Sunday, 24 February 2013

Hard Made Easy In Math

HARD MADE EASY

Lets consider a simple case which shows how our understanding of Rational Numbers is based (mainly) on our knowledge of the integers.

(Generalized) Question 1: If $x$ and $y$ are rational numbers that are not squares of other rational numbers, show that $\sqrt{x} + \sqrt{y}$ is not rational.

This seems hard to approach, because there seem to be a lot of unknowns, and we could be unfamiliar with what rational numbers are. Let’s try the following:

Question 2: If $x$ and $y$ are integers that are not squares of other integers, show that $\sqrt{x} + \sqrt{y}$ is not an integer.

Now, this is more friendly, but still a little tricky. In the spirit of this post, let’s make it even easier.

Question 3: If $x$ is an integer that is not the square of another integer, show that $\sqrt{x}$ is not an integer.

Now this is almost silly, and seems to be just a play on words. The proper way to prove this statement that doesn’t make my head run around in circles, is to show the contrapositive – To show that $P \Rightarrow Q$ , it is equivalent to show the contrapositive which is $\lnot Q \Rightarrow \lnot P$ .

Proof: If $\sqrt{x}$ is an integer $n$ , then $x=n^2$ , which is the square of the integer $n$ . $_\square$

Now, what is the generalized version?

(Generalized) Question 4: If $x$ is a rational number that is not the square of another rational number, show that $\sqrt{x}$ is not a rational number.

Proof: Test Yourself 1. $_\square$

Corollary: If $y$ is an integer such that $\sqrt{y}$ is rational, then $y$ must be the square of an integer.

Proof: From Question 4, we know that y must be the square of a rational $\frac {p}{q}$ with $\gcd(p,q)=1$ . Hence $y = \frac {p^2} {q^2}$ , and the only way for this to be an integer is $q=1$ , so $y=p^2$ . $_\square$

Now, back to question 2.

Proof: Suppose $\sqrt{x} + \sqrt{y} = n$ is an integer. Consider $\sqrt{x} = n - \sqrt{y}$ . Squaring both sides, we obtain that $x = n^2 - 2n\sqrt{y} + y$ , so this means that $\sqrt{y} = \frac {n^2 - x - y}{2n}$ is rational. By the corollary above, $y$ must be an square. Similarly, $x$ must be an square. Hence we are done. $_\square$

And finally, back to question 1.

Proof: Proof by contradiction. Suppose $x = \frac {p_x} {q_x}, y = \frac {p_y} {q_y}$ such that $\sqrt{x} + \sqrt{y} = \frac {p_z} {q_z}$ , where $p_x, p_y, p_z, q_x, q_y, q_z$ are all integers. Then, $\sqrt{ q_y ^2 q_z ^2 p_x q_x} + \sqrt{q_x ^2 q_z ^2 p_y q_y} = q_x q_y p_z$ . This contradicts question 2! $_\square$

Corollary: If $x$ and $y$ are rational numbers such that $\sqrt{x} + \sqrt{y}$ is rational, then $\sqrt{x}$ and $\sqrt{y}$ are both rational.

The slightly surprising result, is that in simplifying Question 1 to Question 2, we ended up using Question 2 to prove Question 1. In fact, as is often the case with rational numbers, it is sufficient to consider the integer case, and then clear out denominators by multiplying throughout.

Get A Try & Test Yourself

1. Complete the proof of Question 4.

2. Prove that if $x$ , $y$ and $z$ are rational numbers such that $\sqrt{x} + \sqrt{y} + \sqrt{z}$ is rational, then $\sqrt{x}$ , $\sqrt{y}$ and $\sqrt{z}$ are all rational. How many terms can you show this for?

3. How many ordered triples of integers $(x, y, z)$ are there such that $\sqrt{x} + \sqrt{y} + \sqrt{z} = \sqrt{2000}$ ?

4. (**) Prove that if $a, b, c, x, y, z$ are rational numbers such that $\sqrt{a} + \sqrt{b} + \sqrt{c} + \sqrt{x} + \sqrt{y} + \sqrt{z}$ is rational, then $\sqrt{a}, \sqrt{b}, \sqrt{c}, \sqrt{x}, \sqrt{y}$ and $\sqrt{z}$ are all rational. Note: This is extremely hard, and not approachable by the methods discussed in this post.

