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Comment by contravariant | Hacker News Reader

I blame this phenomenon for why people feel mathematics is not useful. It is, it's just that reality is more complex than high school mathematics can properly prepare you for. To understand the mathematical solution, should you encounter it, does however require some foundational knowledge so if you don't know any then you can't even begin to understand it (possibly even failing to identify it is mathematical).

I mean take something practical like a mortgage. It's fairly easy to calculate the annuities using a geometric series, but that places it beyond most people's mathematical skills. Sure you could use a special calculator, or if you're adventurous look up the formula (beware though, Wikipedia is bound to lead to errors by defining the monthly rate as yearly rate / 12), and then it probably may not like doing mathematics at all but neither do you really understand what's happening.

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"If people do not believe that mathematics is simple, it is only because they do not realize how complicated life is" - John von Neumann [1]

Saw this quote on an art blog showing SEMs of diatoms [2]

[1] https://butdoesitfloat.com/Diatoms [2] https://en.wikipedia.org/wiki/Diatom

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Yay John von Neumann, genius of geniuses, also said Young man, in mathematics you don't understand things. You just get used to them.

Yep, 10x that for real life.

It's astounding how some of math's first practical applications is to calculate movement of the stars. Absence of frictions and close bodies make physics much easier to approximate than anything on earth. Maybe to have students appreciate math, we should have them predict eclipse one again. God create astronomy to teach humans mathematics.

The oldest writing is numbers... for accounting. I'm pretty sure Pythagoras' theorem was for property boundaries.

Although the oldest mathematical work I'm aware of on accounting is early medieval (al khwarizmi).

Yet you will find that mortage interests charged by the bank don't match your series, because they use an accrual schedule with a special way to measure years fractions, then calculate an equivalent notional for those dates, and remap that to your payment calendar.

Turns out there is a surprinsing amount of details here as well. And you don't have all parameters for the calculation, like often in life.

> (beware though, Wikipedia is bound to lead to errors by defining the monthly rate as yearly rate / 12)

Lead to errors relative to what? The mathematical idealisation or the actual practice?

Are you referring to this page?

https://en.wikipedia.org/wiki/Mortgage_calculator

"Since the quoted yearly percentage rate is not a compounded rate, the monthly percentage rate is simply the yearly percentage rate divided by 12."

Do you think that the explanation above is wrong?

In practice interest payments are calculated in many "wrong" ways, but that's what it is:

https://en.wikipedia.org/wiki/Day_count_convention

“All models are wrong, but some are useful”

https://en.m.wikipedia.org/wiki/All_models_are_wrong

I was looking at: https://en.wikipedia.org/wiki/Mortgage#Principal_and_interes...

It's not necessarily wrong, but it's missing any disclaimer about which interest rates they're talking about so if you don't know what you're doing it will lead to mistakes.

Ok, so defining the monthly rate as yearly rate / 12 will lead to mistakes if the yearly rate is not twelve times the monthly rate.

(Of course any other definition of monthly rate will also lead to mistakes when it's inconsistent with the definition of yearly rate.)

Then you can start layering on the complexity by estimating repayments based on the expected variability of future interest rates, convert to Real Dollars based on inflation and expected wage growth, then compare with alternative options such as renting + investing, etc...

You are correct about the mortgage payments, I had a 1-2 cent error every few months, after a days struggle, I simply added a “Penny” column to add or subtract as necessary so that they would match!

I dunno. It seems to me like high school math is a pretty decent foundation for understanding many important aspects of the world (derivatives, integrals, etc.).

And I think that one can clearly and substantively “get” and intuitively understand and intelligently work with an amortizing loan without needing to understand whatever level of math is required to comprehend why pressing (g) (12i) on an HP 12c is not an atomically precise representation of of the answer out to the 12th decimal place.

> beware though, Wikipedia is bound to lead to errors by defining the monthly rate as yearly rate / 12

Looks like all the calculators do that as well. What's the right way?

My mortgage company seems to use the same formula.

I assume to take compounding into account, you want the 12th root. Instead of 12% → 1%, 1.12^(1/12) gives around 0.95%.

The effective interest rate should be calculated and then converted to the monthly rate:

$$ \left( 1 + \frac{i_a}{n_a} \right)^{n_a} = \left( 1 + \frac{i_b}{n_b} \right)^{n_b} = \left( 1 + \frac{i_{\text{annual}} }{1} \right)^{1} $$

https://www.investopedia.com/terms/e/effectiveinterest.asp

My mortgage company charges interest based on the number of days in the month divided by 365. I have replicated their calculation like this. I'm not sure what they do in leap years - there are at least 3 distinct approaches.

It could be correct, it just depends. Also according to the comments banks do all kinds of weird things.

This is a bit of a rabbit hole. The right way is whatever is agreed, and there are many ways to agree it.

https://en.m.wikipedia.org/wiki/Day_count_convention

Cant you just use 365 instead of using something less accurate like 12 months (months are unevenly distributed)

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