Multiplication of irregular fractions with different denominators. Multiplication of fractions, division of fractions

We will consider the multiplication of ordinary fractions in several ways.

Multiplication of an ordinary fraction by a fraction

This is the simplest case in which you need to use the following multiplication rules for fractions.

To multiply a fraction by a fraction, necessary:

multiply the numerator of the first fraction by the numerator of the second fraction and write their product into the numerator of the new fraction;
the denominator of the first fraction is multiplied by the denominator of the second fraction and their product is written in the denominator of the new fraction;

Before multiplying the numerators and denominators, check if the fractions can be canceled. Reducing the fractions in your calculations will make your calculations much easier.

Multiplying a fraction by a natural number

To the fraction multiply by natural number you need to multiply the numerator of the fraction by this number, and leave the denominator of the fraction unchanged.

If, as a result of multiplication, an incorrect fraction is obtained, do not forget to turn it into a mixed number, that is, select the whole part.

Multiplication of mixed numbers

To multiply mixed numbers, you must first turn them into improper fractions and then multiply according to the rule of multiplication of ordinary fractions.

Another way to multiply a fraction by a natural number

Sometimes, when calculating, it is more convenient to use another method of multiplication common fraction by the number.

To multiply a fraction by a natural number, you need to divide the denominator of the fraction by this number, and leave the numerator the same.

As you can see from the example, this version of the rule is more convenient to use if the denominator of the fraction is divisible without a remainder by a natural number.

Fraction actions

Adding fractions with the same denominator

There are two types of addition of fractions:

Adding fractions with the same denominator
Adding fractions with different denominators

First, let's study the addition of fractions with the same denominators. Everything is simple here. To add fractions with the same denominator, add their numerators and leave the denominator unchanged. For example, add the fractions and. Add the numerators and leave the denominator unchanged:

This example can be easily understood if you think about the pizza, which is divided into four parts. If you add pizzas to pizza, you get pizzas:

Example 2. Add fractions and.

Again, add the numerators, and leave the denominator unchanged:

The answer is an incorrect fraction. If the end of the task comes, then from irregular fractions it is customary to get rid of. To get rid of the incorrect fraction, you need to select the whole part in it. In our case, the whole part is easily distinguished - two divided by two is equal to one:

This example can be easily understood if you think about the pizza, which is divided into two parts. If you add pizza to the pizza, you get one whole pizza:

Example 3... Add fractions and.

This example can be easily understood if you think about the pizza, which is divided into three parts. If you add pizza to the pizza, you get pizza:

Example 4. Find the value of an expression

This example is solved in the same way as the previous ones. The numerators must be added, and the denominator must be left unchanged:

Let's try to depict our solution using a picture. If you add pizzas to the pizza and add pizzas, then you get 1 whole and more pizza.

As you can see, there is nothing difficult in adding fractions with the same denominators. It is enough to understand the following rules:

To add fractions with the same denominator, add their numerators and leave the denominator the same;
If the answer turns out to be an incorrect fraction, then you need to select the whole part in it.

Adding fractions with different denominators

Now let's learn how to add fractions with different denominators. When adding up fractions, the denominators of those fractions should be the same. But they are not always the same.

For example, fractions and can be added, since they have the same denominators.

But fractions cannot be added immediately, since these fractions have different denominators. In such cases, the fractions must be reduced to the same (common) denominator.

There are several ways to bring fractions to the same denominator. Today we will consider only one of them, since the rest of the methods may seem difficult for a beginner.

The essence of this method is that first the least common multiple (LCM) of the denominators of both fractions is sought. Then the LCM is divided by the denominator of the first fraction and the first additional factor is obtained. Do the same with the second fraction - the LCM is divided by the denominator of the second fraction and a second additional factor is obtained.

Then the numerators and denominators of the fractions are multiplied by their additional factors. As a result of these actions, fractions with different denominators are converted into fractions with the same denominators. And we already know how to add such fractions.

Example 1... Add the fractions and

These fractions have different denominators, so you need to bring them to the same (common) denominator.

First of all, we find the least common multiple of the denominators of both fractions. The denominator of the first fraction is 3, and the denominator of the second fraction is 2. The least common multiple of these numbers is 6

LCM (2 and 3) = 6

Now we return to fractions and. First, divide the LCM by the denominator of the first fraction and get the first additional factor. LCM is the number 6, and the denominator of the first fraction is the number 3. Divide 6 by 3, we get 2.

The resulting number 2 is the first additional factor. We write it down to the first fraction. To do this, make a small oblique line above the fraction and write the additional factor found above it:

We do the same with the second fraction. We divide the LCM by the denominator of the second fraction and get the second additional factor. The LCM is the number 6, and the denominator of the second fraction is the number 2. Divide 6 by 2, we get 3.

