Multiply simple fractions. Multiplication and division of fractions

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.

Multiplying a 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.

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

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.

Multiplying a fraction by a number.

Let's start with the rule any number can be represented as a fraction \(\bf n = \frac(n)(1)\) .

Let's use this rule for multiplication.

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

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

In other words, When multiplying a number by a fraction, multiply the number by the numerator and leave the denominator 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 improper 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 the fraction \(\bf \frac(b)(a)\), provided a≠0,b≠0.
The fractions \(\bf \frac(a)(b)\) and \(\bf \frac(b)(a)\) are called reciprocals. The product of reciprocal fractions is 1.
\(\bf \frac(a)(b) \times \frac(b)(a) = 1 \\\)

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

Related questions:
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 to multiply fractions with different denominators?
Answer: it doesn't matter whether they are the same or different denominators for fractions, 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 convert the mixed fraction to 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 product 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 \(\frac(1)(3)\)?
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 mutually inverse fractions be:
a) both proper fractions;
b) simultaneously improper fractions;
c) natural numbers at the same time?

Solution:
a) Let's use an example to answer the first question. The fraction \(\frac(2)(3)\) is proper, its reciprocal will be equal to \(\frac(3)(2)\) - an improper fraction. Answer: no.

b) in almost all enumerations of fractions, this condition is not met, but there are some numbers that fulfill the condition of being simultaneously not proper fraction. For example, the improper fraction is \(\frac(3)(3)\) , its reciprocal is \(\frac(3)(3)\). We get two improper fractions. Answer: not always under certain conditions, when the numerator and denominator are equal.

c) natural numbers are the numbers that we use when counting, for example, 1, 2, 3, .... If we take the number \(3 = \frac(3)(1)\), then its reciprocal will be \(\frac(1)(3)\). The fraction \(\frac(1)(3)\) is not a natural number. If we go through all the numbers, the reciprocal is always a fraction, except for 1. If we take the number 1, then its reciprocal will be \(\frac(1)(1) = \frac(1)(1) = 1\). Number 1 natural number. Answer: they can be simultaneously natural numbers 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 reciprocal numbers be simultaneously mixed numbers?

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

In the course of the average and high school The students went through the topic "Fractions". However, this concept is much broader than given in the learning process. Today, the concept of a fraction is encountered quite often, and not everyone can calculate any expression, for example, multiplying fractions.

What is a fraction?

It so happened historically that fractional numbers appeared due to the need to measure. As practice shows, there are often examples for determining the length of a segment, the volume of a rectangular rectangle.

Initially, students are introduced to such a concept as a share. For example, if you divide a watermelon into 8 parts, then each will get one-eighth of a watermelon. This one part of eight is called a share.

A share equal to ½ of any value is called a half; ⅓ - third; ¼ - a quarter. Entries like 5/8, 4/5, 2/4 are called common fractions. An ordinary fraction is divided into a numerator and a denominator. Between them is a fractional line, or fractional line. A fractional bar can be drawn as either a horizontal or a slanted line. In this case, it stands for the division sign.

The denominator represents how many equal shares the value, object is divided into; and the numerator is how many equal shares are taken. The numerator is written above the fractional bar, the denominator below it.

It is most convenient to show ordinary fractions on a coordinate ray. If a single segment is divided into 4 equal parts, each part is designated with a Latin letter, then as a result you can get an excellent visual aid. So, point A shows a share equal to 1/4 of the entire unit segment, and point B marks 2/8 of this segment.

Varieties of fractions

Fractions are common, decimal, and mixed numbers. In addition, fractions can be divided into proper and improper. This classification is more suitable for ordinary fractions.

A proper fraction is a number whose numerator is less than the denominator. Accordingly, an improper fraction is a number whose numerator is greater than the denominator. The second kind is usually written as a mixed number. Such an expression consists of an integer part and a fractional part. For example, 1½. one - whole part, ½ - fractional. However, if you need to perform some manipulations with the expression (dividing or multiplying fractions, reducing or converting them), the mixed number is converted into an improper fraction.

