## Tag Archives: practical math

#### Degrees and angles | 角度

Today in the fourth grade weekend lesson we learned about angles📐 and degrees. We explained what degrees was first🥇 because angles are classified using degrees📏. Degrees were discovered by Egyptians. They invented the degree symbol ° and also came up with the 360° circle⚪. There is an interesting history of how they connect the movement of the Sun with time. They first divided a year into 360 days, noting that the Sun moved in a circle. Around 1500 BC Egyptians divided 24 hours⏳, though the hours varied with the seasons originally. Then the Greek astronomers made the hours equal. About 300 to 100 BC the Babylonians subdivided the hour into base-60 factions: 60 minutes an hour and 60 seconds in a minute.

We use degrees to measure angles. An angle is a figure formed by two rays🛴 called sides of the angle. In geometry there are three types of angles: an acute angle between 0 and 90 degrees, right angle a 90 degree angle, and an obtuse angle between 90 and 180 degrees. In 1936 a clay tablet was found buried at Shush (Khuzestan region of Iran🌏) some 350km from the ancient city of Babylon on which was inscribed a script that was only translated as late as 1950. The text provided confirmation that the Babylonians measured angles using the figure of 360 to form a circle. The Babylonians knew that the perimeter of a hexagon was exactly equal to six times the radius of a circumscribed circle. This is why they chose to divide a circle in 360 parts⚪. If we did not discover degrees or angles we would not be able to build anything properly🧱. So if we tried building a house without degrees or angles the house would collapse🏚.

We will talk about triangles next time.

#### Welcome to our New Google Classroom!!😃

Hello everybody!😃 Welcome back to Magic Math Mandarin. Since we are staying home because of covid-19, we brought our classes to google classroom so our students can keep on learning even during this pandemic👩🏫. Now we can all communicate with each other online💻. In this classroom we will be learning Chinese🈷, math➗, and programming👩💻. Our teachers will put new assignments everyday about each topic. If you would also like to join our wonderful classrooms then here is the class code **mtxl6j4**

Remember to stay home and don’t get sick!😷 Please join our classroom today!👍💖

#### Wealth distribution | 财富分配 💲

We went onto a journey of wealth discovery in this weekend’s math class. We played with real world data from the Federal Reserve .

We spent the first five minutes just looking quietly at the chart below, and tried to understand its meaning. Hint: read legends and labels.

The area chart shows the time series of total wealth of 4 buckets of US population.

The buckets are defined as 1st percentile, 90th – 99th percentile, 50th – 90th percentile, and the bottom 50% percentile of population by wealth. In other words, if all the people in the US are ranked by wealth, the top 1%, and the top 90% to 99%, …, and the last 50%.

Source: Federal Reserve

The remainder of the class is discussion on what the chart means.

Kudos to Emily. She was quick to point out that the red area of the chart is as thin as a line, and that’s all the wealth of the bottom 50% of Americans.

While the rest of the people are having more and more wealth overtime, the bottom 50% seem to have less and less, relative to the rest.

So is the world fair place? No.

Some questions for the class to think about:

- How to make the world more fair?
- Can the world be more fair?
- Should the world be more fair?
- What does fair mean?

#### Logit transform | 分数对数转换

After we discussed logarithm (‘log’) last week, we explored a bit on some commonly used methods that have log embedded in them. For example, the logit function, or logit transform (using the “natural” logarithm). We explained its definition by the following Python code.

>>> epsilon=0.001

>>> def logit(c):

>>> d = np.log((c+epsilon)/(1+ epsilon-c))

>>> return d

The following is the inverse, which is to bring what was transformed back to what it was before.

>>> def inverse_logit(a):

>>> b = ((1+ epsilon)*np.exp(a) – epsilon)/(np.exp(a)+1)

>>> return b

>> print(logit(0.1)) #-2.1883847407670785

>>> print(inverse_logit(logit(0.1))) #0.09999999999999999

It is much more revealing on what the logit transform is doing by looking at some pictures of how this works. See how fast when it is transformed! Why it is stretched instead of being shrunk? We know that taking log is to do division multiple times (recall log10 of a number is how many times it needs to divide by 10 in order to become 1). But when it applies to numbers between 0 and 1, it gives us the opposite effect. A positive small number less than one has to divide by 10 negative times to become 1. For example, 0.01 needs to be divided by 10 negative two times to be restored to 1. That’s why you see that y axis we have negatives.

On the other hand, we also have positives in the y-axis. That’s because about half of the numbers `(c+epsilon)/(1+ epsilon-c)`

(the odds) are large positive numbers. Play around with it and you will surely get it.

>> x = np.linspace(0,1,1000)

>>> y = logit(x)

>>> plt.scatter(x=x, y=y, alpha=0.3)

>>> title =”logit transform”

>>> plt.title(“%s”%title)

>>> plt.xlabel(“numbers between 0 and 1 (inclusive”)

>>> plt.ylabel(“after logit transform”)

>>> plt.xlim(-6, 6)

>>> plt.ylim(-6, 6)

>>> plt.gca().set_aspect(‘equal’, adjustable=’box’)

>>> plt.draw()

Look at the same plot with the axis scaled differently:

#### Cosmic distance ladder | 宇宙距离

The class has no homework today. We watched the video lecture by Terence Tao (see link below). The name of the video is “Cosmic Distance Ladder”. Quite a mystifying name.

