In this video we will analyze a whole set of linear equations that are solved using the same algorithm - that’s why they are called the simplest.
First, let's define: what is a linear equation and which one is called the simplest?
A linear equation is one in which there is only one variable, and only to the first degree.
The simplest equation means the construction:
All other linear equations are reduced to the simplest using the algorithm:
- Expand parentheses, if any;
- Move terms containing a variable to one side of the equal sign, and terms without a variable to the other;
- Give similar terms to the left and right of the equal sign;
- Divide the resulting equation by the coefficient of the variable $x$.
Of course, this algorithm does not always help. The fact is that sometimes after all these machinations the coefficient of the variable $x$ turns out to be equal to zero. In this case, two options are possible:
- The equation has no solutions at all. For example, when something like $0\cdot x=8$ turns out, i.e. on the left is zero, and on the right is a number other than zero. In the video below we will look at several reasons why this situation is possible.
- The solution is all numbers. The only case when this is possible is when the equation has been reduced to the construction $0\cdot x=0$. It is quite logical that no matter what $x$ we substitute, it will still turn out “zero is equal to zero”, i.e. correct numerical equality.
Now let's see how all this works using real-life examples.
Examples of solving equations
Today we are dealing with linear equations, and only the simplest ones. In general, a linear equation means any equality that contains exactly one variable, and it goes only to the first degree.
Such constructions are solved in approximately the same way:
- First of all, you need to expand the parentheses, if there are any (as in our last example);
- Then bring similar
- Finally, isolate the variable, i.e. move everything connected with the variable—the terms in which it is contained—to one side, and move everything that remains without it to the other side.
Then, as a rule, you need to bring similar ones on each side of the resulting equality, and after that all that remains is to divide by the coefficient of “x”, and we will get the final answer.
In theory, this looks nice and simple, but in practice, even experienced high school students can make offensive mistakes in fairly simple linear equations. Typically, errors are made either when opening brackets or when calculating the “pluses” and “minuses”.
In addition, it happens that a linear equation has no solutions at all, or that the solution is the entire number line, i.e. any number. We will look at these subtleties in today's lesson. But we will start, as you already understood, with the simplest tasks.
Scheme for solving simple linear equations
First, let me once again write the entire scheme for solving the simplest linear equations:
- Expand the brackets, if any.
- We isolate the variables, i.e. We move everything that contains “X’s” to one side, and everything without “X’s” to the other.
- We present similar terms.
- We divide everything by the coefficient of “x”.
Of course, this scheme does not always work; there are certain subtleties and tricks in it, and now we will get to know them.
Solving real examples of simple linear equations
Task No. 1
The first step requires us to open the brackets. But they are not in this example, so we skip this step. In the second step we need to isolate the variables. Please note: we are talking only about individual terms. Let's write it down:
We present similar terms on the left and right, but this has already been done here. Therefore, we move on to the fourth step: divide by the coefficient:
\[\frac(6x)(6)=-\frac(72)(6)\]
So we got the answer.
Task No. 2
We can see the parentheses in this problem, so let's expand them:
Both on the left and on the right we see approximately the same design, but let's act according to the algorithm, i.e. separating the variables:
Here are some similar ones:
At what roots does this work? Answer: for any. Therefore, we can write that $x$ is any number.
Task No. 3
The third linear equation is more interesting:
\[\left(6-x \right)+\left(12+x \right)-\left(3-2x \right)=15\]
There are several brackets here, but they are not multiplied by anything, they are simply preceded by different signs. Let's break them down:
We perform the second step already known to us:
\[-x+x+2x=15-6-12+3\]
Let's do the math:
We carry out the last step - divide everything by the coefficient of “x”:
\[\frac(2x)(x)=\frac(0)(2)\]
Things to Remember When Solving Linear Equations
If we ignore too simple tasks, I would like to say the following:
- As I said above, not every linear equation has a solution - sometimes there are simply no roots;
- Even if there are roots, there may be zero among them - there is nothing wrong with that.
Zero is the same number as the others; you shouldn’t discriminate against it in any way or assume that if you get zero, then you did something wrong.
Another feature is related to the opening of brackets. Please note: when there is a “minus” in front of them, we remove it, but in parentheses we change the signs to opposite. And then we can open it using standard algorithms: we will get what we saw in the calculations above.
Understanding this simple fact will help you avoid making stupid and hurtful mistakes in high school, when doing such things is taken for granted.
