Plot your own SVG Math Graphs

The following are also free offerings. They are either Flash or Java applets.

• Wolfram|Alpha
• Flash Equation Grapher
• GraphSketch.com
• GraphApplet (warning – red text on a black background)
• GCalc

Computer Applications

Here are some of my favorites. In each case you need to download and install the software.

• GeoGebra (Free. I wrote a review of GeoGebra which shows how to get started with it.)
• Graph 4.3 (Free. Small download, from cnet.com)
• Scientific Notebook (US$90, but does much more than a graphics calculator)

Some problems with using graphics software

If you don’t have a good idea of what a function should look like before your use a graphics package, you can have all sorts of problems.

Here’s an example of a function which catches out the unaware.

Let’s graph it on Geogebra.

Hmmm – it appears to be empty. Is there an error? What’s going on?

We zoom out a few times and start to see 2 curves. Once again, is there a mistake? Why 2 curves?

I zoomed out because I had a good idea in my head what the graph should look like. Since I couldn’t see either of the 2 arms of this curve, and because of the 30 in the denominator, I knew I had to zoom out.

If I left my graph at that, I would still not have a good idea of what the function looks like. I have not chosen a view that shows the crucial features of this graph.

The default view in Geogebra was too close to the origin (0, 0) to see any of the curve.

Let’s have another go, this time using Scientific Notebook. This is what I get when I try to graph the curve.

I get a vertical line passing through 2 on the x-axis. But notice the scale on the y-axis. The number "4e+09" actually means 4 × 10⁹ or 4 billion. So Scientific Notebook recognizes that there are very large values of y involved in this function, and has shown us the limits of its internal coding.

This time if we zoom in, we can start to see the graph appearing as before. But I need to know to zoom in, otherwise I would miss it altogether.

What was that vertical line that appeared before? Was it a mistake? Why did it disappear when we zoomed in?

Next, we use my SVG Grapher. It’s similar to Geogebra in that its default view misses the curves, but is also similar to Scientific Notebook in that it shows a vertical line through 2 on the x-axis.

When we zoom out a bit, this is what we get.

We are still not showing all the vital features of the graph, and that vertical line is still there. What’s that about?

Next, I tried the Flash Math Grapher. Once again I needed to do some zooming. The x- and y- scales are strange (multiples of 13.01??, and is that the x-axis or the line y = -2.98?) , but at least I can get a pretty good view.

Next, let’s call in the big guns. Wolfram|Alpha gives us the following 2 graphs when we put our function in their search box (and they give us a lot more information about the function).

This time we get intelligent graphs that have appropriate x- and y- scales and actually show the curves. Note the first one does not have a vertical line through 2, but the second one does. Why?

That vertical line through x = 2

When the software draws the graph, it chooses regularly-spaced x-values and substitutes them into the function, plots the resulting dots and joins them.

In the above examples that have a vertical line, it means the software has chosen a value just slightly less than 2 (which will give a very small, negative value for y) and another one just slightly more than 2 (which gives a very large positive value for y).

Here’s an exaggerated version of what it’s doing (using only a very few data points joined by straight lines).

If the x-value chosen is exactly 2, the software will normally just skip that value (since it will return a "division by 0" error). The (almost) vertical line should not be included in the graph, since we cannot have x = 2 (this function is undefined for x = 2 since the bottom of the fraction would be zero).

This gap in the graph is called a discontinuity. It should be a gap – not joined by a line as above.

Some software handles this situation gracefully (like the Wolfram|Alpha example) while in others, you need to either understand why the vertical line is there, or in some cases, you can elect to include discontinuities or not.

Best answer

Here’s probably the best way to display the graph of this function.

Our graph has 2 asymptotes. When a curve gets closer and closer to a line but does not touch it, that line is called an asymptote

The first asymptote is the x-axis and the other is the vertical line, x = 2, which I drew using a dashed line of a different color (since it is not part of the graph).

I have shown the 2 asymptotes clearly and I have also labeled the x- and y-axes.

Here’s the function again for convenience.

I knew the graph was going to involve asymptotes since the x-variable is in the denominator (and we can’t have 0 in the denominator) and also considering when x gets really big, the value of the function will be really small.

If I didn’t know this (from graphing many of them on paper), I would have made a mess of graphing my function on a computer.

Graphing software and the future

Will graphing software change what we do in classrooms? Should it? Is it really necessary to sketch graphs on paper still?

If you can draw a quick sketch of a function on paper, it certainly helps your understanding for many types of math problems. The conclusion from above is that it is certainly worthwhile to have a good sense of what a graph should look like before graphing it using software, so we can manipulate the settings to show the graph properly.

Related posts:

1. Free math software downloads
2. GeoGebra math software – a review
3. GrafEq math graphing software

Preparing a syringe.

The process of pharmacokinetics has 5 steps:

• Liberation – the drug is released from the formulation.
• Absorption – the drug enters the body.
• Distribution – the drug disperses throughout the body
• Metabolism – the drug is broken down by the body.
• Excretion – the drug is eliminated from the body.

Of course, each drug needs to act on the body in a different way. Some drugs need to be absorbed quickly (like nitroglycerin if we are having a heart attack) and preferably eliminated quickly (otherwise toxins build up in the blood). For other drugs, we want slow absorption so we get maximum benefit and don’t lose a lot of it from excretion.

So when your doctor prescribes (say) "take 2 tablets every meal time", this is based on the desirable levels of drug concentration and known levels of distribution, metabolism and excretion in the body.

What’s the math?

When the nurse first administers the drug, the concentration of the drug in the blood stream is zero. As the drug moves around the body and is metabolized, the concentration of the drug increases.

There comes a point when the concentration no longer increases and begins to decline. This is the period when the drug is fully distributed and metabolism is taking place. As time goes on, the drug concentration gets less and less and falls below a certain effective amount. Time to take some more pills.

We can model such a situation mathematically with a differential equation. It has 2 parts – an absorption part and an elimination part. At first, absorption (increasing drug concentration) takes precedence and over time, elimination (decreasing concentration) is the most important element.

We have the following variables:

D = drug dose given

V = volume distributed in the body

C = concentration of the drug at time t

F = fraction of dose which has been absorbed (also called bioavailability)

A = absorption rate constant

E = elimination rate constant

t = time

Absorption part: This depends on the amount of the drug given, the fraction that has been absorbed and the absorption rate constant. It decreases as time goes on. The expression for absorption is given by:

A × F × D × e^-At

Elimination part: The elimination dynamic is affected by the elimination constant, the volume distributed in the body and the concentration left of the drug. The expression for this part is:

E × V × C

For our model, we need to subtract the elimination part from the absorption part (since the absorption part increases the concentration of drug and the elimination part decreases it). Our differential equation is as follows:

We now substitute some typical values for our variables (without units to keep things simple. Note C is a variable, the one for which we seek an expression in t.)

Solving this differential equation (using a computer algebra system), gives the concentration at time t as:

C(t) = 533.3(e^−0.4t − e^−0.5t)

We can see in the graph the portion where the concentration increases (up to around t = 3) and levels off. The concentration then decreases to almost zero at t = 24.

Pharmacokinetics is yet another interesting “real life” application of math.

[Based on: A First Course in Pharmacokinetics.
Photo credit: Syringe]

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1. Intmath Newsletter – Graphs, pharmacokinetics, color blindness
2. The melting Arctic – a disturbing application of math
3. Today is “e” day