The resulting number 3 is the second additional factor. We write it down to the second fraction. Again, we draw a small oblique line above the second fraction and write the additional factor found above it:

We are now ready to add. It remains to multiply the numerators and denominators of fractions by your additional factors:

Look closely at what we have arrived at. We came to the conclusion that fractions with different denominators turned into fractions with the same denominators. And we already know how to add such fractions. Let's finish this example to the end:

Thus, the example ends. It turns out to add.

Let's try to depict our solution using a picture. If you add pizzas to the pizza, you get one whole pizza and another sixth pizza:

The reduction of fractions to the same (common) denominator can also be depicted using a picture. Reducing fractions and to a common denominator, we got fractions and. These two fractions will be represented by the same pizza slices. The only difference is that this time they will be divided into equal shares (reduced to the same denominator).

The first picture depicts a fraction (four out of six pieces), and the second picture depicts a fraction (three out of six pieces). Putting these pieces together we get (seven pieces out of six). This fraction is incorrect, so we selected the whole part in it. As a result, we got (one whole pizza and another sixth pizza).

Note that we have described this example in too much detail. V educational institutions it is not customary to write so extensively. You need to be able to quickly find the LCM of both denominators and additional factors to them, as well as quickly multiply the found additional factors by your numerators and denominators. While in school, we would have to write this example as follows:

But there is also a downside to the coin. If, at the first stages of studying mathematics, you do not make detailed notes, then questions of the kind begin to appear “Where's that figure come from?” “Why do the fractions suddenly turn into completely different fractions? «.

To make it easier to add fractions with different denominators, you can use the following step-by-step instructions:

Find the LCM of the denominators of fractions;
Divide the LCM by the denominator of each fraction and get an additional factor for each fraction;
Multiply the numerators and denominators of fractions by your additional factors;
Add fractions with the same denominators;
If the answer turns out to be an incorrect fraction, then select its whole part;

Example 2. Find the value of an expression .

Let's use the scheme that we presented above.

Step 1. Find the LCM for the denominators of fractions

Find the LCM for the denominators of both fractions. The denominators of the fractions are the numbers 2, 3 and 4. You need to find the LCM for these numbers:

Step 2. Divide the LCM by the denominator of each fraction and get an additional factor for each fraction

We divide the LCM by the denominator of the first fraction. LCM is the number 12, and the denominator of the first fraction is the number 2. Divide 12 by 2, we get 6. We got the first additional factor 6. We write it over the first fraction:

Now we divide the LCM by the denominator of the second fraction. LCM is the number 12, and the denominator of the second fraction is the number 3. Divide 12 by 3, we get 4. We got the second additional factor 4. We write it over the second fraction:

Now we divide the LCM by the denominator of the third fraction. LCM is the number 12, and the denominator of the third fraction is the number 4. Divide 12 by 4, we get 3. We got the third additional factor 3. We write it over the third fraction:

Step 3. Multiply the numerators and denominators of fractions by your additional factors

We multiply the numerators and denominators by our additional factors:

Step 4. Add fractions with the same denominator

We came to the conclusion that fractions with different denominators turned into fractions with the same (common) denominators. It remains to add these fractions. We add:

The addition did not fit on one line, so we moved the remaining expression to the next line. This is allowed in mathematics. When an expression does not fit on one line, it is transferred to the next line, and it is necessary to put an equal sign (=) at the end of the first line and at the beginning of a new line. The equal sign on the second line indicates that this is a continuation of the expression that was on the first line.

Step 5. If the answer turns out to be an incorrect fraction, then select its whole part

We got the wrong fraction in our answer. We have to select the whole part from it. Highlight:

Received an answer

Subtracting fractions with the same denominator

There are two types of subtraction of fractions:

Subtracting fractions with the same denominator
Subtracting fractions with different denominators

First, let's study the subtraction of fractions with the same denominator. Everything is simple here. To subtract another from one fraction, you need to subtract the numerator of the second fraction from the numerator of the first fraction, and leave the denominator the same.

For example, let's find the value of an expression. To solve this example, you need to subtract the numerator of the second fraction from the numerator of the first fraction, and leave the denominator the same. So let's do it:

This example can be easily understood if you think about the pizza, which is divided into four parts. If you cut pizzas from pizza, you get pizzas:

Example 2. Find the value of the expression.

