Tag: maths

Posted in Science & Nature

Compound Interest

Posted on by Jinavie

When is the best time to invest? Is it when you have sufficient income and savings that you feel that you have a surplus to invest with?

The correct answer is much simpler: yesterday, with the second best time being today. Because of the magic of compound interest, investing early is the best strategy possible.

Thanks to a simple mathematic rule, compound interest rewards early, small investments more than late, large investments.

The way compound interest works is that after a given time interval (e.g. year), interest (as a percentage of the original investment) is paid out. The next year, interest is paid out again but as a percentage of the new amount. As an example:
1000 x 1.08 = 1080 (end of year 1)
1080 x 1.08 = 1166.40 (end of year 2)
1166.40 x 1.08 = 1259.71 (end of year 3)
…until end of year 10

If we use mathematical shortcuts and convert this into a formula, we can express it as:

(A = future value, P = present value, r = interest rate as decimal, n = number of periods/years)

For example, if we invest $1000 (PV) at an interest rate of 5% (r=0.05) for 10 years, then:

$1000 x 1.08^10 = $2158.92,

meaning we have earned $1158.92 over 10 years. Taking it further, in 30 years our investment would have grown to $10062.66 – ten times our original investment.

Because the formula uses exponents (or powers) for the time, your investments grow exponentially with time. This means that the earlier you invest, the greater your returns become disproportionately. This is why within 10 years, we have more than doubled our initial investment despite a reasonable interest rate and not doing anything else.

A rule of thumb for calculating how long it will take your investment to double is to divide 72 by the interest rate in % (e.g. 7). This is the number of years it will take for your investment to double (e.g. 72/7 = 10.3 years).

On top of this, if we invest small amounts every year, then we can benefit even more from the exponential growth of our investment. For example, just by adding in $100 every year, we end up with an additional $564.55 of investment earnings at the end of 10 years – a 50% increase in returns.

Unfortunately, mathematics works both ways and compound interest also applies to certain loans, such as credit cards. This means that your debt will grow exponentially unless you aggressively pay it back, making it seem impossible to pay off your credit card debt sometimes.

(This graph shows that investing early and consistently is the best strategy to maximise your eventual earnings. Compare the grey and purple line and you will see that despite investing a third of what Lyla invests total, Quincy ends up with a higher portfolio by retirement.)
Posted in Science & Nature

Arithmetic

Posted on by Jinavie

Although we all learn mathematics to a high level during our schooling years, most of us find that as working adults, we lose much of our maths skills due to lack of practice. This may be fine for advanced concepts such as calculus and matrices, but we tend to forget even the most basic arithmetic skills, instead choosing to rely on calculators on our phones and computers.

But maths is all around us in day-to-day life. From figuring out how much you save on a sale, to splitting a bill, to calculating tips when you travel in the USA, arithmetic is a handy life skill that many of us have forgotten. As easy as it is to pull out your phone and use the calculator app, here are a few tips to improve your arithmetic skills for quick mental calculations.

If you need to multiply a 2-digit number (e.g. 12 x 17), divide one of the number into its 10’s and 1’s, multiply the other number to each of these numbers then add them.

(e.g. (12 x 10) + (12 x 7) = 120 + 84 = 204)

You can further subdivide the numbers to break it down into easy bite-sized calculations.

e.g. 34 x 26 = (34 x 20) + (34 x 6) = (34 x 2 x 10) + ((30 x 6) + (4 x 6)) = 680 + (180 + 24) = 884

When adding or subtracting large numbers, use 10’s and 100’s for easier calculations. Essentially, you can “fill in the gap” up or down to the nearest 10’s or 100’s, then add/subtract the remainder.

e.g. 64 + 13 -> take 6 away from 13 and add to 64 -> 70 + 7 = 77

You can do this in multiple steps to break a complicated addition or subtraction into simple maths.

Learn to manipulate the decimal point to make multiplication and division simpler. 20% of 68.90 sounds difficult, but if you understand how the decimal point works, you can simply multiply 2 then divide by 10 to get the answer.

e.g. 68.90 x 2 =137.80 / 10 = 13.78

An extension of this is learning basic fractions, such as knowing that 0.5 is half and 0.2 is one-fifth.

e.g. 32 x 15 = 32 x (1.5 x 10) -> so you can add half of 32 to itself (x1.5) then x10 -> 48 x 10 = 480

Lastly, a handy mathematic trick is knowing that X% of Y = Y% of X. This means that if one side of the equation is easier, you can convert it easily. For example, 4% of 25 sounds much more difficult than 25% of 4 (or quarter of 4), yet the answer is the same.