A correct fractional expression is always less than one, and an incorrect one is always greater than or equal to 1.

As for this expression, they understand a record in which any number is represented, the denominator of the fractional expression of which can be expressed through one with several zeros. If the fraction is correct, then the whole part in decimal notation will be equal to zero.

To write a decimal, you must first write the integer part, separate it from the fractional with a comma, and then write the fractional expression. It must be remembered that after the comma the numerator must contain as many numeric characters as there are zeros in the denominator.

Example. Represent the fraction 7 21 / 1000 in decimal notation.

Algorithm for converting an improper fraction to a mixed number and vice versa

It is incorrect to write down an improper fraction in the answer of the problem, so it must be converted to a mixed number:

• divide the numerator by the existing denominator;
• in specific example incomplete quotient - whole;
• and the remainder is the numerator of the fractional part, with the denominator remaining unchanged.

Example. Convert improper fraction to mixed number: 47 / 5 .

Solution. 47: 5. The incomplete quotient is 9, the remainder = 2. Hence, 47 / 5 = 9 2 / 5.

Sometimes you need to represent a mixed number as an improper fraction. Then you need to use the following algorithm:

• the integer part is multiplied by the denominator of the fractional expression;
• the resulting product is added to the numerator;
• the result is written in the numerator, the denominator remains unchanged.

Example. Express the number in mixed form as an improper fraction: 9 8 / 10 .

Solution. 9 x 10 + 8 = 90 + 8 = 98 is the numerator.

Answer: 98 / 10.

Multiplication of ordinary fractions

You can perform various algebraic operations on ordinary fractions. To multiply two numbers, you need to multiply the numerator with the numerator, and the denominator with the denominator. Moreover, the multiplication of fractions with different denominators does not differ from the product fractional numbers with the same denominators.

It happens that after finding the result, you need to reduce the fraction. It is imperative to simplify the resulting expression as much as possible. Of course, it cannot be said that an improper fraction in the answer is a mistake, but it is also difficult to call it the correct answer.

Example. Find the product of two ordinary fractions: ½ and 20/18.

As can be seen from the example, after finding the product, a reducible fractional notation is obtained. Both the numerator and the denominator in this case are divisible by 4, and the result is the answer 5 / 9.

Multiplying decimal fractions

The product of decimal fractions is quite different from the product of ordinary fractions in its principle. So, multiplying fractions is as follows:

• two decimal fractions must be written under each other so that the rightmost digits are one under the other;
• you need to multiply the written numbers, despite the commas, that is, as natural numbers;
• count the number of digits after the comma in each of the numbers;
• in the result obtained after multiplication, you need to count as many digital characters on the right as are contained in the sum in both factors after the decimal point, and put a separating sign;
• if there are fewer digits in the product, then so many zeros must be written in front of them to cover this number, put a comma and assign an integer part equal to zero.

Example. Calculate the product of two decimals: 2.25 and 3.6.

Solution.

Multiplication of mixed fractions

To calculate the product of two mixed fractions, you need to use the rule for multiplying fractions:

• convert mixed numbers to improper fractions;
• find the product of numerators;
• find the product of the denominators;
• write down the result;
• simplify the expression as much as possible.

Example. Find the product of 4½ and 6 2 / 5.

Multiplying a number by a fraction (fractions by a number)

In addition to finding the product of two fractions, mixed numbers, there are tasks where you need to multiply by a fraction.

So, to find the work decimal fraction and a natural number, you need:

• write the number under the fraction so that the rightmost digits are one above the other;
• find the work, despite the comma;
• in the result obtained, separate the integer part from the fractional part using a comma, counting to the right the number of characters that is after the decimal point in the fraction.

To multiply an ordinary fraction by a number, you should find the product of the numerator and the natural factor. If the answer is a reducible fraction, it should be converted.

Example. Calculate the product of 5 / 8 and 12.

Solution. 5 / 8 * 12 = (5*12) / 8 = 60 / 8 = 30 / 4 = 15 / 2 = 7 1 / 2.

