Posted in Life & Happiness

Friendship

We have very little choice about who we have as family. But friends are a different story: we choose who we want to spend time with and call our friends. The beauty of friendship is that it is an active choice. By calling someone our friend, we are telling them that we appreciate their presence in our lives, that we enjoy spending time with them, that we care about their well-being and wish them good fortunes in the future.

That said – as emphasised above – friendship is a choice. We can’t choose our family, but we can choose our friends. If a friendship becomes toxic, burdensome and a major source of stress more than what you gain from it, you have the power to leave the friendship (or vice versa).

We know from psychological studies that one of the most common starts to friendship is the propinquity effect, where we are more likely to like and befriend people who we are regularly exposed to and in close proximity to. This explains why when we are young, our friend group seems to be based around people from school, university and work.

But as we grow older, we learn that proximity and history alone is not enough to maintain friendship. In adulthood, we constantly face pressures such as work demands, romantic relationships and various life stressors. We have so little free time to invest in those around us, so why would we want to use up time to maintain relationships that do not add anything to our lives, let alone those who take from us or bring us down?

Like with any other kind of relationship, friendships should be evaluated and re-evaluated over time. Life is too short to spend time on those who do not earn or deserve our trust, loyalty and love. Life is also too short to take for granted the amazing people around us and to not celebrate the beautiful relationships that build us up and support us in times of need.

When we find that a friendship is becoming toxic, leaving us with a bitter taste in our mouth at the end of each encounter and making us wonder if they add anything to our lives, then we should seriously consider addressing it with communication or distancing our hearts from them.

Similarly, when we find friends who share our values, with whom we can share emotional insights and vulnerable insecurities with just as easily as sharing silly and fun times, or ask for help when we need it and from whom we can learn from and give back to in a meaningful way, then we must acknowledge how rare and precious those connections are.

These are the connections we should be investing our precious little personal time towards, because they are the friendships that amount to something greater, where the sum is greater than the parts, where 1 + 1 = 3.

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.

Posted in Life & Happiness

The Story You Tell Yourself

It is the human condition to be our own worst enemies. Yes, life can get hard and it will throw various obstacles and challenges at us, creating all kinds of stress and distress. However, much of our anguish will come from the stories we tell ourselves.

We often think that we feel emotions as a reaction to a stimulus or a change in our environment. This makes us feel powerless and as if we are slaves to our emotions. In reality though, our emotions are usually reactions to our thoughts.

For example, when a relative or someone close to us dies, we feel sad. This may seem like an automatic response, but we first process the information with our rational mind and tell ourselves the story that we will miss them, or that we will never see them again. Our sadness is a reaction to the thought process rather than a direct result of the event.

In this case, the emotional reaction is highly appropriate. The problem is that it is extremely common for us to tell ourselves the wrong story.

A good example would be insecurities. If you ever notice yourself feeling inexplicably anxious, sad or angry, ask yourself the question: what am I telling myself?

You may find that the reason that you are angry every time your colleague talks to you is because you are telling yourself that they are lazy. You may be frustrated whenever a friend doesn’t reply back to your messages because you think they are avoiding you. You may feel sad whenever you look in the mirror because you tell yourself that you are not physically attractive enough. You may be telling yourself that your partner does not love you whenever they go quiet and withdrawn suddenly.

The importance of understanding this concept is that it lets you be more in control of your emotions and lets you diagnose the problems affecting your mental health. Once you know what story is causing the emotion, you can examine the story. When we run the story through a rational filter, we may find that our reaction was completely irrational.

The “lazy” co-worker may be going through a rough time making it difficult for them to work efficiently. Your friend may be busy at work, hence not able to reply. You may be objectively attractive and in good physical health, but your poor self-confidence may be creating a false story. It could be that withdrawing themselves is your partner’s normal coping mechanism when they are dealing with their own problem and it may have nothing to do with you.

This is also useful in a relationship setting, as you can ask your partner how your actions make them feel and what they are telling themselves in that situation to better break down what the true issue is. This lets you both resolve the issue in a more constructive, peaceful manner.

The bottom line is, to improve our mental health, we must examine and alter the stories we tell ourselves. If you tell yourself the worst stories, it will become reality. So ask yourself: what kind of stories am I telling myself and how is it affecting my life? You may be surprised to see how different life can be when you get your stories straight.

Posted in Science & Nature

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.

Posted in Life & Happiness

Yesterday’s Tomorrow

In life, procrastination feels almost like a base human impulse. It is so easy to put off tasks until tomorrow.
But what is today but yesterday’s tomorrow?

It is difficult to find motivation to do tasks that we find boring, hard or unimportant. But delaying it by a day does nothing to fix that issue. The best approach is to sort out easy tasks early so that they do not accumulate until you feel pressured by the sheer amount of tasks.

A useful rule of thumb is: if something will take you less than 5 minutes to do, do it now.
This might include wiping down the kitchen bench, throwing the garbage out, making your bed, tidying a small pile of mess, replying to an email or writing a bullet journal entry.

By clearing these small tasks as they arise, you have more free time to spend on things you are passionate about or find important.

Doing small tasks also gives you a sense of empowerment and can motivate you to do slightly harder or more complex things, such as vacuuming the house, making an appointment or sorting paperwork. Once the ball is rolling, it is far easier to be productive.