Solving complex linear equations
Let's move on to more complex equations. Now the constructions will become more complex and when performing various transformations a quadratic function will appear. However, we should not be afraid of this, because if, according to the author’s plan, we are solving a linear equation, then during the transformation process all monomials containing a quadratic function will certainly cancel.
Example No. 1
Obviously, the first step is to open the brackets. Let's do this very carefully:
Now let's take a look at privacy:
\[-x+6((x)^(2))-6((x)^(2))+x=-12\]
Here are some similar ones:
Obviously, this equation has no solutions, so we’ll write this in the answer:
\[\varnothing\]
or there are no roots.
Example No. 2
We perform the same actions. First step:
Let's move everything with a variable to the left, and without it - to the right:
Here are some similar ones:
Obviously, this linear equation has no solution, so we’ll write it this way:
\[\varnothing\],
or there are no roots.
Nuances of the solution
Both equations are completely solved. Using these two expressions as an example, we were once again convinced that even in the simplest linear equations, everything may not be so simple: there can be either one, or none, or infinitely many roots. In our case, we considered two equations, both simply have no roots.
But I would like to draw your attention to another fact: how to work with parentheses and how to open them if there is a minus sign in front of them. Consider this expression:
Before opening, you need to multiply everything by “X”. Please note: multiplies each individual term. Inside there are two terms - respectively, two terms and multiplied.
And only after these seemingly elementary, but very important and dangerous transformations have been completed, can you open the bracket from the point of view of the fact that there is a minus sign after it. Yes, yes: only now, when the transformations are completed, we remember that there is a minus sign in front of the brackets, which means that everything below simply changes signs. At the same time, the brackets themselves disappear and, most importantly, the front “minus” also disappears.
We do the same with the second equation:
It is not by chance that I pay attention to these small, seemingly insignificant facts. Because solving equations is always a sequence of elementary transformations, where the inability to clearly and competently perform simple actions leads to the fact that high school students come to me and again learn to solve such simple equations.
Of course, the day will come when you will hone these skills to the point of automaticity. You will no longer have to perform so many transformations each time; you will write everything on one line. But while you are just learning, you need to write each action separately.
Solving even more complex linear equations
What we are going to solve now can hardly be called the simplest task, but the meaning remains the same.
Task No. 1
\[\left(7x+1 \right)\left(3x-1 \right)-21((x)^(2))=3\]
Let's multiply all the elements in the first part:
Let's do some privacy:
Here are some similar ones:
Let's complete the last step:
\[\frac(-4x)(4)=\frac(4)(-4)\]
Here is our final answer. And, despite the fact that in the process of solving we had coefficients with a quadratic function, they canceled each other out, which makes the equation linear and not quadratic.
Task No. 2
\[\left(1-4x \right)\left(1-3x \right)=6x\left(2x-1 \right)\]
Let's carefully perform the first step: multiply each element from the first bracket by each element from the second. There should be a total of four new terms after the transformations:
Now let’s carefully perform the multiplication in each term:
Let’s move the terms with “X” to the left, and those without - to the right:
\[-3x-4x+12((x)^(2))-12((x)^(2))+6x=-1\]
Here are similar terms:
Once again we have received the final answer.
Nuances of the solution
The most important note about these two equations is the following: as soon as we begin to multiply brackets that contain more than one term, this is done according to the following rule: we take the first term from the first and multiply with each element from the second; then we take the second element from the first and similarly multiply with each element from the second. As a result, we will have four terms.
About the algebraic sum
With this last example, I would like to remind students what an algebraic sum is. In classical mathematics, by $1-7$ we mean a simple construction: subtract seven from one. In algebra, we mean the following by this: to the number “one” we add another number, namely “minus seven”. This is how an algebraic sum differs from an ordinary arithmetic sum.
As soon as, when performing all the transformations, each addition and multiplication, you begin to see constructions similar to those described above, you simply will not have any problems in algebra when working with polynomials and equations.
Finally, let's look at a couple more examples that will be even more complex than the ones we just looked at, and to solve them we will have to slightly expand our standard algorithm.
Solving equations with fractions
To solve such tasks, we will have to add one more step to our algorithm. But first, let me remind you of our algorithm:
- Open the brackets.
- Separate variables.
- Bring similar ones.
- Divide by the ratio.