Again, subtract the numerator of the second fraction from the numerator of the first fraction, and leave the denominator the same:

This example can be easily understood if you think about the pizza, which is divided into three parts. If you cut pizzas from pizza, you get pizzas:

Example 3. Find the value of an expression

This example is solved in the same way as the previous ones. From the numerator of the first fraction, you need to subtract the numerators of the remaining fractions:

The answer is an incorrect fraction. If the example is complete, then it is customary to get rid of the incorrect fraction. Let's and we get rid of the wrong fraction in the answer. To do this, select its whole part:

As you can see, there is nothing difficult in subtracting fractions with the same denominators. It is enough to understand the following rules:

To subtract another from one fraction, you need to subtract the numerator of the second fraction from the numerator of the first fraction, and leave the denominator the same;

If the answer turns out to be an incorrect fraction, then you need to select its whole part.

Subtracting fractions with different denominators

For example, you can subtract a fraction from a fraction, since these fractions have the same denominator. But you cannot subtract a fraction from a fraction, since these fractions have different denominators. In such cases, the fractions must be reduced to the same (common) denominator.

The common denominator is found according to the same principle that we used when adding fractions with different denominators. First of all, find the LCM of the denominators of both fractions. Then the LCM is divided by the denominator of the first fraction and the first additional factor is obtained, which is written over the first fraction. Similarly, the LCM is divided by the denominator of the second fraction and a second additional factor is obtained, which is written over the second fraction.

The fractions are then multiplied by their additional factors. As a result of these operations, fractions with different denominators are converted to fractions with the same denominators. We already know how to subtract such fractions.

Example 1. Find the value of an expression:

First, we find the LCM of the denominators of both fractions. The denominator of the first fraction is 3, and the denominator of the second fraction is 4. The least common multiple of these numbers is 12

LCM (3 and 4) = 12

Now back to fractions and

Let's find an additional factor for the first fraction. To do this, we divide the LCM by the denominator of the first fraction. The LCM is the number 12, and the denominator of the first fraction is the number 3. Divide 12 by 3, we get 4. We write the four over the first fraction:

We do the same with the second fraction. We divide the LCM by the denominator of the second fraction. LCM is the number 12, and the denominator of the second fraction is the number 4. Divide 12 by 4, we get 3. Write the three over the second fraction:

We are now ready for subtraction. It remains to multiply the fractions by your additional factors:

We came to the conclusion that fractions with different denominators turned into fractions with the same denominators. We already know how to subtract such fractions. Let's finish this example to the end:

Received an answer

Let's try to depict our solution using a picture. If you cut pizzas from pizza, you get pizza

This is a detailed version of the solution. In school, we would have to solve this example in a shorter way. Such a solution would look like this:

The reduction of fractions and to a common denominator can also be depicted using the figure. Bringing these fractions to a common denominator, we got fractions and. These fractions will be represented by the same pizza slices, but this time they will be divided into equal parts (reduced to the same denominator):

The first drawing depicts a fraction (eight out of twelve pieces), and the second drawing depicts a fraction (three out of twelve pieces). Cutting off three pieces from eight pieces, we get five pieces out of twelve. Fraction and describes these five pieces.

Example 2. Find the value of an expression

These fractions have different denominators, so you first need to bring them to the same (common) denominator.

Find the LCM of the denominators of these fractions.

The denominators of the fractions are 10, 3, and 5. The least common multiple of these numbers is 30

LCM (10, 3, 5) = 30

Now we find additional factors for each fraction. To do this, we divide the LCM by the denominator of each fraction.

Let's find an additional factor for the first fraction. LCM is the number 30, and the denominator of the first fraction is the number 10. Divide 30 by 10, we get the first additional factor 3. We write it over the first fraction:

Now we find an additional factor for the second fraction. Divide the LCM by the denominator of the second fraction. LCM is the number 30, and the denominator of the second fraction is the number 3. Divide 30 by 3, we get the second additional factor 10. We write it over the second fraction:

Now we find an additional factor for the third fraction. Divide the LCM by the denominator of the third fraction. LCM is the number 30, and the denominator of the third fraction is the number 5. Divide 30 by 5, we get the third additional factor 6. We write it over the third fraction:

Everything is now ready for subtraction. It remains to multiply the fractions by your additional factors:

We came to the conclusion that fractions with different denominators turned into fractions with the same (common) denominators. We already know how to subtract such fractions. Let's finish this example.

The continuation of the example will not fit on one line, so we transfer the continuation to the next line. Don't forget about the equal sign (=) on a new line:

The answer turned out to be the correct fraction, and everything seems to suit us, but it is too cumbersome and ugly. It should be made simpler and more aesthetically pleasing. What can be done? You can shorten this fraction. Recall that reducing a fraction is the division of the numerator and denominator by the greatest common divisor of the numerator and denominator.

To correctly reduce the fraction, you need to divide its numerator and denominator by the greatest common divisor (GCD) of the numbers 20 and 30.

GCD should not be confused with NOC. The most common mistake many newbies make. GCD is the greatest common denominator. We find it to reduce the fraction.

And the LCM is the least common multiple. We find it in order to bring fractions to the same (common) denominator.