Discovery Of The Number ' e '

THE DISCOVERY OF THE NUMBER ' e '

The more math and science you encounter, the more you run into the number e. Many of its applications and manifestations are alarmingly beautiful and often abstract. The logarithmic spiral is a shape that appears in nature, commonly in: shells, horns, tusks, sunflowers, and spiral galaxies. Despite the aesthetic wonders of the number e, it was actually first discovered in a pragmatic financial investigation of the behavior of compound interest.

When most of us are first taught about the number $e$ , we are told that it is an irrational, transcendental number that is about 2.7182. Most people simply learn to manipulate $e$ . High school classes rarely mention where $e$ comes from. It is usually introduced when learning about exponents and logarithms as a “special” base that you will use a lot down the road. Early in high school I remember asking a teacher what $e$ was. I received the usual circular answer, that $e$ is the base of the natural logarithm which in turn is the logarithm of an exponent raised to the base of $e$ . This answer did not satisfy me, but I was told that I had to wait for calculus to learn other ways of approaching it’s definition.

$e$ does make profound appearances in calculus. That said, $e$ was first discovered with algebra in the 17th century, a little before calculus was invented. Mathematicians indirectly came close to $e$ many times without directly calculating or recognizing it as anything out of the ordinary. $e$ was almost discovered when logarithms were invented in 1618 by John Napier. Logarithms immensely aided longhand computation by allowing multiplication and division to be carried out by addition and subtraction. These logarithms were not the same as our current conception of a logarithm. They were numbers that aided computation, and were not yet thought of as functions that relate exponents to their bases. Mathematicians effectively computed tables of the natural logarithm but did not know that the number e was anywhere behind them. Historians are not sure who exactly was the first person to calculate $e$ and recognize it as special. Most likely, $e$ was not discovered by a mathematician, but by someone with more worldly motivations. The number $e$ lies at the foundations of one of the most fundamental processes of finance: compound interest.

The 17th century was a time of rapid change. It was the era of the Scientific Revolution, the proliferation of colonialism, the emergence of mass literacy, and an explosion of international trade.The european Age of Exploration(and exploitation) brought disparate cultures of the world in contact, conflict, and business with each other to a degree that none of the large empires of old ever approached. Increasing the scale of commerce increased the demands for capital. Money lending began to play an increasingly large role in the prosperity of individuals, businesses ventures, and nations.

Given the growing presence of finance in the 17th century, historians believe that the first person to calculate $e$ , was most likely a banker or trader exploring the properties of compound interest.

Interest is a fee charged by a lender on a borrower, for the service of providing a loan. Interest fees offset the opportunity cost to the lender of not being able to do anything else with their assets while they are controlled by the borrower. These interest fees accumulate over time, depending on the interest rate. Borrowers think of interest as the cost of having debt, while lenders think of interest as the return on their loan as an investment.

Simple interest is where the interest accumulated is always the same proportion of the initial amount borrowed or invested(this initial amount is called the principal). For instance, if you borrow $1.00 at 5% interest per year, after one year, you will owe $1.05. After two years you will owe $1.10 etc…In effect, simple interest creates an arithmetic progression where a debt or investment grows at the same rate over all periods of time. Simple interest is rare and usually appears only in short term loans.

Compound interest is where interest accumulates on both the principal and the prior interest. Each time the interest rate is applied to the total accumulated debt or investment, it is referred to as compounding. If you invest $1 at 5% annual compounding interest, after the first year you would have $1.05. Unlike simple interest, in the second year it would increase by 5% of $1.05, yielding $1.1025. The general equation for compound interest, compounded once annually is

$S=P(1+r)^{t}$

where $S$ is the total accumulated debt or investment, $P$ is the principal, $r$ is the interest rate and $t$ is the time elapsed in years. Under compound interest, debts and investments grow by a geometric progression.

Interest can be compounded multiple times within a given year or interest rate period. For instance, if you put one dollar into a bank account that returns 5% interest per year compounded biannually, then your account will grow by 2.5% twice in a given year. For multiple compounding intervals within each rate period, the formula for compound interest becomes:

$S=P(1+\frac{r}{n})^{nt}$

where $n$ is the number of times the interest is compounded in a given year and $nt$ is therefore the total number of times it is compounded.

