0.999… = 1?

Such an innocent question. So much controversy around it.

No doubt, you have heard of the usual explanations:

1/3 = 0.333…, therefore 1 = 0.999…
Let x = 0.999…, then 10x – x = 9. Therefore x = 1.
You can’t find a number between 0.999… and 1.
The = sign here merely means “approaches”.
… and many more.

Some of these explanations are valid; some are not. But, even as the rational part of your brain accepts that 0.999… = 1, the intuitive part of your brain might still be complaining “but it doesn’t feel right”. You know why? Because it isn’t right. It isn’t complete. The full statement should be:

If 0.999… is a real number, then it must be equal to 1.

The part of your brain that thinks “but it doesn’t feel right”? It’s trying to say, hey, this doesn’t look like a real number.

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Jianpu (simplified staff)

It’s one of those things that weren’t invented in China but took off in China: 简谱jiǎn pǔ is a way of writing down music using numbers and dots and dashes. It was invented because the standard notation can be difficult to read, especially to beginners:

But this is (arguably) more accessible:

For one thing, there’s no need for “every good boy deserves fun” anymore.

Wikipedia (en, zh) says this notation was invented in France in the 1700s and came to China by way of Germany and Japan in the early 1900s. I haven’t seen it used anywhere outside of China, though — which is a shame, because it’s such a handy tool.

The Notes

简谱 uses the numbers 1 through 7 to denote the seven notes on a major scale — that is, do, re, mi, fa, so, la, si (ti) — and uses a 0 to denote a pause. Very often, this translates to 1 = C, 2 = D, …, 6 = A, 7 = B.

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When is the Chinese New Year?

At this point, most of the English-speaking world has an awareness that the Chinese New Year happens somewhere around January or February. Some might even know that it moves around relative to the western calendar because the Chinese calendar is lunar. But did you know that the Chinese calendar is actually half-lunar, half-solar? It even relies on the same solstices and equinoxes that we know and love in the west.

Through the thousands of years in China’s ancient history, the Chinese calendar has been revised multiple times, so the rules we describe below will not perfectly backdate to ye olden days. The basic principles have always been the same, but the details (leap years and such) haven’t been consistent throughout.

The Chinese calendar currently follows the Chinese national standard GB/T 33661-2017, which was last revised in 2017 (as the name suggests). For some reason, you can purchase the full text through the (American) ANSI webstore for the low, low price of $180 (lol).

Following the GB/T, we’ll reach this conclusion: the Chinese New Year happens on the second, occasionally the third, new moon following the winter solstice.

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Reading Math Out Loud | 数学公式的英语读法

This will be a bilingual post. 本文双语。

When multiple readings are possible, the most commonly used ones are listed on top. 有多种读法时，最常用的读法排在前面。

For Chinese learners
↓

英文读法在这里
↓

Basic arithmetic | 四则zé运yùn算suàn

1 + 2

一加jiā二

一加上shàng二

one plus two

add two to one

1 – 2

一减jiǎn二

一减去qù二

one minus two

subtract two from one

one takeaway two*

* takeaway 稍不正式，一般是教小孩子做减法时这么说。

1 × 2

一乘chéng以yǐ二

二乘一

one multiplied by two

one times two

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Short divisions, etc.

Starbucks has got to have the most annoying drink sizes of all coffee shops. Instead of small, medium, and large, they offer “tall”, “grande”, and “venti”. It’s sometimes English and sometimes Italian, the small size is called “tall” and the medium size means “big”, and the large size forgoes adjectives altogether and just calls itself “twenty”. Twenty what? Twenty fluid ounces, of course, which, as we all know, is the standard unit for measuring coffee drinks worldwide and especially in Italy (yeah, right).