Alas, this wonderful algorithm, for all its effectiveness, turns out to be not entirely appropriate when we have fractions in front of us. And in what we will see below, we have a fraction on both the left and the right in both equations.
How to work in this case? Yes, it's very simple! To do this, you need to add one more step to the algorithm, which can be done both before and after the first action, namely, getting rid of fractions. So the algorithm will be as follows:
- Get rid of fractions.
- Open the brackets.
- Separate variables.
- Bring similar ones.
- Divide by the ratio.
What does it mean to “get rid of fractions”? And why can this be done both after and before the first standard step? In fact, in our case, all fractions are numerical in their denominator, i.e. Everywhere the denominator is just a number. Therefore, if we multiply both sides of the equation by this number, we will get rid of fractions.
Example No. 1
\[\frac(\left(2x+1 \right)\left(2x-3 \right))(4)=((x)^(2))-1\]
Let's get rid of the fractions in this equation:
\[\frac(\left(2x+1 \right)\left(2x-3 \right)\cdot 4)(4)=\left(((x)^(2))-1 \right)\cdot 4\]
Please note: everything is multiplied by “four” once, i.e. just because you have two parentheses doesn't mean you have to multiply each one by "four." Let's write down:
\[\left(2x+1 \right)\left(2x-3 \right)=\left(((x)^(2))-1 \right)\cdot 4\]
Now let's expand:
We seclude the variable:
We perform the reduction of similar terms:
\[-4x=-1\left| :\left(-4 \right) \right.\]
\[\frac(-4x)(-4)=\frac(-1)(-4)\]
We have received the final solution, let's move on to the second equation.
Example No. 2
\[\frac(\left(1-x \right)\left(1+5x \right))(5)+((x)^(2))=1\]
Here we perform all the same actions:
\[\frac(\left(1-x \right)\left(1+5x \right)\cdot 5)(5)+((x)^(2))\cdot 5=5\]
\[\frac(4x)(4)=\frac(4)(4)\]
The problem is solved.
That, in fact, is all I wanted to tell you today.
Key points
Key findings are:
- Know the algorithm for solving linear equations.
- Ability to open brackets.
- Don't worry if you have quadratic functions somewhere; most likely, they will be reduced in the process of further transformations.
- There are three types of roots in linear equations, even the simplest ones: one single root, the entire number line is a root, and no roots at all.
I hope this lesson will help you master a simple, but very important topic for further understanding of all mathematics. If something is not clear, go to the site and solve the examples presented there. Stay tuned, many more interesting things await you!
A+(b + c) can be written without parentheses: a+(b + c)=a + b + c. This operation is called opening parentheses.
Example 1. Let's open the brackets in the expression a + (- b + c).
Solution. a + (-b+c) = a + ((-b) + c)=a + (-b) + c = a-b + c.
If there is a “+” sign in front of the brackets, then you can omit the brackets and this “+” sign while maintaining the signs of the terms in the brackets. If the first term in brackets is written without a sign, then it must be written with a “+” sign.
Example 2. Let's find the value of the expression -2.87+ (2.87-7.639).
Solution. Opening the brackets, we get - 2.87 + (2.87 - 7.639) = - - 2.87 + 2.87 - 7.639 = 0 - 7.639 = - 7.639.
To find the value of the expression - (- 9 + 5), you need to add numbers-9 and 5 and find the number opposite to the resulting sum: -(- 9 + 5)= -(- 4) = 4.
The same value can be obtained in another way: first write down the numbers opposite to these terms (i.e. change their signs), and then add: 9 + (- 5) = 4. Thus, -(- 9 + 5) = 9 - 5 = 4.
To write a sum opposite to the sum of several terms, you need to change the signs of these terms.
This means - (a + b) = - a - b.
Example 3. Let's find the value of the expression 16 - (10 -18 + 12).
Solution. 16-(10 -18 + 12) = 16 + (-(10 -18 + 12)) = = 16 + (-10 +18-12) = 16-10 +18-12 = 12.
To open brackets preceded by a “-” sign, you need to replace this sign with “+”, changing the signs of all terms in the brackets to the opposite, and then open the brackets.
Example 4. Let's find the value of the expression 9.36-(9.36 - 5.48).
Solution. 9.36 - (9.36 - 5.48) = 9.36 + (- 9.36 + 5.48) = = 9.36 - 9.36 + 5.48 = 0 -f 5.48 = 5 ,48.