Answer: 7 1 / 2.

As you can see from the previous example, it was necessary to reduce the resulting result and convert the incorrect fractional expression into a mixed number.

Also, the multiplication of fractions also applies to finding the product of a number in mixed form and a natural factor. To multiply these two numbers, you should multiply the integer part of the mixed factor by the number, multiply the numerator by the same value, and leave the denominator unchanged. If necessary, you need to simplify the result as much as possible.

Example. Find the product of 9 5 / 6 and 9.

Solution. 9 5 / 6 x 9 \u003d 9 x 9 + (5 x 9) / 6 \u003d 81 + 45 / 6 \u003d 81 + 7 3 / 6 \u003d 88 1 / 2.

Answer: 88 1 / 2.

Multiplication by factors 10, 100, 1000 or 0.1; 0.01; 0.001

The following rule follows from the previous paragraph. To multiply a decimal fraction by 10, 100, 1000, 10000, etc., you need to move the comma to the right by as many digit characters as there are zeros in the multiplier after one.

Example 1. Find the product of 0.065 and 1000.

Solution. 0.065 x 1000 = 0065 = 65.

Answer: 65.

Example 2. Find the product of 3.9 and 1000.

Solution. 3.9 x 1000 = 3.900 x 1000 = 3900.

Answer: 3900.

If you need to multiply a natural number and 0.1; 0.01; 0.001; 0.0001, etc., you should move the comma to the left in the resulting product by as many digit characters as there are zeros before one. If necessary, a sufficient number of zeros are written in front of a natural number.

Example 1. Find the product of 56 and 0.01.

Solution. 56 x 0.01 = 0056 = 0.56.

Answer: 0,56.

Example 2. Find the product of 4 and 0.001.

Solution. 4 x 0.001 = 0004 = 0.004.

Answer: 0,004.

So, finding the product of various fractions should not cause difficulties, except perhaps the calculation of the result; In this case, you simply cannot do without a calculator.

Another operation that can be performed with ordinary fractions is multiplication. We will try to explain its basic rules when solving problems, show how an ordinary fraction is multiplied by a natural number and how to correctly multiply three or more ordinary fractions.

Let's write down the basic rule first:

Definition 1

If we multiply one ordinary fraction, then the numerator of the resulting fraction will be equal to the product of the numerators of the original fractions, and the denominator to the product of their denominators. In literal form, for two fractions a / b and c / d, this can be expressed as a b · c d = a · c b · d.

Let's look at an example of how to apply this rule correctly. Let's say we have a square whose side is equal to one numerical unit. Then the area of ​​the figure will be 1 square. unit. If we divide the square into equal rectangles with sides equal to 1 4 and 1 8 of the numerical unit, we get that it now consists of 32 rectangles (because 8 4 = 32). Accordingly, the area of ​​each of them will be equal to 1 32 of the area of ​​the entire figure, i.e. 1 32 sq. units.

We have a shaded fragment with sides equal to 5 8 numerical units and 3 4 numerical units. Accordingly, to calculate its area, it is necessary to multiply the first fraction by the second. It will be equal to 5 8 3 4 square meters. units. But we can simply count how many rectangles are included in the fragment: there are 15 of them, so total area is 1532 square units.

Since 5 3 = 15 and 8 4 = 32 we can write the following equation:

5 8 3 4 = 5 3 8 4 = 15 32

It is a confirmation of the rule we have formulated for multiplying ordinary fractions, which is expressed as a b · c d = a · c b · d. It works the same for both correct and improper fractions; It can be used to multiply fractions with different and the same denominators.

Let's analyze the solutions of several problems for the multiplication of ordinary fractions.

Example 1

Multiply 7 11 by 9 8 .

Solution

To begin with, we calculate the product of the numerators of the indicated fractions by multiplying 7 by 9. We got 63 . Then we calculate the product of the denominators and get: 11 8 = 88 . Let's compose the answer from two numbers: 63 88.

The whole solution can be written like this:

7 11 9 8 = 7 9 11 8 = 63 88

Answer: 7 11 9 8 = 63 88 .

