Negative numbers: subtraction

Mathematicians tell us that there are positive and negative numbers. Only then do they talk about the absolute value of the number (the modulus of the number). I don't trust mathematicians. Are there negative numbers in nature? What do the plus and minus signs mean in math? We'll try to figure this out.

Negative numbers appear in arithmetic when a larger number is subtracted from a smaller one. That's what mathematicians say and that's how they teach us.

If you always subtract a smaller number from a larger number, then negative numbers will never occur. This rule is no better or worse than the first rule. But we are not taught this rule.

Subtraction

The minus signs fell off like dried mud. What remains is a crystal clear addition. As you can see, negative numbers cannot be obtained by subtraction.

I did everything right. That's how they taught me at school. If minus x equals minus one, then x equals one.

-x = -1
x = 1

What is the secret of this trick? Mathematicians solve the problem of subtraction incorrectly. If we know the result of the addition and one of the terms, then the second term can be found by subtracting the known term from the sum. The minus signs indicate that we are confused.

Let's see what my manipulations on the number axis look like.

Subtraction on the number line

On the left side of the figure, the numerical axis is fixed. I change the position of the segments relative to zero. On the right side of the figure, the segments remain motionless. I am changing the position of zero. In any case, the sign of the number changes when passing through zero. In algebra, this is equivalent to moving numbers through the equals sign. In the future, we will return to the numerical axis and consider it carefully. An equals sign and zero is a mathematical boundary where spies become scouts and scouts become spies.

We are done with subtraction. Next, we'll see if negative numbers come up when we borrow something.

Аverage percentage

Average percentage

On Reddit, a person asked an interesting question about percentages:

Help with explaining negative compounding

Hey guys, not a maths teacher and I need to teach a concept related to work where something is negatively compounded.

For example:

Of a 1000 items 20% are rejected, leaving 800. At the next stage of those 800 30% are rejected, leaving 560. At next stage of the 560 all are rejected at 100% rejection.

The "average" rejection rate is NOT (20+30+100)/3 equalling 50%, it's 42% because "maths". I do actually have the formula and I personally understand why it's 42% instead of 50 but can anyone help me try to explain in a manner that makes it simple to understand WHY it's 42 and not 50?

I wrote my comment:

This is an incorrect application of mathematics. The average failure rate can only be calculated within one process. The average percentage between processes is absurd. Mathematically it looks like this:

1000*(1-0,42)=580
580*(1-0,42)=336,4
336,4*(1-0,42)=195,112

I showed that the proposed concept of "average percentage" does not allow you to get zero at the end of the calculation, which is not true. The man explained how he sees his decision:

I'm not sure I understand.

I understand the formula, as you have written, as it is what I use, and what I demonstrated above in a reply to an answer. But I don't understand what you mean about the average percentage between processes being absurd.

If I use my previous example:

First use = 1 (Used once, no rejection)
Second use = 1*(1-0.2) = 0.8 (20% rejected)
Third use = 0.8*(1-0.3) = 0.56 (30% rejected)
Fourth use 0.56*(1-1) = 0 (End of life product, 100% rejected)

Total average = 1/(1+0.8+0.56) = 0.42

In the end, using the average of 0.42, gives us an approximate product rejection, or if we want to call it something else, it's average "life", before it needs a brand new replacement.

Now I will show you what is wrong in this decision.

Let's go back to 1000 items. Absurdity is the summation of numbers with different units of measure. Percentages are taken from different numbers:

I use     100%=1000   1%=10
II use    100%=1000   1%=10
III use   100%=800    1%=8
IV use   100%=560    1%=5.6

To determine the average percentage, we are actually adding numbers with different units of measure. We received 42%, but it is not known from what number these percentages are to be calculated. It's like adding one dress to one gallon. We can get two. But what does the number two mean? Two gallons, two dresses, or two wet dresses?

1[gallon]+1[dress]=2[?]

To get wet dresses, you need to use multiplication:

1[gallon]*1[dress]=1[gallon*dress]

This math matches reality much better. By the way. In mathematics, the absence of a number is denoted by the number "zero". The absence of a letter in the grammar is indicated by a "space". Mathematicians say that zero is a number. Philologists have never said that a space is a letter. In units of measurement, it is more logical to use a space.

1[gallon*dress]=1[wet_dress]

Now let's look at percentages. To be correct, we must count the number of items rejected at each stage as a percentage of the total number of items.

I use      0 items      0%
II use    200 items   20%
III use   240 items   24%
IV use   560 items   56%

It's even better to count as a percentage the number of items that fail after each stage with a cumulative total. This will be a more realistic picture of what is happening.

I use      0 items      0%
II use    200 items   20%
III use   440 items   44%
IV use   1000 items  100%

Personally, I argue like this. In this case, it is not appropriate to calculate the average percentage.

Sine and cosine table

This is the most unusual table of sines and cosines. The idea to make such a table came to me in the process of writing these articles:

Why is the sine of 30 degrees equal to half one?

What is the sine of 60 degrees?

Sine and cosine of 45 degrees

I strongly recommend that you read these articles. Then you will know how easy it is to calculate the values of sine and cosine yourself. I made one sine and cosine table from these three articles.

Sine and cosine table

The table indicates which sides of the triangle to take as a unit. For 0 and 90 degrees, triangles do not exist, these are ordinary segments.