Expanding parentheses and applying commutative and associative properties addition allow you to simplify calculations.
Example 5. Let's find the value of the expression (-4-20)+(6+13)-(7-8)-5.
Solution. First, let's open the brackets, and then find separately the sum of all positive and separately the sum of all negative numbers and, finally, add up the results:
(- 4 - 20)+(6+ 13)-(7 - 8) - 5 = -4-20 + 6 + 13-7 + 8-5 = = (6 + 13 + 8)+(- 4 - 20 - 7 - 5)= 27-36=-9.
Example 6. Let's find the value of the expression
Solution. First, let’s imagine each term as the sum of their integer and fractional parts, then open the brackets, then add the integers and separately fractional parts and finally add up the results:
How do you open parentheses preceded by a “+” sign? How can you find the value of an expression that is the opposite of the sum of several numbers? How to expand parentheses preceded by a “-” sign?
1218. Open the brackets:
a) 3.4+(2.6+ 8.3); c) m+(n-k);
b) 4.57+(2.6 - 4.57); d) c+(-a + b).
1219. Find the meaning of the expression:
1220. Open the brackets:
a) 85+(7.8+ 98); d) -(80-16) + 84; g) a-(b-k-n);
b) (4.7 -17)+7.5; e) -a + (m-2.6); h) -(a-b + c);
c) 64-(90 + 100); e) c+(- a-b); i) (m-n)-(p-k).
1221. Open the brackets and find the meaning of the expression:
1222. Simplify the expression:
1223. Write amount two expressions and simplify it:
a) - 4 - m and m + 6.4; d) a+b and p - b
b) 1.1+a and -26-a; e) - m + n and -k - n;
c) a + 13 and -13 + b; e)m - n and n - m.
1224. Write the difference of two expressions and simplify it:
1226. Use the equation to solve the problem:
a) There are 42 books on one shelf, and 34 on the other. Several books were removed from the second shelf, and as many books were taken from the first shelf as were left on the second. After that, there were 12 books left on the first shelf. How many books were removed from the second shelf?
b) There are 42 students in the first grade, 3 students less in the second than in the third. How many students are there in third grade if there are 125 students in these three grades?
1227. Find the meaning of the expression:
1228. Calculate orally:
1229. Find the greatest value of the expression:
1230. Specify 4 consecutive integers if:
a) the smaller of them is -12; c) the smaller of them is n;
b) the largest of them is -18; d) the greater of them is equal to k.
Parentheses are used to indicate the order in which actions are performed in numeric, literal, and variable expressions. It is convenient to move from an expression with brackets to an identically equal expression without brackets. This technique is called opening brackets.
Expanding parentheses means removing the parentheses from an expression.
One more point deserves special attention, which concerns the peculiarities of recording decisions when opening brackets. We can write the initial expression with brackets and the result obtained after opening the brackets as an equality. For example, after expanding the parentheses instead of the expression
3−(5−7) we get the expression 3−5+7. We can write both of these expressions as the equality 3−(5−7)=3−5+7.
And one more important point. In mathematics, to shorten notations, it is customary not to write the plus sign if it appears first in an expression or in parentheses. For example, if we add two positive numbers, for example, seven and three, then we write not +7+3, but simply 7+3, despite the fact that seven is also a positive number. Similarly, if you see, for example, the expression (5+x) - know that before the bracket there is a plus, which is not written, and before the five there is a plus +(+5+x).
The rule for opening parentheses during addition
When opening brackets, if there is a plus in front of the brackets, then this plus is omitted along with the brackets.
Example. Open the brackets in the expression 2 + (7 + 3) There is a plus in front of the brackets, which means we do not change the signs in front of the numbers in brackets.
2 + (7 + 3) = 2 + 7 + 3
Rule for opening parentheses when subtracting
If there is a minus before the brackets, then this minus is omitted along with the brackets, but the terms that were in the brackets change their sign to the opposite. The absence of a sign before the first term in parentheses implies a + sign.
Example. Expand the parentheses in the expression 2 − (7 + 3)
There is a minus before the brackets, which means you need to change the signs in front of the numbers in the brackets. In parentheses there is no sign before the number 7, this means that seven is positive, it is considered that there is a + sign in front of it.
2 − (7 + 3) = 2 − (+ 7 + 3)
When opening the brackets, we remove from the example the minus that was in front of the brackets, and the brackets themselves 2 − (+ 7 + 3), and change the signs that were in the brackets to the opposite ones.