If in the answer we got a reducible fraction, we need to complete the calculation and perform its reduction. If we get an improper fraction, we need to select the whole part from it.

Example 2

Calculate product of fractions 4 15 and 55 6 .

Solution

According to the rule studied above, we need to multiply the numerator by the numerator, and the denominator by the denominator. The solution entry will look like this:

4 15 55 6 = 4 55 15 6 = 220 90

We have obtained a reduced fraction, i.e. one that has a sign of divisibility by 10.

Let's reduce the fraction: 220 90 GCD (220, 90) \u003d 10, 220 90 \u003d 220: 10 90: 10 \u003d 22 9. As a result, we got an improper fraction, from which we select the whole part and get a mixed number: 22 9 \u003d 2 4 9.

Answer: 4 15 55 6 = 2 4 9 .

For the convenience of calculation, we can also reduce the original fractions before performing the multiplication operation, for which we need to bring the fraction to the form a · c b · d. We decompose the values ​​of the variables into simple factors and cancel the same ones.

Let us explain how this looks like using the data of a specific problem.

Example 3

Calculate the product 4 15 55 6 .

Solution

Let's write the calculations based on the multiplication rule. We will be able to:

4 15 55 6 = 4 55 15 6

Since as 4 = 2 2 , 55 = 5 11 , 15 = 3 5 and 6 = 2 3 , then 4 55 15 6 = 2 2 5 11 3 5 2 3 .

2 11 3 3 = 22 9 = 2 4 9

Answer: 4 15 55 6 = 2 4 9 .

A numerical expression in which the multiplication of ordinary fractions takes place has a commutative property, that is, if necessary, we can change the order of the factors:

a b c d = c d a b = a c b d

How to multiply a fraction with a natural number

Let's write down the basic rule right away, and then try to explain it in practice.

Definition 2

To multiply an ordinary fraction by a natural number, you need to multiply the numerator of this fraction by this number. In this case, the denominator of the final fraction will be equal to the denominator of the original common fraction. The multiplication of some fraction a b by a natural number n can be written as a formula a b · n = a · n b .

It is easy to understand this formula if you remember that any natural number can be represented as an ordinary fraction with a denominator equal to one, that is:

a b n = a b n 1 = a n b 1 = a n b

Let us explain our idea with specific examples.

Example 4

Compute the product of 2 27 by 5 .

Solution

As a result of multiplying the numerator of the original fraction by the second factor, we get 10. By virtue of the rule above, we will get 10 27 as a result. The whole solution is given in this post:

2 27 5 = 2 5 27 = 10 27

Answer: 2 27 5 = 10 27

When we multiply a natural number with a common fraction, we often have to reduce the result or represent it as a mixed number.

Example 5

Condition: Calculate the product of 8 times 5 12 .

Solution

According to the rule above, we multiply a natural number by the numerator. As a result, we get that 5 12 8 = 5 8 12 = 40 12. The final fraction has signs of divisibility by 2, so we need to reduce it:

LCM (40, 12) \u003d 4, so 40 12 \u003d 40: 4 12: 4 \u003d 10 3

Now we only have to select the integer part and write down the finished answer: 10 3 = 3 1 3.

In this entry, you can see the entire solution: 5 12 8 = 5 8 12 = 40 12 = 10 3 = 3 1 3 .

We could also reduce the fraction by factoring the numerator and denominator into prime factors, and the result would be exactly the same.

Answer: 5 12 8 = 3 1 3 .

A numeric expression in which a natural number is multiplied by a fraction also has the displacement property, that is, the order of the factors does not affect the result:

a b n = n a b = a n b

How to multiply three or more common fractions

We can extend to the multiplication of ordinary fractions the same properties that are characteristic of the multiplication of natural numbers. This follows from the very definition of these concepts.

Thanks to the knowledge of the associative and commutative properties, it is possible to multiply three or more ordinary fractions. It is permissible to rearrange the factors in places for greater convenience or arrange the brackets in a way that will make it easier to count.

