One of the most important sciences, the application of which can be seen in disciplines such as chemistry, physics and even biology, is mathematics. Studying this science allows you to develop some mental qualities and improve your ability to concentrate. One of the topics that deserve special attention in the Mathematics course is adding and subtracting fractions. Many students find it difficult to study. Perhaps our article will help you better understand this topic.
How to subtract fractions whose denominators are the same
Fractions are the same numbers with which you can produce various actions. Their difference from whole numbers lies in the presence of a denominator. That is why, when performing operations with fractions, you need to study some of their features and rules. Most simple case is the subtraction of ordinary fractions whose denominators are represented as the same number. Performing this action will not be difficult if you know a simple rule:
- In order to subtract a second from one fraction, it is necessary to subtract the numerator of the subtracted fraction from the numerator of the fraction being reduced. We write this number into the numerator of the difference, and leave the denominator the same: k/m - b/m = (k-b)/m.
Examples of subtracting fractions whose denominators are the same
7/19 - 3/19 = (7 - 3)/19 = 4/19.
From the numerator of the fraction “7” we subtract the numerator of the fraction “3” to be subtracted, we get “4”. We write this number in the numerator of the answer, and in the denominator we put the same number that was in the denominators of the first and second fractions - “19”.
The picture below shows several more similar examples.
Let's consider a more complex example where fractions with like denominators are subtracted:
29/47 - 3/47 - 8/47 - 2/47 - 7/47 = (29 - 3 - 8 - 2 - 7)/47 = 9/47.
From the numerator of the fraction “29” being reduced by subtracting in turn the numerators of all subsequent fractions - “3”, “8”, “2”, “7”. As a result, we get the result “9”, which we write down in the numerator of the answer, and in the denominator we write down the number that is in the denominators of all these fractions - “47”.
Adding fractions that have the same denominator
Adding and subtracting ordinary fractions follows the same principle.
- In order to add fractions whose denominators are the same, you need to add the numerators. The resulting number is the numerator of the sum, and the denominator will remain the same: k/m + b/m = (k + b)/m.
Let's see what this looks like using an example:
1/4 + 2/4 = 3/4.
To the numerator of the first term of the fraction - “1” - add the numerator of the second term of the fraction - “2”. The result - “3” - is written into the numerator of the sum, and the denominator is left the same as that present in the fractions - “4”.
Fractions with different denominators and their subtraction
We have already considered the operation with fractions that have the same denominator. As we see, knowing simple rules, solving such examples is quite easy. But what if you need to perform an operation with fractions that have different denominators? Many secondary school students are confused by such examples. But even here, if you know the principle of the solution, the examples will no longer be difficult for you. There is also a rule here, without which solving such fractions is simply impossible.
- 2/3 - one three and one two are missing in the denominator:
2/3 = (2 x 3 x 2)/(3 x 3 x 2) = 12/18. - 7/9 or 7/(3 x 3) - the denominator is missing a two:
7/9 = (7 x 2)/(9 x 2) = 14/18. - 5/6 or 5/(2 x 3) - the denominator is missing a three:
5/6 = (5 x 3)/(6 x 3) = 15/18. - The number 18 is made up of 3 x 2 x 3.
- The number 15 is made up of 5 x 3.
- The common multiple will be the following factors: 5 x 3 x 3 x 2 = 90.
- 90 divided by 15. The resulting number “6” will be a multiplier for 3/15.
- 90 divided by 18. The resulting number “5” will be a multiplier for 4/18.
- Convert all fractions that have an integer part to improper ones. Speaking in simple words, remove the whole part. To do this, multiply the number of the integer part by the denominator of the fraction, and add the resulting product to the numerator. The number that comes out after these actions is the numerator of the improper fraction. The denominator remains unchanged.
- If fractions have different denominators, they should be reduced to the same denominator.
- Perform addition or subtraction with the same denominators.
- When receiving an improper fraction, select the whole part.
To subtract fractions from different denominators, it is necessary to reduce them to the same lowest denominator.
We will talk in more detail about how to do this.
Property of a fraction
In order to bring several fractions to the same denominator, you need to use the main property of a fraction in the solution: after dividing or multiplying the numerator and denominator by the same number, you get a fraction equal to the given one.
