Posted in Science & Nature

Grandi’s Series

In 1703, Italian mathematician and monk Guido Grandi posed a deceptively simple-sounding question:

What is the sum of the following infinite series?
1 – 1 + 1 – 1 + 1 – 1 + 1 – 1…

With simple arithmetic, we can easily divide the series using parentheses (brackets):

(1 – 1) + (1 – 1) + (1 – 1) + (1 – 1)… = 0 + 0 + 0 + 0 +… = 0

But what if we changed the way we used the parentheses?

1 + (-1 + 1) + (-1 + 1) + (-1 + 1)… = 1 + 0 + 0 + 0 +… = 1

Because of the way negative numbers work, this solution is equally feasible. Ergo, both 0 and 1 are acceptable answers.

How can one series possibly have two different answers? Grandi used the fact that both 0 and 1 are possible from his series as proof that God exists, as something (1) can be made from nothing (0).

Grandi’s series becomes even stranger when a more advanced technique is applied.

Let us say that Grandi’s series is denoted by S (S = 1 – 1 + 1 – 1…).
We can then break down the series as 1 – (1 + 1 -1 + 1…), because the plus and minus signs can be inverted together.
Ergo, S = 1 – S → 2S = 1 → S = ½

Now we have three answers to Grandi’s question: 0, 1 and ½.
For over 150 years, mathematicians fiercely debated the answer to Grandi’s question. By the 19th century, mathematics had evolved and mathematicians had figured out better ways to solve infinite series.

The classic example is the solution to the series: 1 + ½ + ¼ + ⅛…
To solve this, you can add the partial sums, where you add each number to the sum of the previous numbers to see what number you are approaching (the limit).

1 → 1.5 → 1.75 → 1.875 → 1.9375… until we infinitely approach 2 (or 1.9999999…)

If we apply this method to Grandi’s series, we do not approach a single number because we keep swinging between 0 and 1. (1 → 0 → 1 → 0 → 1…)

So we can apply another method, where we average the partial sums as we go instead of adding.

e.g. 1 → ½(1 + 1.5) = 1.25 → ⅓(1 + 1.5 + 1.75) = 1.416 → ¼(1 + 1.5 + 1.75 + 1.875) = 1.531… until we approach 2.

Using this method on Grandi’s series:

1 → ½(1 + 0) = ½ → ⅓(1 + 0 + 1) = ⅔ → ¼(1 + 0 + 1 + 0) = ½…

Eventually, the series appears to converge on ½, showing that the answer to Grandi’s series seems to be ½.

The problem with this method is that Grandi’s series does not actually have a limit, but we are applying a solution as if it has a limit. This is similar to using a divide by 0 trick to prove that 1 + 1 = 3. In mathematics, when rules are bent, we end up with weird, paradoxical results.

To show this empirically, consider the thought experiment of Thomson’s Lamp:

Imagine a lamp that is turned on after 1 minute, turned off after ½ minute, turned on again after ¼ minute ad infinitum.
This incorporates both infinite series discussed above.
Ergo, we know that the sum of time is 2 minutes.
So, at the end of 2 minutes, is the lamp on or off?
If Grandi’s series solves to 0, the light is off; if it is 1, the light is on.
Then what does it mean if Grandi’s series solves to ½?
Is the light on or off?

Posted in Science & Nature

Hurricane

We often hear on the news of cataclysmic storms with oddly common names such as Hurricane Sandy, Katrina and Harvey. It seems weird that we give such devastating forces of nature a basic name, let alone naming them human names at all.

A hurricane is the name given to tropical storms that occur in the Atlantic Ocean. For reference, a hurricane is essentially the same as a cyclone or typhoon. The history of naming hurricanes dates back over a hundred years, with residents of the Caribbean Islands naming hurricanes after the saint of the day from the Catholic calendar. Initially, American meteorologists named hurricanes by the geographic location that the storm originated in.

