The title is admittedly clickbait. Or a joke. Or a provocation. It depends on who you are talking to or who is reading it with what expectation. Well, I can’t influence any of that. I can only say what I mean, namely as an entertaining collection of simple truths that later turn out to be not as simple as expected from a scientific point of view. Some of them could even lead to heated debates within the scientific community, so maybe I should say: from my point of view. So before you get upset or even angry about what’s to come, please remember to take it with a large grain of salt and be entertained and not feel educated.

## # 10. The sum of all the angles in a triangle is 180 degrees.

We all live on a globe, a huge ball. The angles of a triangle on this ball add up to a number greater than ## 180 ° ##.

And the amount by which the sum ## 180 ° ## extends is not even constant. It depends on the size of the triangle. The flat triangle with the sum of the angles ## 180 ° ## is the exception, not the norm. The real world is crooked.

## # 9. What is 360?

Measuring angles in degrees is confusing at best. Even the calculator on the computer allows three versions of a full angle: ## 360 °, 400 °, 2 pi ##. And who used the ## 400 ° ##? Anyway, ## 2 pi ## is what it should be: the ratio of the perimeter of a circle with radius ## 1 ## to its radius ## 1 ##. This is how angles are used in math: multiples of ## pi ##. Degrees should be treated like Roman numerals: a historical aside.

## # 8. Parallels do not overlap.

Have you ever stood on a railroad track? Nobody would deny that the rails are parallel. Do they overlap? Certainly not soon, because locomotives don’t usually derail. However, you have probably been looking towards the horizon on the rails. And – surprisingly – they cross on the horizon or mathematically: at infinity. But infinity on a ball we live on is not infinite. It’s actually somewhere. We see that there are possible geometries in which parallels intersect.

## # 7. What is a tangent?

Yes, it is the derivative of ## y. ## But what does that mean? Obviously we have a function ## x longmapsto y = y (x) ## and a derivative $$ y ‘= y’ (x) = dfrac {dy} {dx} = left. dfrac {d} {dx} right | _ {x = a} y (x) = y (a + h) -J (h) -r (h) = y ‘(a) $$ It is not now’ It is not at all clear what is meant: the function ## x longmapsto y ‘(x) ##, the value of the slope ## y’ (a) ## or the linear map ## J, ## of the Jacobi matrix, the tangent in a certain way? The fact is that they are all needed depending on the situation. I’m not saying we should be teaching tangent bundles and sections, but a little more accuracy would smooth the step to calculus in college.

## # 6. ## 0.999999999… = 0. Bar {9} = 1 ##

That’s not really a problem at school. It’s a problem of understanding. Yes, the two numbers are actually the same. They are just written in a different representation, e.g. B. ## pi = 3.1415926… ## Notes pi and everyone knows that it doesn’t end with ## 6 ## or ever. Nevertheless we write ‘## = ##’ and not ‘## approx ##’ because we mean the actual number, not its approximation.

begin {align *} 0. bar {9} = 0.999999999… & = dfrac {9} {10} + dfrac {9} {100} + dfrac {9} {1000} + ldots \[10pt] & = sum_ {i = 0} ^ infty dfrac {9} {10 ^ {i + 1}} = 9 cdot sum_ {i = 0} ^ infty dfrac {1} {10} cdot dfrac {1} {10 ^ {i}} stackrel {[10pt] } {=} dfrac {9} {10} cdot lim_ {n to infty} sum_ {i = 0} ^ n dfrac {1} {10 ^ {i}} \

& = dfrac {9} {10} cdot lim_ {n to infty} dfrac {1- left ( dfrac {1} {10} right) ^ {i + 1}} {1- dfrac {1} {10}} stackrel {(**)} {=} dfrac {9} {10} cdot dfrac {1-0} {1- dfrac {1} {10}} = dfrac {9} {10} cdot dfrac {10} {9} = 1 end {align *}

The crucial points are the limits. While the first ##

## ## is simply a translation for ‘and so on’, ie the points in ‘## 0.999999999… ##’ that shouldn’t be controversial, the misunderstanding starts with the second ## (**) ##. A limit is not a process, it is a number! And ## 0. Bar {9} ## is also not a process, but a number. Number 1.##

