? Why is the area of a circle π·r² (pi r squared)?

The area of a circle vs a square

$$A=\pi r^2$$

The above is perhaps the best known formula and is also rarely understood. Although the formula for the area of ​​a circle was already known in Ancient Greece, its justification is not easy at all. So it’s a great topic to enrich the “Why?” series.

⭐️ The area of a circle – formula

The area of a circle vs a square

As you can see above – a square and a circle of the same area are not “somehow intuitively easy” related. Moreover, it has even been shown that squaring a circle (a procedure performed using a compass and a ruler without a scale) is impossible! And here comes the brilliant idea with a rectangle. Before I tell you what it is, let’s take a look at what the π·r² formula really says.

The area of a circle - 3+ squares

So there are slightly more than 3 squares of the side r in the circle with radius r.

⭐️ The area of a circle – proof by animation

The area of a circle - proof by animation

I worked hard on this animation, I hope you like it. Let me know in the comments 🙂

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? Why 0! = 1 (zero factorial is one)?

Scalar Calculator - Why 0! = 1

Recently, I was thinking about various justifications for the definition of 0! (factorial of zero) which is

$$0!=1$$

The assumed value of 1 may seem quite obvious if you consider the recursive formula. However, it did not satisfy me “mathematically”. That’s why I decided to write these few sentences. I will give motivations for the less advanced ones, but there will also be motivations for slightly more insiders.

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? Why negative times a negative is a positive?

Scalar Calculator - Negative times Negative

Surely everyone knows that the result of multiplying two negative numbers is positive. The formula “minus times a minus is a plus” or “negative times a negative is a positive” was put into our heads during the early school years. However, the teachers have forgotten to explain why this is the case, and to pass the motivation of mathematicians who defined the arithmetic of negative numbers.

⭐️ Multiplication as a short notation of repeated addition

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? User functions definition – the unique convenience provided by the Scalar Calculator

Scalar Calculator - User Functions

Functions for mathematics are what elementary particles are for physics. For this reason, the Scalar Calculator and its mathematical engine, provide the syntax for defining user functions, which is as close to natural as possible. Just enter f (x) = x^2 and you are ready to go. Stay tuned to learn a bit more details! 🙂

⭐ Basic user defined functions

The syntax for defining user functions is as follows.

FunctionName(param1 <, param2, …>) = expression

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? Fun fact – Scalar – digits clicks distribution – real data

Scalar Calculator - digits clicks distribution - by digit

After few months of Scalar being present in the Google Play Store, we decided to check what is the distribution of digits (buttons) used by our users directly in the application. Below we present gathered data that say something, but we do not know exactly what. Maybe you will help in the interpretation? Do you see any relation to the Benford’s law?

⭐ Scalar – digits clicks distribution – ordered by digit

The most frequently used numbers are 2 and 5. It is also well visible the effect of multi-use of the digit 0 considering particular user perspective. The digit, that is used the least often, is 7.

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