Now we will find the greatest common divisor (GCD) of numbers 20 and 30.

So, we find the GCD for the numbers 20 and 30:

GCD (20 and 30) = 10

Now go back to our example and divide the numerator and denominator of the fraction by 10:

We got a nice answer

Multiplying a fraction by a number

To multiply a fraction by a number, you need to multiply the numerator of this fraction by this number, and leave the denominator the same.

Example 1... Multiply the fraction by 1.

Multiply the numerator of the fraction by 1

Recording can be understood as taking half 1 time. For example, if you take pizzas 1 time, you get pizzas

From the laws of multiplication, we know that if the multiplier and the multiplier are reversed, then the product will not change. If the expression is written as, then the product will still be equal. Again, the rule for multiplying an integer and a fraction works:

This record can be understood as taking half of one. For example, if there is 1 whole pizza and we take half of it, then we will have pizza:

Example 2... Find the value of an expression

Multiply the numerator of the fraction by 4

The expression can be understood as taking two quarters 4 times. For example, if you take pizzas 4 times, you get two whole pizzas.

And if we swap the multiplier and the multiplier in places, we get the expression. It will also be equal to 2. This expression can be understood as taking two pizzas from four whole pizzas:

Multiplication of fractions

To multiply fractions, you need to multiply their numerators and denominators. If the answer turns out to be an incorrect fraction, you need to select the whole part in it.

Example 1. Find the value of the expression.

We got an answer. It is desirable to shorten this fraction. The fraction can be reduced by 2. Then the final decision will take the following form:

The expression can be understood as taking pizza from half of the pizza. Let's say we have half a pizza:

How to get two thirds of this half? First, you need to divide this half into three equal parts:

And take two from these three pieces:

We'll make pizza. Remember what a pizza looks like when divided into three parts:

One slice from this pizza and the two slices we took will have the same dimensions:

In other words, it comes about the same size of pizza. Therefore, the value of the expression is

Example 2... Find the value of an expression

We multiply the numerator of the first fraction by the numerator of the second fraction, and the denominator of the first fraction by the denominator of the second fraction:

The answer is an incorrect fraction. Let's select the whole part in it:

Example 3. Find the value of an expression

The answer is a correct fraction, but it will be good if you reduce it. To reduce this fraction, it must be divided by the GCD of the numerator and denominator. So, let's find the GCD of numbers 105 and 450:

GCD for (105 and 150) is 15

Now we divide the numerator and denominator of our answer to the GCD:

Fraction representation of an integer

Any integer can be represented as a fraction. For example, the number 5 can be represented as. From this, the five will not change its value, since the expression means "the number five divided by one", and this, as you know, is equal to five:

Reverse numbers

Now we will get acquainted with a very interesting topic in mathematics. It is called "back numbers".

Definition. The inverse of the number a is a number that, when multiplied by a gives one.

Let's substitute in this definition instead of a variable a number 5 and try to read the definition:

The inverse of the number 5 is a number that, when multiplied by 5 gives one.

Can you find a number that, when multiplied by 5, gives one? It turns out you can. Let's represent the five as a fraction:

Then multiply this fraction by itself, just swap the numerator and denominator. In other words, multiply the fraction by itself, only inverted:

What will be the result of this? If we continue to solve this example, we get one:

This means that the inverse of 5 is a number, since 5 is multiplied by one.

The reciprocal can also be found for any other integer.

the inverse of 3 is the fraction
the inverse of 4 is the fraction

You can also find the reciprocal for any other fraction. To do this, just turn it over.

In the fifth century BC, the ancient Greek philosopher Zeno of Elea formulated his famous aporias, the most famous of which is the aporia "Achilles and the tortoise." This is how it sounds:

Let's say Achilles runs ten times faster than a turtle and is a thousand steps behind it. During the time it takes Achilles to run this distance, the turtle will crawl a hundred steps in the same direction. When Achilles has run a hundred steps, the turtle will crawl ten more steps, and so on. The process will continue indefinitely, Achilles will never catch up with the turtle.

This reasoning came as a logical shock to all subsequent generations. Aristotle, Diogenes, Kant, Hegel, Hilbert ... All of them, in one way or another, considered Zeno's aporias. The shock was so strong that " ... discussions continue at the present time, the scientific community has not yet managed to come to a common opinion about the essence of paradoxes ... mathematical analysis, set theory, new physical and philosophical approaches were involved in the study of the issue; none of them has become a generally accepted solution to the question ..."[Wikipedia, Zeno's Aporia"]. Everyone understands that they are being fooled, but no one understands what the deception is.