I can’t do anything about the fluid ounces, but the “tall” name actually has an explanation: Starbucks offers five drink sizes, not three:

Size Name	Fluid Ounces
Short	8 fl oz
Tall	12 fl oz
Grande	16 fl oz
Venti	20 fl oz for cold drinks, 24 hot
Trenta	31 fl oz

As of October 2021

It certainly looks like they planned to offer the first three sizes — short, tall, and grande — as the standard sizes, and venti and trenta were meant to be “super sized” drinks. Who knows, maybe the short cup is too short, maybe they underestimated the American appetite, it came to be that the middle three sizes became the standard. You can still ask for a short at Starbucks; they just don’t list that size on the menu anymore.

This venti-sized intro leads us to today’s main topic: we all know long division from school. Is there a short division?

Yes. Yes there is.

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Mortgage math

What’s the monthly payment on a $360,000 mortgage, at an interest rate of 3%, to be repaid over 30 years?

Just let me quickly have a Google…

No! What’s the fun in that? Let’s re-derive the formulas from scratch. We will, in the end, reach this conclusion:

$p = \frac{L\cdot r}{1-\frac{1}{(1+r)^n}}$

Although this post talks about a mortgage, this formula applies to any fixed-rate installment loans, such as a car loan or a purchase through Affirm.

Note: This is not a money blog. None of this is financial advice. This post will touch on a few minor financial topics, but only to clarify the math.

Principal and Interest

For most mortgages, interest does not compound. Let me say that again: mortgage interest does not compound. Unless your mortgage has a “negative amortization feature” (which, in the US, would be clearly indicated on page 4 of your Closing Disclosure), your monthly payment will cover the entirety of interest accrued in that month.

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Math with units

What’s the average fuel economy of two cars, one with 40 mpg, and the other with 10 mpg? It all depends on what you mean by “average”. (That’s 6 and 24 liters per 100 km.)

Mathematicians often deal with pure math — that is, numbers without units. (40 + 10) / 2 = 25 might be a correct equation, but it might not be the appropriate one for our problem. In real life, most numbers have a unit: 3 days, 5 cars, 8 people, 11 dollars. When numbers come with units, there are additional rules for doing math.

Addition and Subtraction

Adding is counting. Subtracting is counting backwards. When counting, one must stay with the same unit. “1 apple, 2 apples, 3 apples” is counting. “1 apple, 2 oranges, 3 pears” is a shopping list.

1 apple + 2 apples = 3 apples
5 hours – 3 hours = 2 hours
1 apple + 1 hour = ??? (doesn’t make sense)

You are, of course, allowed to convert units, if they’re convertible:

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Hindsight is 5.0

As the saying goes, “hindsight is 20/20”, meaning “things are clearer in retrospect”. 20/20 is a measure of a person’s visual acuity: you can see from 20 feet away what a “normal” person can also see from 20 feet away. In other words, your eyesight is normal (but you can still be colorblind, etc).

20/20 isn’t perfect vision; people with good eyesights often measure 20/15 or even 20/10 — they can see from 20 feet away what a “normal” person can only see from 15 or 10 feet away. D. R. on the other hand is severely nearsighted and measures worse than 20/200 without correction — without my glasses on, I can only see at 20 feet what “normal” people can see from 200 feet or even further away. 20/20 is the low end of “normal”; if you see at least 20/20, then an eye doctor wouldn’t prescribe you any glasses.

Decimals

What’s special about “20 feet”? Not much, really. It’s just been a standard testing distance since the Dutch ophthalmologist (= eye scientist) Herman Snellen came up with this measurement. In metric countries (i.e. outside the US), optometrists (= eye doctors) write 6/6 (meters) instead of 20/20. 6.00m = 19.7ft, but humans like round numbers.

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PEMDAS is incomplete

Parentheses, Exponents, Multiplication, Division, Addition, Subtraction. Order of operations. 4th grade math.

Elsewhere in the world, parentheses are called brackets, and exponents could be called orders or indices. So depending on where you went to school, you might have learned BEDMAS, BODMAS, BIDMAS, or something similar.

Hang on — is it PEMDAS or BODMAS? Is it MD or DM?

Equal Precedence

My first bone to pick with PEMDAS (and the like) is that on the first glance, it’s six things, when it should only be four.

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