However, during World War II, military meteorologists in the Pacific started using women’s names for hurricanes. This made communication much easier as hurricanes could be identified by name and much easier to say. There are some apocryphal stories about the origin of women’s names for hurricanes, such as wishing that the hurricane will be calmer and of better temperament, or that they were named after the meteorologists’ wives and girlfriends. This practice soon spread to the rest of USA and became the default method of naming hurricanes. From 1979, it was decided that the gender of the names would be alternated.

In the present, there is a rolling six-year roster of 21 names each year in alphabetical order that is used to name hurricanes (see below for list). For example, the first hurricane of 2019 was called Andrea, the second Barry, the third Chantal and so on. In 2020, the first hurricane will be named Arthur, then Bertha, et cetera. The same names would be used in 2025 and 2026.

The one exception to this rule is that when a hurricane is particularly devastating and results in many deaths, the name is “retired” in honour of those who have lost their lives or livelihoods to the hurricane. For example, there will be no more hurricanes named Katrina or Harvey in the future.

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Grasshopper Mouse

Grasshopper mice are a species of New World mice found in deserts throughout North America. They are small-to-medium sized, growing up to 13cm in size and weighing around 40-50g.

Despite their cute appearances, they are carnivorous, ferocious hunters. They feast on various insects, but are also known to hunt other mice.

Grasshopper mice have interesting adaptations that make them seem more like a miniature wolf or mongoose rather than a mouse. For example, they stalk their prey like a cat before pouncing. They hunt highly venomous insects such as scorpions and centipedes because they have evolved to convert the deadly toxins of a scorpion sting into harmless chemicals.

An interesting feature of the grasshopper mouse is that they often howl like a wolf to ward off competitors and to communicate with each other. It has been nicknamed the werewolf mouse because they are known to howl in the night with their heads thrown back, communicating over the vast desert with a high-pitched howl.

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Compound Interest

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.)
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The Closest Planet

Which planet is closest to Earth? If we look at a typical model of the Solar System with each planet neatly lined up, we can see that Venus approaches Earth closer than any other planet. However, this is only one interpretation of the question.

Technically, Venus is the planet that comes closest to Earth. However, as they do not orbit in synchrony, this approximation happens about once a year. At other times, Venus will orbit away from Earth and can go on the other side of the Sun, making the distance between Earth and Venus vast. In those times, Mars may seem like the next obvious choice to be closest to Earth.

But then again, Mars has the same issue where it and Earth are often on opposite sides of the Sun. Because of the nature of circular orbits, the distances between the planets swing and fluctuate, meaning that the real question should be:

Which planet is closest to the Earth most of the time on average?

The answer to this question happens to be Mercury. If we look at a “top-down” model of the Solar System, we can see that Mercury – being closest to the Sun – orbits rapidly around the Sun and often lies between Earth and the two other planets, Venus and Mars. If we plot the distance between each of these three planets and Earth, we can see that on average, Mercury is closer to Earth because the distance fluctuates less.

Interestingly, if we take this question further, we find that Mercury is also Mars and Venus’ closest neighbour on average. This is a property of the Solar System being formed of concentric circles, meaning that Mercury’s smallest orbit makes it average a closer distance to all of these planets.

Fascinatingly, if we go even further than that, we find that the same pattern holds for every other planet in the Solar System, despite the vast distance between Mars and Jupiter due to the Asteroid Belt. Even Pluto (not formally a planet anymore) with its massive elliptical orbit has Mercury as its closest neighbour on average compared to the other planets, due to the unique property of concentric circles.

No matter the distance, if you are orbiting the Sun, Mercury is the closest planet to you.

This video from CGP Grey explains it in a concise and informative way, complete with clean diagrams and animations!
Posted in Science & Nature

Shooting Star

When an object from outer space enters the Earth’s atmosphere, it starts to burn up and creates a brilliant streak in the sky, which we call a meteor or shooting star. Contrary to popular belief, this is not due to friction with the air in the atmosphere.

An object entering the atmosphere is typically travelling at extraordinary speeds. Most meteors are travelling around 20km/s (or 72000km/h) when they hit the atmosphere. At these speeds, air molecules do not have a chance to move out of the way. The meteor will instead collide into the air molecules, pushing them closer and closer to each other, compressing the air in front of it.