From the point of view of mathematics, Zeno in his aporia clearly demonstrated the transition from magnitude to. This transition implies application instead of constants. As far as I understand, the mathematical apparatus for applying variable units of measurement either has not yet been developed, or it has not been applied to Zeno's aporia. Applying our usual logic leads us into a trap. We, by inertia of thinking, apply constant units of measurement of time to the reciprocal. From a physical point of view, it looks like time dilation until it stops completely at the moment when Achilles is level with the turtle. If time stops, Achilles can no longer overtake the turtle.

If we turn over the logic we are used to, 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 infinitely quickly catch up with the turtle."

How can you avoid this logical trap? Stay in constant time units and do not go backwards. In Zeno's language, it looks like this:

During the time during which Achilles will run a thousand steps, the turtle will crawl a hundred steps in the same direction. Over the next interval of time, equal to the first, Achilles will run another thousand steps, and the turtle will crawl a hundred steps. Now Achilles is eight hundred steps ahead of the turtle.

This approach adequately describes reality without any logical paradoxes. But this is not a complete solution to the problem. Einstein's statement about the insuperability of the speed of light is very similar to the Zeno aporia "Achilles and the Turtle". 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 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 rests at different points in space, which, in fact, is motion. Another point should be noted here. From a single 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 the fact of the movement of the car, two photographs are needed, taken from the same point at different points in time, but the distance cannot be determined from them. To determine the distance to the car, you need two photographs taken from different points in space at the same time, but it is impossible to determine the fact of movement from them (of course, additional data are still needed 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, 4 July 2018

The distinction between set and multiset is very well described in Wikipedia. We look.

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". Such logic of absurdity will never be understood by rational beings. This is the level of talking parrots and trained monkeys, who lack intelligence from the word "completely". Mathematicians act as ordinary trainers, preaching their absurd ideas to us.

Once the engineers who built the bridge were in a boat under the bridge during the tests of the bridge. If the bridge collapsed, the incompetent engineer died under the rubble of his creation. If the bridge could withstand the load, a talented engineer would build other bridges.

No matter how mathematicians hide behind the phrase "chur, 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 the mathematicians themselves.

We studied mathematics very well and now we are sitting at the cash desk, giving out salaries. Here comes a mathematician for his money. We count the entire amount to him and lay out on our table into different piles, in which we put bills of the same denomination. Then we take one bill from each pile and hand the mathematician his “mathematical set of salary”. Let us explain the mathematics that he will receive the rest of the 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: "You can apply it to others, you can not apply it to me!" Further, we will begin to assure us that there are different denomination numbers on bills of the same denomination, which means that they cannot be considered the same elements. Okay, let's count the salary in coins - there are no numbers on the coins. Here the mathematician will start to frantically remember physics: different coins have different amounts of dirt, the crystal structure and arrangement of atoms in each coin is unique ...

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 did not lie anywhere near here.

Look here. We select football stadiums with the same pitch. The area of the fields is the same, which means we have got a multiset. But if we consider the names of the same stadiums, we get a lot, because the names are different. As you can see, the same set of elements is both a set and a multiset at the same time. How is it correct? And here the mathematician-shaman-schuller takes a trump ace out of his sleeve and begins to tell us either about the set or about the 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 "thinkable as not a single whole" or "not thinkable as a whole."

Sunday, 18 March 2018

The sum of the digits of the 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 is why they are shamans in order to teach their descendants their skills and wisdom, otherwise shamans will simply die out.

Need proof? Open Wikipedia and try to find the Sum of Digits of a Number page. It doesn't exist. There is no formula in mathematics by which you can 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 - it is elementary.

Let's see 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 should be done in order to find the sum of the digits of this number? Let's go through all the steps in order.

1. We write down the number on a piece of paper. What have we done? We have converted the number to the graphic symbol of the number. This is not a mathematical operation.

2. We cut one resulting picture into several pictures containing separate numbers. Cutting a picture is not a mathematical operation.

3. Convert individual graphic symbols to numbers. This is not a mathematical operation.

4. Add up the resulting numbers. Now that's mathematics.

The sum of the digits of 12345 is 15. These are the "cutting and sewing courses" from shamans used by mathematicians. But that is not all.

From the point of view of mathematics, it does not matter in which number system we write the 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 do not want to fool my head, consider the number 26 from the article about. Let's write this number in binary, octal, decimal and hexadecimal number systems. We will not look at every step under a microscope, we have already done that. Let's see 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 would get completely different results when you determined the area of a rectangle in meters and centimeters.

Zero in all number systems looks the same and has no sum of digits. This is another argument for the fact that. A question for mathematicians: how is something that is not a number designated in mathematics? What, for mathematicians, nothing but numbers exists? For shamans, I can allow this, but for scientists - no. Reality is not all 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 their comparison, then this has nothing to do with mathematics.

What is real mathematics? This is when the result of a mathematical action does not depend on the magnitude of the number, the unit of measurement used and on who performs this action.