In a similar vein, setting up routines such as setting aside an hour or two on the weekend as “life admin” time, or having a to do list in your journal help form healthy habits to fight against procrastination.

At the end of the day, procrastination is like taking a loan out from your future free time, with interest added. It will simply rob you of quality time you can spend on your passions and loved ones.

So think to yourself “What is a simple task I need to do that I can sort out right now?” and just do it. You will find that you had the motivation to be productive and efficient all along.

Original video, worth watching if you don’t feel motivated: https://www.youtube.com/watch?v=ZXsQAXx_ao0
Posted in Life & Happiness

Elephant Riding

Psychologist Jonathan Haidt described the relationship between our rational and emotional minds as that of a person riding an elephant. The rational person can guide the elephant using reins, but if the elephant really wants to go a certain way, it will easily overpower the rider. Fighting against the elephant risks it throwing you off or going on a destructive rampage.

This analogy is helpful in making us understand that emotions are natural and powerful. Fighting against emotions (particularly strong, negative emotions) can be pointless and harmful. The best thing is to let yourself feel the emotion, so that it can resolve rather than build up.

This may sound defeatist, because it feels as if we cannot ever control our emotions and we are slaves to it. However, as the analogy points out, our thoughts are the riders atop our elephants of emotions.

Thoughts and perception lead to emotions, as emotions are typically a reaction to an internal or external stimulus. For example, if someone is rude to us, then we feel angry. If we have doubts or insecurities about ourselves, we feel anxious and sad. If we perceive ourselves to be loved, then we feel happy.

And there we have the secret to controlling our emotions. We cannot choose what we feel, but we can choose what to think. By changing the way we think about or perceive something, we can directly influence how often or how intensely we feel certain emotions.

Take road rage as a common example. It is so easy and automatic to think that someone cut in front of us, or going too slow, or too distracted because they are terrible people or stupid. This thought and perception makes us enraged and frustrated, creating stress and sometimes even making us engage in risky behaviour such as tailgating or aggressively overtaking. But if we try to think of it from their perspective, they may be an inexperienced driver, in a rush or having a horrible day. At the very least, we can think of the times we have done the same thing to other people unintentionally. This change of perspective helps us suffer less from our emotional outbursts and overall reduces our stresses.

Take anxiety as another case, where if we stop and think rationally, many of our worries and doubts can be settled. The problem is that because we give less attention to our thoughts, our emotions take over and drag us down into a negative spiral. When that happens, our emotions override our thoughts and we powerless, feeling that we have no control over either our emotions or thoughts.

To counteract this, we need to have a paradigm shift where we know that we have the power to think freely. When we feel an emotion that we do not like, then we can approach it with mindfulness by recognising that we are feeling the emotion, then trying to diagnose the problem. To do this, ask yourself the following questions.

  • What am I thinking?
  • Why am I choosing to think this?
  • How does this thought make me feel?

The point of these questions is to figure out what thought is making you feel that way so you can fix the thought rather than the emotion. Even if you can’t, it puts you in the habit of forming a link between thought and emotion, leading to a healthier connection to your feelings and giving you back some control over them.

Posted in Science & Nature

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.)
Posted in History & Literature

Unspeakable Names

An important part of the Harry Potter story is the infamous villain Voldemort, who is so fearsome that the general populace are too afraid to say his name out loud. Instead, they call him “He who must not be named” or “You-know-who“.

The phenomenon of taboo avoidance of names is fascinating and examples can be found all around the world.
In ancient China and Japan, it was forbidden by law to say the emperor’s name, to the point that the names of some historical figures have been forgotten.
Some Australian Aboriginal cultures do not refer to their dead by name during the mourning period, but instead use titles such as kunmanara, translating to “what’s his name”.
In cultures speaking Highland East Cushitic languages such as some parts of Ethiopia, women practice ballishsha – a system where they avoid pronouncing any words beginning with the same syllable as the name of their mother or father-in-law.

This is called avoidance speech and it is typically used as a sign of respect or fear. For example, there are cases of cultures avoiding saying the name of demons or other evil creatures in fear that calling its name may summon it.

Perhaps the best example of this is the bear.
The old word for bear is arkto (note that Arctic comes from the same Latin roots, as the North is associated with the constellation Ursa Major and Minor – the bear). However, the bear is a fearsome wild beast and it was thought that saying its name would summon it, which would be particularly problematic if you were a hunter.
So instead, they used the word bear, which comes from the Proto-Indo-European word for “the brown one“. This practice became so commonplace that this euphemism became the present formal name for this animal.

Posted in History & Literature, Uncategorized

Meese

The plural for goose is geese. But the plural for moose is not meese: it is just moose. Why is this the case? This is because English is formed from words of various origins, all following different rules.

Goose is an old word that derives from Old English with Germanic roots. Typically in Old English, words were pluralised (turned into plurals) by a process called mutation, where the vowel sounds are changed to an adjacent sound (e.g. “oo” to “ee”). This explains why goose becomes geese, foot becomes feet and tooth becomes teeth.

However, the word moose traces its roots back to a Northeastern Algonquian language – a subfamily of Native American languages. This means that it does not follow the Old English rules of mutation. Furthermore, because Algonquian languages do not pluralise, the plural for moose is just “moose”.