As we know from physics class, compression increases temperature in gases as per the ideal gas law (PV=nRT). The impressive entry speed of these meteors result in so much air compression that their surface can heat up to 1650 degrees Celsius.

The heat boils and breaks apart the contents of the meteor, turning it into superheated plasma that gives off a glow. This is the streak of light that we see in the night sky when we wish upon a shooting star.

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Oobleck

If you mix 1 part water to 1.5-2 part corn starch, you create a strange mixture called “oobleck“, named after a Dr. Seuss story. It is so simple to make, yet it exhibits some very strange properties that makes it a popular science experiment.

Oobleck is what is known as a non-Newtonian fluid, where the viscosity (or “thickness”) changes with how much stress it is under. If you press your finger gently into it, it will feel like water, but if you strike it with a hammer, it will behave as a solid. It will stiffen when you stir it, but run when you swirl it.

Related image

You can even run over a tub of oobleck as long as you change steps quickly enough to apply enough pressure to keep the fluid under your feet solid. This is because oobleck becomes very viscous under high stress, making it behave more solidly (shear thickening).

We can learn from oobleck not only some interesting physics principles, but also how to interact with people.

Much like a non-Newtonian fluid, people will tend to react stiffly and with more resistance if you apply stress or force. But if you apply gentle pressure and be assertive, you will find people generally react more softly and fluidly.

This simple change in your approach will lead to much better conflict resolution and constructive outcomes when dealing with other people.

Image result for oobleck run gif

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Pringles

Pringles are a beloved snack well-known for its addictiveness (“Once you pop, you can’t stop“). There are a few other interesting factors that set Pringles aside from other potato chips.

Firstly, Pringles have been called many things, because it is not strictly a potato chip. When it first debuted, other snack companies complained that it was not technically a potato chip as they were made from dried potatoes, so they were labelled “potato crisps“. Ironically, the company successfully argued in 2008 that Pringles were not “potato crisps”, using the logic that they were not of natural shapes and only contained 42% potato as they are made from potato-based dough. This was so that they could avoid the British tax on potato crisps.

Secondly, Pringles chips have a characteristic saddle-shape, known in mathematics as a hyperbolic paraboloid. This creates a uniform shape, meaning they can be stacked neatly in a tubular container for efficient and reliable packaging, as opposed to most potato chips that are packaged in bags. Furthermore, the shape is structurally sound, preventing the chips from breaking under the weight of the stack.

Finally, the inventor of the cylindrical container was a chemist named Fredric Baur, who started the process of making Pringles. His dying wish was to have his ashes buried in a Pringles can and this wish was respected by his children.

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Hexapod

Dragons are fantastic creatures of our imaginations, so they do not follow many of the rigid laws of natural science. They breathe unlimited amounts of fire, can endure extreme heat and they can fly despite their massive size. But perhaps the most unrealistic feature of dragons is the fact that they have an unnatural number of limbs.

All vertebrate animals on Earth follow a simple rule: they are four-legged creatures, also called tetrapods. The limbs may have devolved away such as in whales and snakes, but they remain as vestigial structures or still encoded for in the genes. Birds and bats have adapted their upper limbs into wings to fly, but the total number of limbs is still four.

How many limbs does a dragon have? They have four legs that they stand on, but also two large membranous wings like a bat. This means that they have a total of six limbs. The only other animals that share this trait are insects and other mythical creatures such as the centaur and pegasus.

To be a vertebrate with six limbs, a dragon must have evolved from an ancestor separate to Tetrapodomorpha, an ancient fish-like creature with four limbs that is the common ancestor to all four-legged beasts. Alternatively, the wings may not be true “limbs” and be similar to flying lizards that evolved to have a rib jut out with a membrane attached to act as a glider.

Unlike the scientifically inaccurate dragon, a wyvern obeys nature’s four-leg rule. Furthermore, unlike the traditional Western dragon that we have been describing, dragons of the Far East have no wings and four limbs, also obeying the law.

As ridiculous as it may sound, applying scientific principles to our imagination allows us to learn more about how our world works.

Posted in Science & Nature

Arithmetic

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.

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