Sign on the door Opens the door and says:

Ouch! Isn't this a women's toilet?
- Young woman! This is a laboratory for the study of the indiscriminate holiness of souls during the ascension to heaven! Halo on top and up arrow. What other toilet?

Female ... The nimbus above and the down arrow is male.

If a piece of design art like this flashes before your eyes several times a day,

Then it is not surprising that in your car you suddenly find a strange icon:

Personally, I make an effort on myself so that in a pooping person (one picture), I can see minus four degrees (a composition of several pictures: minus sign, number four, degrees designation). And I don’t think this girl is a fool who doesn’t know physics. She just has a stereotype of perception of graphic images. And mathematicians constantly teach us this. 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 the number and the letter as one graphic symbol.

Multiplication and division of fractions.

Attention!
There are additional
materials in Special Section 555.
For those who are very "not very ..."
And for those who "very much ...")

This operation is much nicer than addition-subtraction! Because it's easier. Let me remind you: to multiply a fraction by a fraction, you need to multiply the numerators (this will be the numerator of the result) and the denominators (this will be the denominator). That is:

For example:

Everything is extremely simple... And please don't search common denominator! Don't need him here ...

To divide a fraction into a fraction, you need to flip second(this is important!) fraction and multiply them, i.e .:

For example:

If you come across multiplication or division with integers and fractions - that's okay. As with addition, we make a fraction with one in the denominator out of an integer - and off we go! For example:

In high school, you often have to deal with three-story (or even four-story!) Fractions. For example:

How to bring this fraction to a decent look? It's very simple! Use two-point division:

But don't forget the division order! Unlike multiplication, this is very important here! Of course, 4: 2, or 2: 4, we will not confuse. But in a three-story fraction it is easy to make a mistake. Note, for example:

In the first case (expression on the left):

In the second (expression on the right):

Do you feel the difference? 4 and 1/9!

And what determines the order of division? Or brackets, or (as here) the length of horizontal bars. Develop an eye. And if there are no brackets or dashes, like:

then we divide-multiply in order, from left to right!

And another very simple and important trick. In actions with degrees, it will come in handy for you! Divide the unit by any fraction, for example, by 13/15:

The fraction has turned over! And it always does. When dividing 1 by any fraction, the result is the same fraction, only inverted.

That's all for fractions. The thing is quite simple, but it gives more than enough errors. Note practical advice, and there will be fewer (errors)!

Practical advice:

1. The most important thing when working with fractional expressions is accuracy and care! These are not general words, not good wishes! This is a dire necessity! Do all calculations on the exam as a full-fledged task, with concentration and clarity. It is better to write two extra lines in a draft than to mess it up when calculating in your head.

2. In examples with different types of fractions - go to ordinary fractions.

3. All fractions are reduced to stop.

4. Multi-storey fractional expressions are reduced to ordinary ones, using division through two points (watch the order of division!).

5. Divide the unit into a fraction mentally, simply turning the fraction over.

Here are the tasks that you must definitely solve. Answers are given after all tasks. Use the materials on this topic and practical advice. Consider how many examples you were able to solve correctly. The first time! No calculator! And make the right conclusions ...

Remember - the correct answer is received from the second (all the more - the third) time - does not count! This is a harsh life.

So, we solve in exam mode ! This is already preparation for the exam, by the way. We solve the example, check it, solve the next one. We decided everything - checked again from the first to the last. But only after look at the answers.

Calculate:

Have you solved it?

We are looking for answers that match yours. I deliberately wrote them down in a mess, away from temptation, so to speak ... Here they are, the answers, separated by semicolons.

0; 17/22; 3/4; 2/5; 1; 25.

And now we draw conclusions. If everything worked out, I'm glad for you! Basic calculations with fractions are not your problem! You can do more serious things. If not...

So you have one of two problems. Or both at once.) Lack of knowledge and / or inattention. But this solvable Problems.

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. Instant validation testing. Learning - with interest!)

you can get acquainted with functions and derivatives.

To correctly multiply a fraction by a fraction or a fraction by a number, you need to know simple rules. We will now analyze these rules in detail.

Multiplication of an ordinary fraction by a fraction.

To multiply a fraction by a fraction, you need to calculate the product of the numerators and the product of the denominators of these fractions.

\ (\ bf \ frac (a) (b) \ times \ frac (c) (d) = \ frac (a \ times c) (b \ times d) \\\)

Let's consider an example:
We multiply the numerator of the first fraction with the numerator of the second fraction, and we also multiply the denominator of the first fraction with the denominator of the second fraction.