The stories, which Terence Tao told in the lecture, were about philosophers and astronomers from ancient times, such as Aristotle and others, and those who were closer to us in history. What all of them have in common is that they were able to use good observations and ingenious reasonings to indirectly measure the distance between the Earth and the Moon, and the Sun, and the distance of the galaxies, without any technology (the earliest did not even know the number Pi), with amazing accuracy (as verified by what we know today).

You should definitely watch the video a few times. Think about this: compare with human observation and reasonings, what computers can do is still just technology and tools. The computers can’t do indirect reasonings that connect the dots from disparate information. It makes zero sense to believe computers (including phones) are smarter than you are.

So, use your great mind. Let your mind observe and reason, and make computers help you along the way.

#### Logarithm | 对数

As we had explored in previous classes, division is subtraction again and again and again, multiplication is adding again and again. Exponentiation is multiply again and again and again— They are all **inventions to simplify repeated computation**.

So is the invention of logarithm: taking log is division again and again and again. They were invented by John Napier who was a Scottish mathematician, physicist, and astronomer in 1614 as a means to simplify calculations.

🙂 Today’s Python **numpy **class summary:

Log10 means how many times divide by 10 will return you to 1. log10(100) will give you 2 because 100 divide by 10 twice returns us to one.

>>> np.log10(100)

One trillion divide by 10 twelve times returns it to 1.

>>> np.log10(1000000000000)

>>> np.linspace(0.0, 3.0, num=4)

Out: array([0., 1., 2., 3.])

>>> np.logspace(0.0, 3.0, num=4)

Out: array([ 1., 10., 100., 1000.])

>>> np.linspace(0.0, 12.0, num=13)

Out: array([ 0., 1., 2., 3., 4., 5., 6., 7., 8., 9., 10., 11., 12.])

>>> np.logspace(0.0, 12.0, num=13)

Out: array([1.e+00, 1.e+01, 1.e+02, 1.e+03, 1.e+04, 1.e+05, 1.e+06, 1.e+07, 1.e+08, 1.e+09, 1.e+10, 1.e+11, 1.e+12])

Bonus: Did you know that Engineers and scientists used to use a tool called “slide rule” (计算尺) to do logarithmic computations until 1970s when electronic computer and calculators came into use. You should go and check it out if any of your grandparents have one of these.

#### What is social credit score | 什么是社会信用分

Social credit score 社(shè)会(huì)信(xìn)用(yòng)分(fēn) means rating how trustworthy you are based on your spending habits, social connections and your online behavior on social media. Traditional credit scores are what lending institutions use to judge how likely you will pay back the money before they lend you. Traditional credit score use information such as whether and how long you have a job, how much money you owe relative to how much you earn, and so on. Social credit is being used in similar ways with a different set of data. Tencent Credit 腾讯信用分 and Sesame Credit 芝麻信用分 are the prime examples.

For example, social credit score can be like a test score number that ranges between 300 and 850 and made up of five dimensions/categories 5 个维度:

- social connections
- consumption behavior
- security
- wealth
- compliance

WSJ reported in 2016 on social credit in China with an article title “China’s New Tool for Social Control: A Credit Rating for Everything“. The words “social control” and “Big Brother” have bad connotations. However, politics aside, we do appreciate people who have good social credit.

What can social credit scores be used for? Traditional credit scores are used by banks or other lenders to approve loans, used by employers to screen candidates, and some other kind of approvals. Likewise, social credit scores can be used for similar purposes. Alibaba’s affiliate Ant Financial opened a strictly online bank called MyBank that serves small businesses and individuals. This online bank takes deposits the same way as Synchrony or AlyBank in the US do. But it also gives out unsecured loans (without any collateral) up to $850,000. That is a lot of money to lend without collateral (by the way, mortgages are secured loans collateralized by the house). Underlying the decision to lend or not to lend is social credit score, calculated based on huge amount of online transaction data.

Back to the US, which companies may have data that can generate a full or partial social credit score? I think Airbnb, Amazon, Facebook, Lyft, UBER and etc. all have huge amount of data that can be used for credit scoring. Due to the high regulatory cost (banks are highly regulated), it is not likely that any of them would want to become a bank. But they can take some of the businesses that banks always have been doing. For example, Amazon has Amazon Cash and variations of it in various countries such as the U.K. Also, UBER has recently offered UBER cash. These are banking businesses: they allow people to shop or ride without a credit or debit card as long as you load your account with cash.

#### What does aggregate mean?

In math and especially statistic, aggregate means to get some ideas on a set of numbers or measures.

For example, let’s say we are having lunch at the school cafeteria. You may want to know some aggregated measures, such as:

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