So, for example, the fraction 2/3 can have denominators such as “6”, “9”, “12”, etc., that is, it can have the form of any number that is a multiple of “3”. After we multiply the numerator and denominator by “2”, we get the fraction 4/6. After we multiply the numerator and denominator of the original fraction by “3”, we get 6/9, and if we perform a similar operation with the number “4”, we get 8/12. One equality can be written as follows:
2/3 = 4/6 = 6/9 = 8/12…
How to convert multiple fractions to the same denominator
Let's look at how to reduce multiple fractions to the same denominator. For example, let's take the fractions shown in the picture below. First you need to determine which number can become the denominator for all of them. To make things easier, let's factorize the existing denominators.
The denominator of the fraction 1/2 and the fraction 2/3 cannot be factorized. The denominator 7/9 has two factors 7/9 = 7/(3 x 3), the denominator of the fraction 5/6 = 5/(2 x 3). Now we need to determine which factors will be the smallest for all these four fractions. Since the first fraction has the number “2” in the denominator, it means that it must be present in all denominators; in the fraction 7/9 there are two triplets, which means that both of them must also be present in the denominator. Taking into account the above, we determine that the denominator consists of three factors: 3, 2, 3 and is equal to 3 x 2 x 3 = 18.
Let's consider the first fraction - 1/2. There is a “2” in its denominator, but there is not a single “3” digit, but there should be two. To do this, we multiply the denominator by two triples, but, according to the property of a fraction, we must multiply the numerator by two triples:
1/2 = (1 x 3 x 3)/(2 x 3 x 3) = 9/18.
We perform the same operations with the remaining fractions.
All together it looks like this:
How to subtract and add fractions that have different denominators
As mentioned above, in order to add or subtract fractions that have different denominators, they must be reduced to the same denominator, and then use the rules for subtracting fractions that have the same denominator, which have already been discussed.
Let's look at this as an example: 4/18 - 3/15.
Finding the multiple of numbers 18 and 15:
After the denominator has been found, it is necessary to calculate the factor that will be different for each fraction, that is, the number by which it will be necessary to multiply not only the denominator, but also the numerator. To do this, divide the number that we found (the common multiple) by the denominator of the fraction for which additional factors need to be determined.
The next stage of our solution is to reduce each fraction to the denominator “90”.
We have already talked about how this is done. Let's see how this is written in an example:
(4 x 5)/(18 x 5) - (3 x 6)/(15 x 6) = 20/90 - 18/90 = 2/90 = 1/45.
If fractions have small numbers, then you can determine the common denominator, as in the example shown in the picture below.
The same is true for those with different denominators.
Subtraction and having integer parts
We have already discussed in detail the subtraction of fractions and their addition. But how to subtract if a fraction has an integer part? Again, let's use a few rules:
There is another way in which you can add and subtract fractions with whole parts. To do this, actions are performed separately with whole parts, and actions with fractions separately, and the results are recorded together.
The example given consists of fractions that have the same denominator. In the case when the denominators are different, they must be brought to the same value, and then perform the actions as shown in the example.
Subtracting fractions from whole numbers
Another type of operation with fractions is the case when a fraction must be subtracted from. At first glance, such an example seems difficult to solve. However, everything is quite simple here. To solve it, you need to convert the integer into a fraction, and with the same denominator that is in the subtracted fraction. Next, we perform a subtraction similar to subtraction with identical denominators. In an example it looks like this:
7 - 4/9 = (7 x 9)/9 - 4/9 = 53/9 - 4/9 = 49/9.
The subtraction of fractions (grade 6) presented in this article is the basis for solving more complex examples that are covered in subsequent grades. Knowledge of this topic is subsequently used to solve functions, derivatives, and so on. Therefore, it is very important to understand and understand the operations with fractions discussed above.
Fractions are ordinary numbers and can also be added and subtracted. But because they have a denominator, they require more complex rules than for integers.
Let's consider the simplest case, when there are two fractions with the same denominators. Then:
To add fractions with the same denominators, you need to add their numerators and leave the denominator unchanged.
To subtract fractions with the same denominators, you need to subtract the numerator of the second from the numerator of the first fraction, and again leave the denominator unchanged.
Within each expression, the denominators of the fractions are equal. By definition of adding and subtracting fractions we get:
As you can see, it’s nothing complicated: we just add or subtract the numerators and that’s it.