2 − (+ 7 + 3) = 2 − 7 − 3
Expanding parentheses when multiplying
If there is a multiplication sign in front of the brackets, then each number inside the brackets is multiplied by the factor in front of the brackets. In this case, multiplying a minus by a minus gives a plus, and multiplying a minus by a plus, like multiplying a plus by a minus, gives a minus.
Thus, the parentheses in the products are expanded in accordance with the distributive property of multiplication.
Example. 2 (9 - 7) = 2 9 - 2 7
When you multiply a bracket by a bracket, each term in the first bracket is multiplied with each term in the second bracket.
(2 + 3) · (4 + 5) = 2 · 4 + 2 · 5 + 3 · 4 + 3 · 5
In fact, there is no need to remember all the rules, it is enough to remember only one, this: c(a−b)=ca−cb. Why? Because if you substitute one instead of c, you get the rule (a−b)=a−b. And if we substitute minus one, we get the rule −(a−b)=−a+b. Well, if you substitute another bracket instead of c, you can get the last rule.
Opening parentheses when dividing
If there is a division sign after the brackets, then each number inside the brackets is divided by the divisor after the brackets, and vice versa.
Example. (9 + 6) : 3=9: 3 + 6: 3
How to expand nested parentheses
If an expression contains nested parentheses, they are expanded in order, starting with the outer or inner ones.
In this case, it is important that when opening one of the brackets, do not touch the remaining brackets, simply rewriting them as is.
Example. 12 - (a + (6 - b) - 3) = 12 - a - (6 - b) + 3 = 12 - a - 6 + b + 3 = 9 - a + b
“Opening parentheses” - Mathematics textbook, grade 6 (Vilenkin)
Short description:
In this section you will learn how to expand parentheses in examples. What is it for? Everything is for the same thing as before - to make it easier and simpler for you to count, to make fewer mistakes, and ideally (the dream of your mathematics teacher) in order to solve everything without mistakes.
You already know that parentheses are placed in mathematical notation if two mathematical signs appear in a row, if we want to show the combination of numbers, their regrouping. Expanding parentheses means getting rid of unnecessary characters. For example: (-15)+3=-15+3=-12, 18+(-16)=18-16=2. Do you remember the distributive property of multiplication relative to addition? Indeed, in that example we also got rid of brackets to simplify calculations. The named property of multiplication can also be applied to four, three, five or more terms. For example: 15*(3+8+9+6)=15*3+15*8+15*9+15*6=390. Have you noticed that when you open the brackets, the numbers in them do not change sign if the number in front of the brackets is positive? After all, fifteen is a positive number. And if you solve this example: -15*(3+8+9+6)=-15*3+(-15)*8+(-15)*9+(-15)*6=-45+(- 120)+(-135)+(-90)=-45-120-135-90=-390. We had a negative number minus fifteen in front of the brackets, when we opened the brackets all the numbers began to change their sign to another - the opposite - from plus to minus.
Based on the above examples, two basic rules for opening parentheses can be stated:
1. If you have a positive number in front of the brackets, then after opening the brackets all the signs of the numbers in the brackets do not change, but remain exactly the same as they were.
2. If you have a negative number in front of the brackets, then after opening the brackets the minus sign is no longer written, and the signs of all absolute numbers in the brackets suddenly change to the opposite.
For example: (13+8)+(9-8)=13+8+9-8=22; (13+8)-(9-8)=13+8-9+8=20. Let's complicate our examples a little: (13+8)+2(9-8)=13+8+2*9-2*8=21+18-16=23. You noticed that when opening the second brackets, we multiplied by 2, but the signs remained the same as they were. Here’s an example: (3+8)-2*(9-8)=3+8-2*9+2*8=11-18+16=9, in this example the number two is negative, it’s before the brackets stands with a minus sign, so when opening them, we changed the signs of the numbers to the opposite ones (nine was with a plus, became a minus, eight was with a minus, became a plus).
Let's say Achilles runs ten times faster than the tortoise and is a thousand steps behind it. During the time it takes Achilles to run this distance, the tortoise will crawl a hundred steps in the same direction. When Achilles runs a hundred steps, the tortoise crawls another ten steps, and so on. The process will continue ad infinitum, Achilles will never catch up with the tortoise.
