Thread by @mathsppblog

“Be consistent”, they say.

But why?

Have you ever heard of the compounding effect?

Or exponential growth?

I'll show you how all these are connected, with real-life examples 👇🧵

I'll let you in on a little secret: I have a personal goal on Twitter.

The goal is to grow by 1% every day.

That's just that.

Every day, I want my follower count to increase by 1%

If you are anything like me, you might be thinking:

“Isn't 1% too little? You're gonna take ages to grow!”

Well...

Is 1% that little?

I'll show you it isn't!

First, here's a screenshot of my growth in the beginning of my Twitter journey 👇

Let's work with those numbers...

On the 6th of August, I woke up to 257 followers.

If I had grown 1% on that day, how many followers would I gain?

1% of 257 is 2.57, so I'd get 2 or 3 followers...

Not much, right?

Except...

Those 2 or 3 followers represent a growth of 1%.

1% is a relative measure, not an absolute one.

1% is always 1%, regardless of whether I have 257 followers or 257k.

So, if I keep growing at the rate of 1%/day, where will I be by the end of August?

Hm, growing at 1%/day,

I'd wake up to 257 × 1.01 followers on the 7th of August.

Then, I'd wake up to 257 × 1.01 × 1.01 followers on the 8th.

Then, I'd wake up to 257 × 1.01³ followers on the 9th.

,,,

And I'd wake up to 257 × 1.01²⁶ ≈ 333 followers on the 1st of September.

Well, still not much, right?

We can even look at the growth graph, and it'd look like this 👇

Pretty much a line.

The thing is, we are too close to the graph!

Let's zoom out a bit in the next one...

If I keep growing at the mere rate of 1%/day...

Where will I be in one year?

How many followers will I have on the 6th of August of 2022?

Well, let's plot it! 👇

After 1 year, I'd be at ≈9,710 followers!

How come ⁉⁉

That's just 257 × 1.01³⁶⁵

What seemed like a slow start, actually picks up the pace quite fast!

Just look at the graph again 👇

It starts as a “slow line”, but suddenly it “curves up” and explodes!

That's the power of consistency and compound growth.

Small improvements build on top of themselves.

1% improvement looks like a small thing.

But if you do 365 small improvements of 1%, one on top of the other, do you know how much better you get?

If you do 365 of those 1% improvements, you end up 37× better at whatever you were doing!

After just one year!

Isn't that amazing?

E.g., if you increase your follower count by 1% every day, you'll 37× your followers in one year!

Now, here comes the “paradoxical” part...

At +1%/day, how many days does it take to double your follower count?

Care to guess?

Or, rather, try computing the answer.

I won't spoil the answer for you yet.

👇

You need 70 days to double your follower count.

The graph below 👇 shows how the “improvement multiplier” grows with the days passing by.

(This is just the 1.01 to the power of the number of days elapsed.)

Doesn't the 70 look like “too much”?

Like, in 365 days my followers grow by 37× but it takes SEVENTY days to grow 2×?

But it makes sense, if you think of it...

In 70 days, followers × 2, so...

In 140 days, that's followers × 2 × 2, which is followers × 4.

Then, in 280 days, that's followers × 4 × 4, which is followers × 16.

See how these things quickly stack up?

Isn't maths amazing? ✨

So, we can see something here.

This compounding effect translates into using exponentiation when we do calculations.

That's why this is called exponential growth.

Because the growth is modelled by an exponential function.

For my 257 follower example, what's that function?

In Python 🐍, we could implement it as follows 👇

It's just the initial number of followers, with `× 1.01` written in front of it a bunch of times.

“A bunch of times” is as many times as days elapsed, and we can condense that as an exponentiation operation.

Making sense..?

I hope so!

Here's another very-real-life-example.

Suppose you have $100 with you and you decide you want to invest it.

You find that, for example, stocks can easily fluctuate 3% in a regular day, so you settle for a “conservative” investment goal:

You want to grow your money 0.5% each day.

That's not that much, right?

If you do manage to grow your money 0.5% each day, and started with $100, how much money will you have after 1 year?

That'd be approximately $617.47 😲

And after 2 years?

That'd be approximately $3,812.63 😱😱😱

Ok, ok, ok.

At this +0.5% / day rate, how long would it take for you to have $1M?

Can you figure the answer out, without using trial and error?

I'm effectively asking you to find the value of `n` in this code 👇

If we want to go from $100 to $1,000,000, our money needs to grow by ×10,000.

Hence, we need that 1.005 to the power of `n` be greater than or equal to 10,000.

We can find `n` using the logarithm function 👇

(We round up with `ceil` to figure out a whole number of days.)

If you don't know the `log` (logarithm) function, think of it as the function that plays a game with you.

Say you want a number, call it `base`.

Now you decide a `target`.

You want to exponentiate `base` to get to `target`...

E.g., how to exponentiate 2 to get to 32?
2⁵ = 32

So `log(32, 2)` is 5.

The logarithm just tells you how to reach the target!

In maths notation, we'd write log₂ 32 = 5.

My `base` was 1.005 and I wanted to reach the 10,000 multiplier.

That's why I did `log(10_000, 1.005)` to get my `n`, the answer to the challenge.

`n` was 1847 days, which is just a little over 5 years!

That sounds ridiculous, doesn't it?

Starting with $100, if you grow your money 0.5% / day, you become a millionaire in a little over 5 years.

Well, IMPORTANT disclaimer 🚨:

⚠ Be advised that the example above only went through the mathematics of investment growth.

It is theoretically possible to do what I described.

But it is ALMOST impossible to do it in practice.

Invest money at your own risk.

Enough with that, one final (SHORT) example:

Say you go to the gym to lift weights 🏋️

You are out of shape, so you start with a mere 10kg when exercising the legs.

But you want to see evolution, so you decide to increase weight by 1% each training session.

Sure, if you can do 10kg, ...

I'm SURE you'll be able to do 10.1kg the next day.

What's 100g? 20 sheets of paper?

You won't even feel the difference.

So you set the goal. Increase the weight by 1% each day.

In 70 days you'll be lifting 20kg with little effort.

But what about in 1 year..? 🤔

In 1 year you'd be lifting more than 370kg...

And in 2 years you'd be lifting more than 14,000kg.

That's more than 10 average cars 😂😂😂

What's the moral of this story?

Just because calculations work, doesn't mean that real life behaves like that:

Some things CANNOT grow exponentially...

For example, the amount of weight you will be able to lift probably looks like a different type of curve, a logistic curve 👇

Its start LOOKS like an exponential, but eventually stagnates.

Maybe I'll tweet about the logistic curve some other time 😊

That's it for now!

If you like concepts like this explained in simple terms and with good examples, follow me for more: @mathsppblog

Also, it'd help a lot if you retweeted the beginning 🙏

https://twitter.com/mathsppblog/status/1443533848411787266

Finally, thank you, @guilatrova, for the picture you took and tweeted, which inspired me to write this thread.

I also got one of the images above from @unsplash.

Here's a TL;DR:

- small gains accumulate over time;
- ↑ is explained mathematically with exponentials;
- logarithms do the inverse operation;
- 1%/day means ×2 after 70 days and more than 37× after 365 days; and
- don't forget to check reality before doing calculations!