But even in such simple actions people manage to make mistakes. What is most often forgotten is that the denominator does not change. For example, when adding them, they also begin to add up, and this is fundamentally wrong.
Get rid of bad habit Adding the denominators is quite simple. Try the same thing when subtracting. As a result, the denominator will be zero, and the fraction will (suddenly!) lose its meaning.
Therefore, remember once and for all: when adding and subtracting, the denominator does not change!
Many people also make mistakes when adding several negative fractions. There is confusion with the signs: where to put a minus and where to put a plus.
This problem is also very easy to solve. It is enough to remember that the minus before the sign of a fraction can always be transferred to the numerator - and vice versa. And of course, don’t forget two simple rules:
- Plus by minus gives minus;
- Two negatives make an affirmative.
Let's look at all this with specific examples:
Task. Find the meaning of the expression:
In the first case, everything is simple, but in the second, let’s add minuses to the numerators of the fractions:
What to do if the denominators are different
You cannot add fractions with different denominators directly. At least, this method is unknown to me. However, the original fractions can always be rewritten so that the denominators become the same.
There are many ways to convert fractions. Three of them are discussed in the lesson “Reducing fractions to a common denominator”, so we will not dwell on them here. Let's look at some examples:
Task. Find the meaning of the expression:
In the first case, we reduce the fractions to a common denominator using the “criss-cross” method. In the second we will look for the NOC. Note that 6 = 2 · 3; 9 = 3 · 3. The last factors in these expansions are equal, and the first ones are relatively prime. Therefore, LCM(6, 9) = 2 3 3 = 18.
What to do if a fraction has an integer part
I can please you: different denominators in fractions are not the biggest evil. Much more errors occur when the whole part is highlighted in the addend fractions.
Of course, there are own addition and subtraction algorithms for such fractions, but they are quite complex and require a long study. Better use the simple diagram below:
- Convert all fractions containing an integer part to improper ones. We obtain normal terms (even with different denominators), which are calculated according to the rules discussed above;
- Actually, calculate the sum or difference of the resulting fractions. As a result, we will practically find the answer;
- If this is all that was required in the problem, we perform the inverse transformation, i.e. We get rid of an improper fraction by highlighting the whole part.
The rules for moving to improper fractions and highlighting the whole part are described in detail in the lesson “What is a numerical fraction”. If you don’t remember, be sure to repeat it. Examples:
Task. Find the meaning of the expression:
Everything is simple here. The denominators inside each expression are equal, so all that remains is to convert all fractions to improper ones and count. We have:
To simplify the calculations, I have skipped some obvious steps in the last examples.
A small note about the last two examples, where fractions with the integer part highlighted are subtracted. The minus before the second fraction means that the entire fraction is subtracted, and not just its whole part.
Re-read this sentence again, look at the examples - and think about it. This is where beginners make a huge number of mistakes. They love to give such problems on tests. You will also encounter them several times in the tests for this lesson, which will be published shortly.
Summary: general calculation scheme
In conclusion, I will give a general algorithm that will help you find the sum or difference of two or more fractions:
- If one or more fractions have an integer part, convert these fractions to improper ones;
- Bring all the fractions to a common denominator in any way convenient for you (unless, of course, the writers of the problems did this);
- Add or subtract the resulting numbers according to the rules for adding and subtracting fractions with like denominators;
- If possible, shorten the result. If the fraction is incorrect, select the whole part.
Remember that it is better to highlight the whole part at the very end of the task, immediately before writing down the answer.
Actions with fractions.
Attention!
There are additional
materials in Special Section 555.
For those who are very "not very..."
And for those who “very much…”)
So, what are fractions, types of fractions, transformations - we remembered. Let's get to the main issue.
What can you do with fractions? Yes, everything is the same as with ordinary numbers. Add, subtract, multiply, divide.
All these actions with decimal working with fractions is no different from working with whole numbers. Actually, that’s what’s good about them, decimal ones. The only thing is that you need to put the comma correctly.
Mixed numbers, as I already said, are of little use for most actions. They still need to be converted to ordinary fractions.