This reasoning became a logical shock for all subsequent generations. Aristotle, Diogenes, Kant, Hegel, Hilbert... They all considered Zeno's aporia in one way or another. The shock was so strong that " ... discussions continue to this day; the scientific community has not yet been able to come to a common opinion on the essence of paradoxes ... mathematical analysis, set theory, new physical and philosophical approaches were involved in the study of the issue; none of them became a generally accepted solution to the problem..."[Wikipedia, "Zeno's Aporia". Everyone understands that they are being fooled, but no one understands what the deception consists of.
From a mathematical point of view, Zeno in his aporia clearly demonstrated the transition from quantity to . This transition implies application instead of permanent ones. As far as I understand, the mathematical apparatus for using variable units of measurement has either not yet been developed, or it has not been applied to Zeno’s aporia. Applying our usual logic leads us into a trap. We, due to the inertia of thinking, apply constant units of time to the reciprocal value. From a physical point of view, this looks like time slowing down until it stops completely at the moment when Achilles catches up with the turtle. If time stops, Achilles can no longer outrun the tortoise.
If we turn our usual logic around, everything falls into place. Achilles runs at a constant speed. Each subsequent segment of his path is ten times shorter than the previous one. Accordingly, the time spent on overcoming it is ten times less than the previous one. If we apply the concept of “infinity” in this situation, then it would be correct to say “Achilles will catch up with the turtle infinitely quickly.”
How to avoid this logical trap? Remain in constant units of time and do not switch to reciprocal units. In Zeno's language it looks like this:
In the time it takes Achilles to run a thousand steps, the tortoise will crawl a hundred steps in the same direction. During the next time interval equal to the first, Achilles will run another thousand steps, and the tortoise will crawl a hundred steps. Now Achilles is eight hundred steps ahead of the tortoise.
This approach adequately describes reality without any logical paradoxes. But this is not a complete solution to the problem. Einstein’s statement about the irresistibility of the speed of light is very similar to Zeno’s aporia “Achilles and the Tortoise”. We still have to study, rethink and solve this problem. And the solution must be sought not in infinitely large numbers, but in units of measurement.
Another interesting aporia of Zeno tells about a flying arrow:
A flying arrow is motionless, since at every moment of time it is at rest, and since it is at rest at every moment of time, it is always at rest.
In this aporia, the logical paradox is overcome very simply - it is enough to clarify that at each moment of time a flying arrow is at rest at different points in space, which, in fact, is motion. Another point needs to be noted here. From one photograph of a car on the road it is impossible to determine either the fact of its movement or the distance to it. To determine whether a car is moving, you need two photographs taken from the same point at different points in time, but you cannot determine the distance from them. To determine the distance to a car, you need two photographs taken from different points in space at one point in time, but from them you cannot determine the fact of movement (of course, you still need additional data for calculations, trigonometry will help you). What I want to draw special attention to is that two points in time and two points in space are different things that should not be confused, because they provide different opportunities for research.
Wednesday, July 4, 2018
The differences between set and multiset are described very well on Wikipedia. Let's see.
As you can see, “there cannot be two identical elements in a set,” but if there are identical elements in a set, such a set is called a “multiset.” Reasonable beings will never understand such absurd logic. This is the level of talking parrots and trained monkeys, who have no intelligence from the word “completely”. Mathematicians act as ordinary trainers, preaching to us their absurd ideas.
Once upon a time, the engineers who built the bridge were in a boat under the bridge while testing the bridge. If the bridge collapsed, the mediocre engineer died under the rubble of his creation. If the bridge could withstand the load, the talented engineer built other bridges.
No matter how mathematicians hide behind the phrase “mind me, I’m in the house,” or rather, “mathematics studies abstract concepts,” there is one umbilical cord that inextricably connects them with reality. This umbilical cord is money. Let us apply mathematical set theory to mathematicians themselves.
We studied mathematics very well and now we are sitting at the cash register, giving out salaries. So a mathematician comes to us for his money. We count out the entire amount to him and lay it out on our table in different piles, into which we put bills of the same denomination. Then we take one bill from each pile and give the mathematician his “mathematical set of salary.” Let us explain to the mathematician that he will receive the remaining bills only when he proves that a set without identical elements is not equal to a set with identical elements. This is where the fun begins.
First of all, the logic of the deputies will work: “This can be applied to others, but not to me!” Then they will begin to reassure us that bills of the same denomination have different bill numbers, which means they cannot be considered the same elements. Okay, let's count salaries in coins - there are no numbers on the coins. Here the mathematician will begin to frantically remember physics: different coins have different amounts of dirt, the crystal structure and arrangement of atoms is unique for each coin...