The third column of the table shows how to use the Pythagorean theorem to calculate the desired value of the sine or cosine. For a cosine of 30 degrees and a sine of 60 degrees, the calculations are the same.

At the bottom of the table there are hints on how to calculate the tangent and cotangent values if you know the sine and cosine values. If you forgot how to divide one fraction by another, use one more hint. You need to multiply the first fraction by the reciprocal of the second fraction.

Sine and cosine of 45 degrees

We have already considered what the sine of 30 degrees is and what the sine of 60 degrees is equal to. Now we will see how to find the sine and cosine of 45 degrees and why they are equal.

Last time we considered an equilateral triangle. Now we will consider an isosceles right triangle. A very rare beast in a herd of triangles. Mathematicians have known it for many thousands of years, and it is simply boring for mathematicians to tinker with it.

Isosceles right triangle

We will consider a tired isosceles right triangle. Why is the triangle tired? He lay down on his side to rest.

Isosceles right triangle

The legs of this triangle are equal to one. Once again I repeat that both legs have the same length. We do not know the length of the hypotenuse of this triangle, but we can easily calculate it using the Pythagorean theorem.

Sine and cosine of 45 degrees

Smart people came to us and said that the sine and cosine of an angle are equal to the ratio of the legs to the hypotenuse. Since the legs of a right triangle with an angle of 45 degrees are equal, then the value of the sine of 45 degrees is equal to the value of the cosine of 45 degrees. We take the math in hand and calculate this value.

Sine and cosine of 45 degrees

Why did I multiply the numerator and denominator of a fraction by the square root of two? Small children do not like lumps in porridge. Mathematicians don't like square roots in denominators. Very capricious uncles and aunts.

What is the sine of 60 degrees?

Before we look for the answer to the question: "What is the sine of 60 degrees?", I strongly recommend reading the post "Why is the sine of 30 degrees equal to half one?".

Height of an equilateral triangle

I again take an equilateral triangle with a side equal to one. I draw height.

Height of an equilateral triangle

After that, I will not turn anything. I'll just remove half of the drawing.

Cosine 60 degrees

I have a right-angled triangle, the diagonal of which is equal to one.

Cosine 60 degrees

The base of this triangle is equal to 1/2 and is also equal to the cosine of an angle of 60 degrees.

Sine 60 degrees

The height of this triangle is equal to the sine of the angle of 60 degrees. We calculate the height using the Pythagorean theorem.

Sine 60 degrees

The sine of 60 degrees is equal to the square root of three divided by two. It is calculated in the same way as the cosine of 30 degrees.

Where does math end?

I was offered to sell my Russian site for 45.000 rubles. This prompted me to a very interesting question: "Where does mathematics end?". And here's my answer: "Math ends where money begins". Try any mathematical theory to apply to money and you will learn a lot of interesting things about this theory.

I love math trolling. The sentence contained the phrase: "The price may be revised upwards". My response was: "$45.000 Price may be revised down". From the point of view of mathematics, the equality 45.000=45.000 is beyond doubt. As soon as I add different units of money to this equality, it disappears.

Where does math end?

We use numbers to measure money. Numbers are written on coins and banknotes. Have you heard of banknote or a coin with the number "zero"? Zero is not a number. The more I study this question, the more I am convinced of the correctness of my statement.

Zero dollars

I'm not even talking about the minus sign on money. Through the prism of money, I considered set theory. The result is very interesting.

Why is the sine of 30 degrees equal to half one?

The answer to the question "why is the sine of 30 degrees equal to half one?" can be searched in the history of mathematics. These are ancient Mesopotamia, ancient Greece and other ancient civilizations. I am not an expert in this area. Obviously, knowledge has evolved from simpler to more complex. I will show you the most obvious answer to this question as I see it.

Equilateral triangle

Equilateral triangle

I have drawn an equilateral triangle with sides equal to one. I so want. The sum of the angles of a triangle is 180 degrees. An equilateral triangle has three 60 degree angles. I don't know trigonometry yet.

Height of an equilateral triangle

Height of an equilateral triangle

I drew the height in an equilateral triangle. The height is always perpendicular to the base of the triangle. If the height is drawn through the vertex of such a triangle, it will divide it into two equal right-angled triangles. This is always the case in isosceles triangles. An equilateral triangle is a special case of an isosceles triangle, in which the base is equal to the sides.

What happened as a result? The height divided the angle at the apex into two equal angles 60=30+30, it divided the base into two equal segments 1=(1/2)+(1/2). I still haven't heard anything about trigonometric functions.

Sine 30 degrees

After that, someone came up with trigonometric functions. I was told that the sine of an angle in a right triangle is the ratio of the opposite leg to the hypotenuse. How do I find the sine value for a 30 degree angle? I just flip the picture 90 degrees and remove all unnecessary.

Sine 30 degrees

The hypotenuse is equal to one. Any number divided by one does not change. So the length of the opposite leg in my triangle is equal to the sine of the angle of 30 degrees, that is, 1/2.

Cosine 30 degrees

The cosine of 30 degrees I can easily find from the Pythagorean theorem. We take the Pythagorean theorem in our hands and count.

Cosine 30 degrees

The cosine of 30 degrees turned out to be equal to the square root of three, divided by two.

Here's how easy it is to calculate. No tables needed.