The common theme of these tips is using shortcuts and breaking down complicated equations into bite-sized steps so that your brain can solve simple arithmetic in sequence. This may be asking for too much in a time when all of us seem to have minimal attention spans, but you never know when basic maths will come in handy.

MRW My girlfriend says "You know why I'm mad."
Posted in Science & Nature

Sudoku

Posted on by Jinavie

Sudoku is a mathematic puzzle that has gained considerable popularity in the 21st century, rivalling the classic puzzle that is the crossword. You are given a 9×9 table divided into 9 equal squares, filled with a certain number of digits. Your goal is to fill in the table so that each row, column and subsquare (of 9 small squares) contains every digit from 1 to 9. You are not allowed to have the same number appear on the same row, column or subsquare, as there are not enough spaces for spare digits.

The more digits (“clues”) that you are given at the start of the puzzle, the easier it is to solve it. This begs the question: what is the minimum number of clues that you need to solve a sudoku puzzle?

Sudoku puzzles with 17 clues have been completed traditionally. We know that 7 clues is not enough as the last 2 digits can be interchanged, creating puzzles with more than one solution. Using mathematics, we know that if we can solve a puzzle with n clues, then a puzzle with n+1 clues can be solved as well. Ergo, the answer lies somewhere between 8 and 16.

In 2012, Gary McGuire, Bastian Tugemann and Gilles Civario tackled this problem using one of the oldest tricks in mathematical analysis: brute force. The total number of possible sudoku puzzles that can be generated is 6,670,903,752,021,072,936,960, or 6.67 x 10²¹. After accounting for symmetry arguments (meaning that two puzzles may be essentially identical, but just rotated or flipped), we are left with 5,472,730,538 possible unique solutions.

The team used supercomputers to analyse all of these possibilities to see if any puzzle can be solved with just 16 clues, as the conventional thought was that 17 was the minimum number of clues possible from traditional methods. After a year of calculations, the computer found no sudoku puzzle could be solved with only 16 clues. This was confirmed by another team from Taiwan a year later, proving that the minimum number of clues required for sudoku is indeed 17.

Posted in Science & Nature

Rain

Posted on by Jinavie

Let’s imagine that you are walking outside, when rain clouds catch you by surprise and suddenly pour down on you. Assuming that you have no umbrella or anything to cover yourself with, is it best to run back home or walk back? Or to elaborate, should you walk and spend more time in the rain, or should you run, which means you will run into rain sideways?

There are two ways you can get wet in the rain: it will either fall on top of your head, or you will run into it from the side. The amount of rain that falls on your head is constant whether you are walking or raining, as the entire field you are travelling through is full of raindrops. Therefore, one would naturally think that running would not add much benefit as you run into more rain by moving faster, as you essentially hit a wall of raindrops.

But this is not true. No matter how fast you travel, the amount of rain you hit sideways is constant. The only variable that affects the amount of rain you hit sideways is the distance you travel. This is because the amount of raindrops in the space between you and your destination is constant.

Summarising this, the wetness from rain you receive is:

(wetness falling on your head per second x time spent in rain) + (wetness you run into per meter x distance travelled).

Since you cannot really change how far you are from your destination, the best way to minimise getting wet is to run as fast as you can to minimise the time you spend in the rain.

Then again, this is only the most practical option to keep you dry. If you are feeling particularly romantic or blue, then feel free to stroll through the rain, savouring the cold drops on your face (or wallow in the sadness that is your life).

(Here’s a very good video explaining the maths/science of it all: http://www.youtube.com/watch?v=3MqYE2UuN24)

image

Posted in Science & Nature

Mathematical Beauty

Posted on by Jinavie

What is the most “beautiful” mathematical equation? For millenia, many mathematical formulas and concepts have been described as beautiful (and some defining beauty, as the golden ratio does). In the mathematical world, the adjective “beautiful” is used in the sense that certain mathematical concepts, despite the fact they are rational and objective, are so pure, simple and elegant that they can only be described as art.

One such formula is Euler’s identity:

image

Renowned physicist Richard Feynman described it as “the most remarkable formula in mathematics”. What makes this array of symbols and numbers so beautiful? Firstly, it contains the three basic arithmetic operations exactly once each: addition, multiplication and exponentiation. It also connects five fundamental mathematical constants with nothing other than themselves and the arithmetic operations.

