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

In this Newsletter

1. Math tip (a) – Functions
2. Math tip (b) – Gini Coefficient
3. Math in the news – Math Anxiety
4. Latest from the Math Blog
5. Final thought – Attention

1. Math tip (a) – Functions

Math functions is a topic that confuses a lot of people, but is very important.

In this article I discuss a problem sent to me by a reader. He got stuck because he was not sure about function notation.

[Confession time: When I was a young math student, I remember struggling with the concept of functions for a few months. Now I realize the problem was the notation, not with me!]

Check out the article: Functions

2. Math tip (b) – Gini Coefficient

Here’s an interesting "real life" math topic – Gini Coefficient.

In some countries, wealth (and income) is distributed reasonably evenly (for example, Denmark), while in others, a small group of people hold most of the money (many African and South American countries).

The Gini Coefficient is one way to measure income equality.

Some of the math concepts in this article include:

• Statistics
• Area
• Integration (area under a curve)

Here’s the link: The Gini Coefficient of wealth distribution

3. Math in the news – Math Anxiety

[Photo credit]

(a) Anxiety affects basic counting: It’s been known for some time that anxiety about math causes a conflict in our brains and reduces math performance. But it was thought this only affected higher-level math.

A new study from Canada’s Waterloo University, "Mathematics Anxiety Affects Counting But Not Subitizing During Visual Enumeration" shows math anxiety affects even basic counting. You can read a (simple English) summary here:

New Waterloo study shows math anxiety hinders basic counting

(b) Girls learn math anxiety from their teachers: A study by the University of Chicago examined 52 boys and 65 girls in elementary school classes taught by 17 different women.

The more math anxious the female teachers were, the more likely their female students became anxious about math. For more, see:

Girls’ math anxiety a learned response?

4. Latest from the Math Blog

	Beauty is defined by a new “Golden Ratio”, new research shows. Is she beautiful? The new Golden Ratio
	Here’s a video about population trends. I’m not sure I agree with all he’s saying, but it’s interesting to think about. Friday Math movie – Addressing Population Myths
	One of life’s toughest decisions is choosing the right mate. Math to the rescue! How math helps to choose the right marriage partner

5. Final thought – Attention and math

The articles about math anxiety reminded me of the following quote. We like to think we can multi-task, but in fact we can only really pay attention to one thing at a time. Yes, that applies to both men and women!

However, women tend to have a better ability at what I call "parallel-tasking", that is, keeping track of several things at once.

Math anxiety affects counting (and all levels of math) because it takes up valuable attention. If we’re concentrating on our fears of ridicule, or our performance anxiety, or what our parents will say when we fail, it "clogs" the attention we can give to the math.

I’ll leave you with this quote:

“There is no such thing as not paying attention; the brain is always paying attention to something.”
(Patricia Wolfe)

Until next time, enjoy whatever you learn.

Related posts:

1. The Gini Coefficient of wealth distribution
2. IntMath Newsletter – Math exam preparation, periodic functions in the body
3. Database of Happiness

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.]

Related posts:

1. Death, taxes, birth, marriage and blogging
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.]

Related posts:

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

In this Newsletter

1. Math tip (a) – Graphs using free math software
2. Math tip (b) – Math of drugs and bodies (pharmacokinetics)
3. Latest IntMath Poll – math applications
4. Latest from the Math Blog
5. Final thoughts

1. Math tip (a) – Graphs using free math software

The latest IntMath Poll asked readers how they normally draw math graphs.

The response from 1900 users was interesting:

65% said they use paper; 20% said graphics calculator and 15% said computer software.

There are many free online and downloadable graphics programs out there and it surprises me so few people use them to draw their math graphs.

In this tip, I show how to avoid some of the pitfalls of using software to draw graphs. Go to:

Graphs using free math software

2. Math tip (b) – Math of drugs and bodies (pharmacokinetics)

Several people have written asking me to write an introduction to pharmacokinetics. This is the process where the body absorbs and metabolizes drugs (or food, or any chemical).

The math involves diffrential equations, but I think everyone will find it an interesting read. Go to:

Math of drugs and bodies (pharmacokinetics)

3. Latest IntMath Poll – Math applications

One of the most common questions from math students is, "When are we ever going to use this stuff?"

This month’s IntMath Poll asks readers whether they feel they get a good understanding of how math is applied in the "real world".

Please add your vote – you can do so on any page in:

Interactive Mathematics.

4. Latest from the Math Blog

A) Math and color blindness

What’s the best way to present math so color blind people can read it? Are you color blind? I’d love to hear your reaction to this article.

B) Camera purchase decisions – how math helps
A site selling electronics uses math concepts to help customers decide.

C) Friday math movie – George Dyson at the birth of the computer
The story of one of the most important inventions ever.

D) Math graphs on the Web without images
Here’s one way to plot good looking graphs on the Web – use ASCIIsvg.

5. Final thoughts

a. Practice

Everyone tells us to practice math and you will become an expert. Here’s another take on that advice.

Practice does not make perfect.
Only perfect practice makes perfect. [Vince Lombardi.]

b. Biodiversity

2010 is the International Year of Biodiversity. The rise and decline of populations is a very interesting math topic.

What can you do to study – and help – endangered species in your area?

Until next time, enjoy whatever you learn.

Related posts:

1. IntMath Newsletter – Drawing graphs, fear of math tests
2. Math and color blindness
3. Math of drugs and bodies (pharmacokinetics)