? 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 🙂

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

? 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.

Continue reading “? Why 0! = 1 (zero factorial is one)?”

? 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

Continue reading “? Why negative times a negative is a positive?”