Compounding more times in a given time period causes your debt or investment to grow more often, but at a smaller rate each time it is compounded. As you can see, it would not take much imagination for an ambitious banker to wonder how much money could be made if the interest rate was really high and it was compounded as much as possible(daily, hourly, infinitely etc…). If more compounding intervals make investments grow more often, would more compounding result in investments growing faster?

Historians believe that a businessman or banker likely beat the mathematicians to the thought experiment: if $1 is invested at 100% interest over one year, how much more money will be made if it is compounded often as opposed to only compounding it a few times in a year??

The equation for compound interest in this special case would simplify to:

$S=(1+\frac{1}{n})^{n}$

It turns out that compounding weekly barely yields any more money than compounding monthly and at higher values of $n$ , it gets closer and closer to what we recognize as the number $e$ . As you can see, compounding more often does yield more money up to a point, but rapidly reaches an upper bound where growing more often, does not yield faster growth.Mathematicians would later go on to define $e$ as the limit:

$\lim_{n \to \infty} (1+1/n)^{n}=e$

With $n$ going to infinity, growth occurs continuously, at every possible instant no matter how small the time interval. Simple interest reflects an arithmetic growth progression, compound interest reflects a discrete geometric growth progression, and infinitely compounding interest reflects a continuous growth progression(exponential growth). The number $e$ is thought of as the base that represents the growth of processes or quantities that grow continuously in proportion to their current quantity. This is why e appears so often in modeling the exponential growth or decay of everything from bacteria to radioactivity.

Saturday, 29 December 2012

Magical Age Cards

Tell the age of people (beween 0 and 63) from the cards they pick.

Some traditional magic age cards forgo the numbers 61, 62 and 63 (so that only 29 or 30 numbers per card are required, which are printed in a 5 by 6 pattern, with or without a star in the 30th position). Full-range cards (with 32 numbers printed on each card) are more satisfying. Here are those 6 cards:

32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63	16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63	08 09 10 11 12 13 14 15 24 25 26 27 28 29 30 31 40 41 42 43 44 45 46 47 56 57 58 59 60 61 62 63
04 05 06 07 12 13 14 15 20 21 22 23 28 29 30 31 36 37 38 39 44 45 46 47 52 53 54 55 60 61 62 63	02 03 06 07 10 11 14 15 18 19 22 23 26 27 30 31 34 35 38 39 42 43 46 47 50 51 54 55 58 59 62 63	01 03 05 07 09 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47 49 51 53 55 57 59 61 63

Effect : A spectator thinks of a number (up to 63) and tells you on what cards it is.
You call the exact number!

Secret : The weight of each card is the smallest number printed on it. Any number is equal to the sum of the weights of the cards it appears on. For example:

52 = 32 + 16 + 4

Magical Mind Reader - Multiples Of Nine

Pick a 2-digit number...

Add the two digits together.
Subtract that sum of digits from the original number.
Look up the symbol corresponding to the result in a special table.

How can the magician predict what that symbol is? The trick will become boring (or obvious) if the same table is used repeatedly. Thus, a new table must be provided each time. Several online implementation do this quite effectively, with nice graphics. Examples:

Magic Gopher (British Council)

How fast can you discover the secret which makes this work? [ Answer ]

Answer :

The difference between a number and the sum of its digits
is always divisible by 9. The same symbol appears at all positions that are multiples of 9.

Mysterious Magical Trick - 1089

Pick a 3-digit number where the first and last digits differ by 2 or more...
Consider the "reverse" number, obtained by reading it backwards. Subtract the smaller of these two numbers from the larger one. Add the result to its own reverse.
Why is this always equal to 1089 ?
: This is one of the better tricks of its kind, because the effect of reversing the digits is not obvious to most people at first... If the 3-digit number reads abc, it's equal to 100a+10b+c and the second step gives the following result:

| (100a+10b+c) - (100c+10b+a) | = 99 | a-c |

The quantity | a-c | is between 2 and 9, so the above is a 3-digit multiple of 99, namely: 198, 297, 396, 495, 594, 693, 792 or 891. The middle digit is always 9, while the first and last digits of any such multiple add up to 9. Thus, adding the thing and its reverse gives 909 plus twice 90, which is 1089, as advertised.