\ (\ frac (6) (7) \ times \ frac (2) (3) = \ frac (6 \ times 2) (7 \ times 3) = \ frac (12) (21) = \ frac (4 \ times 3) (7 \ times 3) = \ frac (4) (7) \\\)

The fraction \ (\ frac (12) (21) = \ frac (4 \ times 3) (7 \ times 3) = \ frac (4) (7) \\\) has been reduced by 3.

Multiplication of a fraction by a number.

First, let's remember the rule any number can be represented as a fraction \ (\ bf n = \ frac (n) (1) \).

Let's use this rule when multiplying.

\ (5 \ times \ frac (4) (7) = \ frac (5) (1) \ times \ frac (4) (7) = \ frac (5 \ times 4) (1 \ times 7) = \ frac (20) (7) = 2 \ frac (6) (7) \\\)

Irregular fraction \ (\ frac (20) (7) = \ frac (14 + 6) (7) = \ frac (14) (7) + \ frac (6) (7) = 2 + \ frac (6) ( 7) = 2 \ frac (6) (7) \\\) was converted to a mixed fraction.

In other words, when multiplying a number by a fraction, the number is multiplied by the numerator, and the denominator is left unchanged. Example:

\ (\ frac (2) (5) \ times 3 = \ frac (2 \ times 3) (5) = \ frac (6) (5) = 1 \ frac (1) (5) \\\\\) \ (\ bf \ frac (a) (b) \ times c = \ frac (a \ times c) (b) \\\)

Multiplication of mixed fractions.

To multiply mixed fractions, you must first represent each mixed fraction as an incorrect fraction, and then use the multiplication rule. The numerator is multiplied with the numerator, the denominator is multiplied with the denominator.

Example:
\ (2 \ frac (1) (4) \ times 3 \ frac (5) (6) = \ frac (9) (4) \ times \ frac (23) (6) = \ frac (9 \ times 23) (4 \ times 6) = \ frac (3 \ times \ color (red) (3) \ times 23) (4 \ times 2 \ times \ color (red) (3)) = \ frac (69) (8) = 8 \ frac (5) (8) \\\)

Multiplication of reciprocal fractions and numbers.

The fraction \ (\ bf \ frac (a) (b) \) is the inverse of \ (\ bf \ frac (b) (a) \), provided a ≠ 0, b ≠ 0.
The fractions \ (\ bf \ frac (a) (b) \) and \ (\ bf \ frac (b) (a) \) are called reciprocal fractions. The product of reciprocal fractions is 1.
\ (\ bf \ frac (a) (b) \ times \ frac (b) (a) = 1 \\\)

Example:
\ (\ frac (5) (9) \ times \ frac (9) (5) = \ frac (45) (45) = 1 \\\)

Questions on the topic:
How to multiply a fraction by a fraction?
Answer: The product of ordinary fractions is the multiplication of the numerator with the numerator, the denominator with the denominator. To get the product of mixed fractions, you need to convert them to an improper fraction and multiply according to the rules.

How do I multiply fractions with different denominators?
Answer: it doesn't matter if fractions have the same or different denominators, multiplication occurs according to the rule of finding the product of the numerator with the numerator, the denominator with the denominator.

How to multiply mixed fractions?
Answer: first of all, you need to translate the mixed fraction into an improper fraction and then find the product according to the rules of multiplication.

How to multiply a number by a fraction?
Answer: we multiply the number with the numerator, and leave the denominator the same.

Example # 1:
Calculate the product: a) \ (\ frac (8) (9) \ times \ frac (7) (11) \) b) \ (\ frac (2) (15) \ times \ frac (10) (13) \ )

Solution:
a) \ (\ frac (8) (9) \ times \ frac (7) (11) = \ frac (8 \ times 7) (9 \ times 11) = \ frac (56) (99) \\\\ \)
b) \ (\ frac (2) (15) \ times \ frac (10) (13) = \ frac (2 \ times 10) (15 \ times 13) = \ frac (2 \ times 2 \ times \ color ( red) (5)) (3 \ times \ color (red) (5) \ times 13) = \ frac (4) (39) \)

Example # 2:
Calculate the products of a number and a fraction: a) \ (3 \ times \ frac (17) (23) \) b) \ (\ frac (2) (3) \ times 11 \)

Solution:
a) \ (3 \ times \ frac (17) (23) = \ frac (3) (1) \ times \ frac (17) (23) = \ frac (3 \ times 17) (1 \ times 23) = \ frac (51) (23) = 2 \ frac (5) (23) \\\\\)
b) \ (\ frac (2) (3) \ times 11 = \ frac (2) (3) \ times \ frac (11) (1) = \ frac (2 \ times 11) (3 \ times 1) = \ frac (22) (3) = 7 \ frac (1) (3) \)

Example # 3:
Write the reciprocal of the fraction \ (\ frac (1) (3) \)?
Answer: \ (\ frac (3) (1) = 3 \)

Example # 4:
Calculate the product of two reciprocal fractions: a) \ (\ frac (104) (215) \ times \ frac (215) (104) \)

Solution:
a) \ (\ frac (104) (215) \ times \ frac (215) (104) = 1 \)

Example # 5:
Can reciprocal fractions be:
a) at the same time with regular fractions;
b) at the same time with incorrect fractions;
c) simultaneously natural numbers?