And now I have the most interesting question: where is the line beyond which the elements of a multiset turn into elements of a set and vice versa? Such a line does not exist - everything is decided by shamans, science is not even close to lying here.
Look here. We select football stadiums with the same field area. The areas of the fields are the same - which means we have a multiset. But if we look at the names of these same stadiums, we get many, because the names are different. As you can see, the same set of elements is both a set and a multiset. Which is correct? And here the mathematician-shaman-sharpist pulls out an ace of trumps from his sleeve and begins to tell us either about a set or a multiset. In any case, he will convince us that he is right.
To understand how modern shamans operate with set theory, tying it to reality, it is enough to answer one question: how do the elements of one set differ from the elements of another set? I'll show you, without any "conceivable as not a single whole" or "not conceivable as a single whole."
Sunday, March 18, 2018
The sum of the digits of a number is a dance of shamans with a tambourine, which has nothing to do with mathematics. Yes, in mathematics lessons we are taught to find the sum of the digits of a number and use it, but that’s why they are shamans, to teach their descendants their skills and wisdom, otherwise shamans will simply die out.
Do you need proof? Open Wikipedia and try to find the page "Sum of digits of a number." She doesn't exist. There is no formula in mathematics that can be used to find the sum of the digits of any number. After all, numbers are graphic symbols with which we write numbers, and in the language of mathematics the task sounds like this: “Find the sum of graphic symbols representing any number.” Mathematicians cannot solve this problem, but shamans can do it easily.
Let's figure out what and how we do in order to find the sum of the digits of a given number. And so, let us have the number 12345. What needs to be done in order to find the sum of the digits of this number? Let's consider all the steps in order.
1. Write down the number on a piece of paper. What have we done? We have converted the number into a graphical number symbol. This is not a mathematical operation.
2. We cut one resulting picture into several pictures containing individual numbers. Cutting a picture is not a mathematical operation.
3. Convert individual graphic symbols into numbers. This is not a mathematical operation.
4. Add the resulting numbers. Now this is mathematics.
The sum of the digits of the number 12345 is 15. These are the “cutting and sewing courses” taught by shamans that mathematicians use. But that is not all.
From a mathematical point of view, it does not matter in which number system we write a number. So, in different number systems the sum of the digits of the same number will be different. In mathematics, the number system is indicated as a subscript to the right of the number. With the large number 12345, I don’t want to fool my head, let’s consider the number 26 from the article about. Let's write this number in binary, octal, decimal and hexadecimal number systems. We won't look at every step under a microscope; we've already done that. Let's look at the result.
As you can see, in different number systems the sum of the digits of the same number is different. This result has nothing to do with mathematics. It’s the same as if you determined the area of a rectangle in meters and centimeters, you would get completely different results.
Zero looks the same in all number systems and has no sum of digits. This is another argument in favor of the fact that. Question for mathematicians: how is something that is not a number designated in mathematics? What, for mathematicians nothing exists except numbers? I can allow this for shamans, but not for scientists. Reality is not just about numbers.
The result obtained should be considered as proof that number systems are units of measurement for numbers. After all, we cannot compare numbers with different units of measurement. If the same actions with different units of measurement of the same quantity lead to different results after comparing them, then this has nothing to do with mathematics.
What is real mathematics? This is when the result of a mathematical operation does not depend on the size of the number, the unit of measurement used and on who performs this action.
Oh! Isn't this the women's restroom?
- Young woman! This is a laboratory for the study of the indephilic holiness of souls during their ascension to heaven! Halo on top and arrow up. What other toilet?
Female... The halo on top and the arrow down are male.
If such a work of design art flashes before your eyes several times a day,
Then it’s not surprising that you suddenly find a strange icon in your car:
Personally, I make an effort to see minus four degrees in a pooping person (one picture) (a composition of several pictures: a minus sign, the number four, a designation of degrees). And I don’t think this girl is a fool who doesn’t know physics. She just has a strong stereotype of perceiving graphic images. And mathematicians teach us this all the time. Here's an example.
1A is not “minus four degrees” or “one a”. This is "pooping man" or the number "twenty-six" in hexadecimal notation. Those people who constantly work in this number system automatically perceive a number and a letter as one graphic symbol.