# 5. The rational are numbers.

You are not. They are equivalence classes. My favorite example is that it makes a big difference whether you take a cake from the bakery or ## 12/12 ## pieces of cake home with you. The amount of cakes and the prices would be the same, their appearance is not. Of course we treat ## 1 = frac {12} {12} ## the same because we are interested in its value, but they are only the same because ## 1 cdot 12 = 12 cdot 1. ## It becomes clearer in its general form:

$$ dfrac {a} {b} sim dfrac {c} {d} Longleftrightarrow a cdot d = b cdot c $$

Strictly speaking, ‘## sim ##’ is an equivalence relationship. It really collects a lot of quotients under one name

$$ 1 = left {1, dfrac {2} {2}, dfrac {-3} {- 3}, dfrac {12} {12}, ldots right } $$

## The same applies to all other quotients. We take them as the same and write ‘## = ##’ instead of ‘## sim ##’ because we are only interested in their values. But ## 1 neq dfrac {12} {12}. ## You can literally see it is different: ## 5 ## symbols instead of ## 1. ##

# 4. Reduce an equation.

Hopefully this time I agree with the teachers. The standard procedure is

$$ ab = ac ; Longrightarrow ; b = c $$

Not only is it sloppy, it’s wrong sometimes and a standard class error. If we all took the time to write it down in more detail, this could really avoid the error: begin {align *} ab = ac ; Longrightarrow ; ab-ac = 0 ; Longrightarrow ; a cdot (bc) = 0 ; Longrightarrow ; a = 0 quad text {or} quad b = c end {align *}

## The second possibility is suddenly obvious. The advice behind this is simple: avoid divisions for as long as possible. And if so, make sure you are allowed to. What brings us to …

# 3. You cannot divide by ## 0 ##.

## It always sounded to me as if there was an obscure math police forbidding us from doing this. Why? Well, there is actually a simple reason: ## 0 ## has nothing to do with multiplication and even less with division. The question of dividing by ## 0 ## doesn’t even arise! The neutral element for the multiplication is ## 1, ## not ## 0 ##. That’s part of it. And there is only one way to combine the two, namely through the distribution law ## a cdot (b + c) = a cdot b + b cdot c. ## Let us imagine that there was a reasonable way to ## define m = a: 0. ## Then $$ a = m cdot 0 = m cdot (1-1) = m cdot 1 – m cdot 1 = mm = 0 $$ So we can at least get ## divide a = 0 ## from ## 0 ## and ## m ##? I’m afraid not. Look at $$ m cdot a = dfrac {0} {0} cdot a = dfrac {0 cdot a} {0 cdot 1} = dfrac {0} {0} = m Longrightarrow a = 1 $$ or $$ 2m = (1 + 1) cdot m = m + m = dfrac {0} {0} + dfrac {0} {0} = dfrac {0 + 0} {0 } = dfrac {0} {0} = m $$ and we could only multiply ## m ## by ## 1 ##. Even if we define ## m = 0/0 ##, we wouldn’t get anything useful in the sense that it would be connected to whatever number we normally use for calculations.

## # 2.1 + 1 = 2.

1 + 1 = 2

## 1 + 1 = 2

[sqrt{-5}] # 1. A prime number is only divisible by ## 1 ## and itself.

This is wrong. Yes and no. Strictly speaking, this definition describes irreducibility. And irreducibility is different from primality. A prime number is actually a number that if it divides a product it must divide one of the factors. $$ 7 , | , 28 = 2 times 14 ; Longrightarrow ; 7 , | , 2 text {or} 7 , | , 14 $$ $$ 4 , | , 12 = 2 cdot 6 text {but} 4 , nmid , 2 text {and} 4 , nmid , 6 $$ It’s a little more complicated, but it’s the right definition. However, irreducible integers are prime numbers and vice versa, which is why the correct definition in school is replaced by the more practical one. However, there are areas where this is not the case. The standard example is ## mathbb {Z}

## where $$ 6 = 2 cdot 3 = (1+ sqrt {-5}) cdot (1- sqrt {-5}) $$ is a decomposition into irreducible factors that are not prime numbers.

Masters in mathematics, minor in economics, and always worked in the periphery of IT. Often as a programmer in ERP systems on different platforms and in different languages, as a software designer, project, network, system or database administrator, maintenance and even as a CIO.