But the actions with ordinary fractions they will be more cunning. And much more important! Let me remind you: all actions with fractional expressions with letters, sines, unknowns, and so on and so forth are no different from actions with ordinary fractions! Operations with ordinary fractions are the basis for all algebra. It is for this reason that we will analyze all this arithmetic in great detail here.
Adding and subtracting fractions.
Everyone can add (subtract) fractions with the same denominators (I really hope!). Well, let me remind those who are completely forgetful: when adding (subtracting), the denominator does not change. The numerators are added (subtracted) to give the numerator of the result. Type:
In short, in general view:
What if the denominators are different? Then, using the basic property of a fraction (here it comes in handy again!), we make the denominators the same! For example:
Here we had to make the fraction 4/10 from the fraction 2/5. For the sole purpose of making the denominators the same. Let me note, just in case, that 2/5 and 4/10 are the same fraction! Only 2/5 are uncomfortable for us, and 4/10 are really okay.
By the way, this is the essence of solving any math problems. When we from uncomfortable we do expressions the same thing, but more convenient for solving.
Another example:
The situation is similar. Here we make 48 from 16. By simple multiplication by 3. This is all clear. But we came across something like:
How to be?! It's hard to make a nine out of a seven! But we are smart, we know the rules! Let's transform every fraction so that the denominators are the same. This is called “reduce to a common denominator”:
Wow! How did I know about 63? Very simple! 63 is a number that is divisible by 7 and 9 at the same time. Such a number can always be obtained by multiplying the denominators. If we multiply a number by 7, for example, then the result will certainly be divisible by 7!
If you need to add (subtract) several fractions, there is no need to do it in pairs, step by step. You just need to find the denominator common to all fractions and reduce each fraction to this same denominator. For example:
And what will be the common denominator? You can, of course, multiply 2, 4, 8, and 16. We get 1024. Nightmare. It’s easier to estimate that the number 16 is perfectly divisible by 2, 4, and 8. Therefore, from these numbers it’s easy to get 16. This number will be the common denominator. Let's turn 1/2 into 8/16, 3/4 into 12/16, and so on.
By the way, if you take 1024 as the common denominator, everything will work out, in the end everything will be reduced. But not everyone will get to this end, because of the calculations...
Complete the example yourself. Not some kind of logarithm... It should be 29/16.
So, the addition (subtraction) of fractions is clear, I hope? Of course, it is easier to work in a shortened version, with additional multipliers. But this pleasure is available to those who worked honestly in the lower grades... And did not forget anything.
And now we will do the same actions, but not with fractions, but with fractional expressions. New rake will be revealed here, yes...
So, we need to add two fractional expressions:
We need to make the denominators the same. And only with the help multiplication! This is what the main property of a fraction dictates. Therefore, I cannot add one to X in the first fraction in the denominator. (that would be nice!). But if you multiply the denominators, you see, everything grows together! So we write down the line of the fraction, leave an empty space at the top, then add it, and write the product of the denominators below, so as not to forget:
And, of course, we don’t multiply anything on the right side, we don’t open the parentheses! And now, looking at the common denominator on the right side, we realize: in order to get the denominator x(x+1) in the first fraction, you need to multiply the numerator and denominator of this fraction by (x+1). And in the second fraction - to x. This is what you get:
Note! Here are the parentheses! This is the rake that many people step on. Not parentheses, of course, but their absence. The parentheses appear because we are multiplying all numerator and all denominator! And not their individual pieces...
In the numerator of the right side we write the sum of the numerators, everything is as in numerical fractions, then we open the brackets in the numerator of the right side, i.e. We multiply everything and give similar ones. There is no need to open the parentheses in the denominators or multiply anything! In general, in denominators (any) the product is always more pleasant! We get:
So we got the answer. The process seems long and difficult, but it depends on practice. Once you solve the examples, get used to it, everything will become simple. Those who have mastered fractions in due time do all these operations with one left hand, automatically!
And one more note. Many smartly deal with fractions, but get stuck on examples with whole numbers. Like: 2 + 1/2 + 3/4= ? Where to fasten the two-piece? You don’t need to fasten it anywhere, you need to make a fraction out of two. It's not easy, but very simple! 2=2/1. Like this. Any whole number can be written as a fraction. The numerator is the number itself, the denominator is one. 7 is 7/1, 3 is 3/1 and so on. It's the same with letters. (a+b) = (a+b)/1, x=x/1, etc. And then we work with these fractions according to all the rules.