0 is the additive identity, as adding it to another number results in the original number. 1 is the multiplicative identity for the same reason as 0. Pi(π) is one of the most important mathematical constants in the history of mathematics that is ubiquitous in Euclidean geometry and trigonometry. Euler’s number(e) is the base of natural logarithms and is used widely in mathematical and scientific analysis. i(√-1) is the imaginary unit of complex numbers, a field of imaginary numbers that are not “real”, allowing for the calculation of all roots of polynomials. Euler’s identity neatly sums up the relation between these five numbers that are so crucial in the field of mathematics. It is also interesting to note that these five numbers were discovered at different points in history spanning over 3000 years.

Some people describe mathematics as a distinct language in itself. Not only that, but mathematics is considered the universal language as it is both universal and ubiquitous. If that is the case, than Euler’s identity can be considered an extremely pithy literary masterpiece.

Posted in Science & Nature

Rock-Paper-Scissors

Posted on by Jinavie

Rock-paper-scissors is a game with a long history. The earliest example of the game is a Chinese game called huoquan, which follows a cyclic rule where the frog eats the slug, the slug dissolves the snake and the snake eats the frog. The reason why rock-paper-scissors has been saved throughout history is because of the uncertainty it contains. Any hand you choose, the chance of winning is the same. Ergo, there is no single best choice and there is no move that will always win. But this is still a game played by people. It is not a game played by emotionless machines, meaning that you can use human psychology, the surfacing of emotion and specific signs and movements to help deduce your opponent’s hand. Mentalist Derren Brown can read tiny flickering of muscles in the opponent and microexpressions to pull off his “undefeatable rock-paper-scissors trick”, but this is near impossible for a normal person to try. However, you can use the following strategies to improve your odds.

  1. Use paper on a beginner: Statistically, people prefer using rock. Males especially have a strong tendency to play rock.
  2. Use scissors on an experienced player: People who know the first trick can be defeated by going one step further.
  3. Use a hand that loses to the hand your opponent played: This uses the psychology of the opponent wanting to mix up hands and wanting to beat the hand you last played (which is the same as theirs as you drew).
  4. Say what you will play and play that hand: In a competitive situation like rock-paper-scissors, people tend not to trust others. Thus, if you say you will play a certain hand, they will think is a trap and not play the hand that defeats that hand. For example, if you said you will play scissors, the opponent will play paper or scissors and you will either win or draw.
  5. Do not give the opponent a chance to think: People have a subconscious tendency to play a hand that beats the hand that they played before. Without time to think, the subconscious takes action meaning that you can predict their move. If you do the same as strategy 3 and play a hand that loses against the opponent’s previous hand, you will win.
  6. Suggest a certain hand: This is a form of hypnosis where you suggest something to the opponent’s subconscious. To use this trick, pretend to go over the rules by saying “rock, paper, scissors” then play a certain hand. The opponent will likely play the hand that the subconscious last saw.
  7. If you keep drawing, use paper: This is the same as strategy 1.

Unfortunately, rock-paper-scissors has an equal probability of a win and a draw, meaning draws are rather common. Thus, a computer engineer called Samuel Kass devised a game where two additional hands are added: rock-paper-scissors-lizard-Spock. Lizard is played by making your hand into the shape of an animal’s head, while Spock is played using the Vulcan Salute from the science fiction show Star Trek, where you make a V-shape with two fingers on each side. The rules are as follows.

Scissors cut paper. Paper covers rock. Rock crushes lizard. Lizard poisons Spock. Spock smashes scissors. Scissors decapitate lizard. Lizard eats paper. Paper disproves Spock. Spock vaporizes rock. Rock crushes scissors.

As each hand has two ways of winning, the odds of winning is 10/25, or 2/5 and the odds of drawing is 5/25, or 1/5. As you can see, you have double the chance of winning compared to drawing, making the game much faster to play than the original game.

Posted in Science & Nature

Marriageable Age

Posted on by Jinavie

When is the right time to get married? According to Professor Tony Dooley, you can use an equation to find the right age for proposing. To do this, take “the youngest age you want to marry” and minus it from “the oldest age you want to marry” then times 0.368. Add this number to the youngest age. For example, if you would consider getting married from age 21 onwards and at the latest 30, your ideal age to marry is: (30 – 21) x 0.368 = 3.312 + 21 = 24.312, thus about 24 years and 4 months old. 

This equation is very practical as it is a modified version of equations used in financial and medical fields. This equation is used to maximise profit while minimising loss using mathematics. It may not sound romantic, but according to Professor Dooley, after you reach the calculated age you should not waste time and ask the hand of the next person you date in marriage.

(Source: http://soulofautumn87.deviantart.com/art/All-We-Need-Is-A-4-Letter-Word-111260511)