Solution:
a) to answer the first question, let's give an example. The fraction \ (\ frac (2) (3) \) is correct, its reciprocal will be \ (\ frac (3) (2) \) is an improper fraction. The answer is no.

b) for almost all enumeration of fractions, this condition is not met, but there are some numbers that satisfy the condition to be at the same time an improper fraction. For example, the improper fraction \ (\ frac (3) (3) \), its reciprocal is \ (\ frac (3) (3) \). We get two irregular fractions. Answer: not always under certain conditions, when the numerator and denominator are equal.

c) natural numbers are numbers that we use when counting, for example, 1, 2, 3,…. If we take the number \ (3 = \ frac (3) (1) \), then its reciprocal is \ (\ frac (1) (3) \). The fraction \ (\ frac (1) (3) \) is not a natural number. If we iterate over all the numbers, getting the reciprocal is always a fraction, except 1. If we take the number 1, then its reciprocal will be \ (\ frac (1) (1) = \ frac (1) (1) = 1 \). Number 1 is a natural number. Answer: they can be natural numbers at the same time only in one case, if this number is 1.

Example # 6:
Perform the product of mixed fractions: a) \ (4 \ times 2 \ frac (4) (5) \) b) \ (1 \ frac (1) (4) \ times 3 \ frac (2) (7) \)

Solution:
a) \ (4 \ times 2 \ frac (4) (5) = \ frac (4) (1) \ times \ frac (14) (5) = \ frac (56) (5) = 11 \ frac (1 )(5)\\\\ \)
b) \ (1 \ frac (1) (4) \ times 3 \ frac (2) (7) = \ frac (5) (4) \ times \ frac (23) (7) = \ frac (115) ( 28) = 4 \ frac (3) (7) \)

Example # 7:
Can two mutually inverse numbers be mixed numbers at the same time?

Let's look at an example. Take a mixed fraction \ (1 \ frac (1) (2) \), find its reciprocal, for this we convert it to an improper fraction \ (1 \ frac (1) (2) = \ frac (3) (2) \). Its inverse fraction will be \ (\ frac (2) (3) \). The fraction \ (\ frac (2) (3) \) is a regular fraction. Answer: two mutually inverse fractions cannot be mixed numbers at the same time.

) and the denominator by the denominator (we get the denominator of the product).

The formula for multiplying fractions:

For example:

Before you start multiplying the numerators and denominators, you need to check for the possibility of reducing the fraction. If you can reduce the fraction, then it will be easier for you to make further calculations.

Division of an ordinary fraction into a fraction.

Division of fractions with the participation of a natural number.

It's not as scary as it sounds. As in the case of addition, convert an integer to a fraction with one in the denominator. For example:

Multiplication of mixed fractions.

The rules for multiplying fractions (mixed):

converting mixed fractions to irregular ones;
multiply the numerators and denominators of fractions;
we reduce the fraction;
if you got an incorrect fraction, then convert the incorrect fraction to a mixed one.

Note! To multiply a mixed fraction by another mixed fraction, you first need to bring them to the form of improper fractions, and then multiply according to the rule of multiplication of ordinary fractions.

The second way to multiply a fraction by a natural number.

It may be more convenient to use the second method of multiplying an ordinary fraction by a number.

Note! To multiply a fraction by a natural number, you must divide the denominator of the fraction by this number, and leave the numerator unchanged.

From the above example, it is clear that this option is more convenient to use when the denominator of the fraction is divided without a remainder by a natural number.

Multi-storey fractions.

In high school, three-story (or more) fractions are often found. Example:

To bring such a fraction to its usual form, use division through 2 points:

Note! In the division of fractions, the order of division is very important. Be careful, it is easy to get confused here.

Note, for example:

When dividing one by any fraction, the result will be the same fraction, only inverted:

Practical tips for multiplying and dividing fractions:

1. The most important thing in working with fractional expressions is accuracy and attentiveness. Do all calculations carefully and accurately, with concentration and clarity. It is better to write a few extra lines in the draft than to get confused in the calculations in your head.

2. In tasks with different types of fractions - go to the form of ordinary fractions.

3. Reduce all fractions until it becomes impossible to reduce.

4. Multi-storey fractional expressions are converted into ordinary ones, using division through 2 points.

5. Divide the unit into a fraction mentally, simply turning the fraction over.