Well, the knowledge of addition and subtraction of fractions was refreshed. Converting fractions from one type to another was repeated. You can also get checked. Shall we settle it a little?)
Calculate:
Answers (in disarray):
71/20; 3/5; 17/12; -5/4; 11/6
Multiplication/division of fractions - in the next lesson. There are also tasks for all operations with fractions.
If you like this site...
By the way, I have a couple more interesting sites for you.)
You can practice solving examples and find out your level. Testing with instant verification. Let's learn - with interest!)
You can get acquainted with functions and derivatives.
You can perform various operations with fractions, for example, adding fractions. Addition of fractions can be divided into several types. Each type of addition of fractions has its own rules and algorithm of actions. Let's look at each type of addition in detail.
Adding fractions with like denominators.
Let's look at an example of how to add fractions with a common denominator.
The tourists went on a hike from point A to point E. On the first day they walked from point A to B or \(\frac(1)(5)\) of the entire path. On the second day they walked from point B to D or \(\frac(2)(5)\) the whole way. How far did they travel from the beginning of the journey to point D?
To find the distance from point A to point D, you need to add the fractions \(\frac(1)(5) + \frac(2)(5)\).
Adding fractions with like denominators means that you need to add the numerators of these fractions, but the denominator will remain the same.
\(\frac(1)(5) + \frac(2)(5) = \frac(1 + 2)(5) = \frac(3)(5)\)
In literal form, the sum of fractions with the same denominators will look like this:
\(\bf \frac(a)(c) + \frac(b)(c) = \frac(a + b)(c)\)
Answer: the tourists walked \(\frac(3)(5)\) the entire way.
Adding fractions with different denominators.
Let's look at an example:
You need to add two fractions \(\frac(3)(4)\) and \(\frac(2)(7)\).
To add fractions with different denominators, you must first find, and then use the rule for adding fractions with like denominators.
For denominators 4 and 7, the common denominator will be the number 28. The first fraction \(\frac(3)(4)\) must be multiplied by 7. The second fraction \(\frac(2)(7)\) must be multiplied by 4.
\(\frac(3)(4) + \frac(2)(7) = \frac(3 \times \color(red) (7) + 2 \times \color(red) (4))(4 \ times \color(red) (7)) = \frac(21 + 8)(28) = \frac(29)(28) = 1\frac(1)(28)\)
In literal form we get the following formula:
\(\bf \frac(a)(b) + \frac(c)(d) = \frac(a \times d + c \times b)(b \times d)\)
Adding mixed numbers or mixed fractions.
Addition occurs according to the law of addition.
For mixed fractions, we add the whole parts with the whole parts and the fractional parts with the fractions.
If the fractional parts of mixed numbers have same denominators, then we add the numerators, but the denominator remains the same.
Let's add the mixed numbers \(3\frac(6)(11)\) and \(1\frac(3)(11)\).
\(3\frac(6)(11) + 1\frac(3)(11) = (\color(red) (3) + \color(blue) (\frac(6)(11))) + ( \color(red) (1) + \color(blue) (\frac(3)(11))) = (\color(red) (3) + \color(red) (1)) + (\color( blue) (\frac(6)(11)) + \color(blue) (\frac(3)(11))) = \color(red)(4) + (\color(blue) (\frac(6 + 3)(11))) = \color(red)(4) + \color(blue) (\frac(9)(11)) = \color(red)(4) \color(blue) (\frac (9)(11))\)
If the fractional parts of mixed numbers have different denominators, then we find the common denominator.
Let's perform the addition of mixed numbers \(7\frac(1)(8)\) and \(2\frac(1)(6)\).
The denominator is different, so we need to find the common denominator, it is equal to 24. Multiply the first fraction \(7\frac(1)(8)\) by an additional factor of 3, and the second fraction \(2\frac(1)(6)\) by 4.
\(7\frac(1)(8) + 2\frac(1)(6) = 7\frac(1 \times \color(red) (3))(8 \times \color(red) (3) ) = 2\frac(1\times \color(red) (4))(6\times \color(red) (4)) =7\frac(3)(24) + 2\frac(4)(24 ) = 9\frac(7)(24)\)
Related questions:
How to add fractions?
Answer: first you need to decide what type of expression it is: fractions have the same denominators, different denominators or mixed fractions. Depending on the type of expression, we proceed to the solution algorithm.
How to solve fractions with different denominators?
Answer: you need to find the common denominator, and then follow the rule of adding fractions with the same denominators.
How to solve mixed fractions?
Answer: we add integer parts with integers and fractional parts with fractions.
Example #1:
Can the sum of two result in a proper fraction? Improper fraction? Give examples.
\(\frac(2)(7) + \frac(3)(7) = \frac(2 + 3)(7) = \frac(5)(7)\)
The fraction \(\frac(5)(7)\) is a proper fraction, it is the result of the sum of two proper fractions \(\frac(2)(7)\) and \(\frac(3)(7)\).
\(\frac(2)(5) + \frac(8)(9) = \frac(2 \times 9 + 8 \times 5)(5 \times 9) =\frac(18 + 40)(45) = \frac(58)(45)\)
The fraction \(\frac(58)(45)\) is an improper fraction, it is the result of the sum of the proper fractions \(\frac(2)(5)\) and \(\frac(8)(9)\).
Answer: The answer to both questions is yes.
Example #2:
Add the fractions: a) \(\frac(3)(11) + \frac(5)(11)\) b) \(\frac(1)(3) + \frac(2)(9)\).
a) \(\frac(3)(11) + \frac(5)(11) = \frac(3 + 5)(11) = \frac(8)(11)\)
b) \(\frac(1)(3) + \frac(2)(9) = \frac(1 \times \color(red) (3))(3 \times \color(red) (3)) + \frac(2)(9) = \frac(3)(9) + \frac(2)(9) = \frac(5)(9)\)
Example #3:
Write it down mixed fraction as a sum natural number and proper fraction: a) \(1\frac(9)(47)\) b) \(5\frac(1)(3)\)
a) \(1\frac(9)(47) = 1 + \frac(9)(47)\)
b) \(5\frac(1)(3) = 5 + \frac(1)(3)\)
Example #4:
Calculate the sum: a) \(8\frac(5)(7) + 2\frac(1)(7)\) b) \(2\frac(9)(13) + \frac(2)(13) \) c) \(7\frac(2)(5) + 3\frac(4)(15)\)
a) \(8\frac(5)(7) + 2\frac(1)(7) = (8 + 2) + (\frac(5)(7) + \frac(1)(7)) = 10 + \frac(6)(7) = 10\frac(6)(7)\)
b) \(2\frac(9)(13) + \frac(2)(13) = 2 + (\frac(9)(13) + \frac(2)(13)) = 2\frac(11 )(13) \)
c) \(7\frac(2)(5) + 3\frac(4)(15) = 7\frac(2\times 3)(5\times 3) + 3\frac(4)(15) = 7\frac(6)(15) + 3\frac(4)(15) = (7 + 3)+(\frac(6)(15) + \frac(4)(15)) = 10 + \frac (10)(15) = 10\frac(10)(15) = 10\frac(2)(3)\)
Task #1:
At lunch we ate \(\frac(8)(11)\) from the cake, and in the evening at dinner we ate \(\frac(3)(11)\). Do you think the cake was completely eaten or not?
Solution:
The denominator of the fraction is 11, it indicates how many parts the cake was divided into. At lunch we ate 8 pieces of cake out of 11. At dinner we ate 3 pieces of cake out of 11. Let’s add 8 + 3 = 11, we ate pieces of cake out of 11, that is, the whole cake.
\(\frac(8)(11) + \frac(3)(11) = \frac(11)(11) = 1\)
Answer: the whole cake was eaten.
In the fifth century BC, the ancient Greek philosopher Zeno of Elea formulated his famous aporias, the most famous of which is the “Achilles and the Tortoise” aporia. Here's what it sounds like: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...were involved in the study of the issue mathematical analysis, set theory, new physical and philosophical approaches; 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 point out Special attention, 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. Applicable mathematical theory sets to the 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: on different coins there is different quantities dirt, crystal structure and atomic arrangement of each coin is unique...
And now I have the most interest Ask: 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 the help of 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 a large number 12345 I don’t want to fool my head, let’s look at 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, it means it 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.