Let's show an example how this is done.

Example 6

Multiply four common fractions 1 20 , 12 5 , 3 7 and 5 8 .

Solution: First, let's record the work. We get 1 20 12 5 3 7 5 8 . We need to multiply all the numerators and all the denominators together: 1 20 12 5 3 7 5 8 = 1 12 3 5 20 5 7 8 .

Before we start multiplication, we can make it a little easier for ourselves and decompose some numbers into prime factors for further reduction. This will be easier than reducing the finished fraction resulting from it.

1 12 3 5 20 5 7 8 = 1 (2 2 3) 3 5 2 2 5 5 7 (2 2 2) = 3 3 5 7 2 2 2 = 9 280

Answer: 1 12 3 5 20 5 7 8 = 9280.

Example 7

Multiply 5 numbers 7 8 12 8 5 36 10 .

Solution

For convenience, we can group the fraction 7 8 with the number 8 and the number 12 with the fraction 5 36 , since this will make future reductions clear to us. As a result, we will get:
7 8 12 8 5 36 10 = 7 8 8 12 5 36 10 = 7 8 8 12 5 36 10 = 7 1 2 2 3 5 2 2 3 3 10 = = 7 5 3 10 = 7 5 10 3 = 350 3 = 116 2 3

Answer: 7 8 12 8 5 36 10 = 116 2 3 .

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

This reasoning became a logical shock for all subsequent generations. Aristotle, Diogenes, Kant, Hegel, Gilbert... All of them, in one way or another, considered Zeno's aporias. The shock was so strong that " ... discussions continue at the present time, to come to a common opinion about the essence of paradoxes scientific community has not yet succeeded ... mathematical analysis, set theory, new physical and philosophical approaches; none of them became a universally accepted solution to the problem ..."[Wikipedia," Zeno's Aporias "]. 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 the value to. This transition implies applying instead of constants. As far as I understand, the mathematical apparatus for applying variable units of measurement has either not yet been developed, or it has not been applied to Zeno's aporia. The application of our usual logic leads us into a trap. We, by the inertia of thinking, apply constant units of time to the reciprocal. From a physical point of view, it looks like time slowing down to a complete stop at the moment when Achilles catches up with the tortoise. If time stops, Achilles can no longer overtake the tortoise.

If we turn the logic we are used to, everything falls into place. Achilles runs at a constant speed. Each subsequent segment of its 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 overtake the tortoise."

How to avoid this logical trap? Remain in constant units of time and do not switch to reciprocal values. In Zeno's language, it looks like this:

In the time it takes Achilles to run a thousand steps, the tortoise crawls 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 one hundred steps. Now Achilles is eight hundred paces 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 insurmountability of the speed of light is very similar to Zeno's aporia "Achilles and the tortoise". We have yet 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 of a flying arrow:

A flying arrow is motionless, since at each 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 the flying arrow rests at different points in space, which, in fact, is movement. There is another point 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 the fact of the movement of the car, two photographs taken from the same point at different points in time are needed, but they cannot be used to determine the distance. To determine the distance to the car, you need two photographs taken from different points in space at the same time, but you cannot determine the fact of movement from them (naturally, you still need additional data for calculations, trigonometry will help you). What I want to point out in particular is that two points in time and two points in space are two different things that should not be confused, because they provide different possibilities for research.

Wednesday, July 4, 2018

Very well the differences between set and multiset are described in Wikipedia. We look.

As you can see, "the set cannot have two identical elements", but if there are identical elements in the set, such a set is called a "multiset". Similar logic of absurdity sentient beings never understand. This is the level talking parrots and trained monkeys, in which the mind is absent from the word "completely." Mathematicians act as ordinary trainers, preaching their absurd ideas to us.

Once upon a time, the engineers who built the bridge were in a boat under the bridge during the tests of 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 desk, paying salaries. Here a mathematician comes to us for his money. We count the entire amount to him and lay it 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 it to the math " mathematical set We explain the mathematics that he will receive the rest of the banknotes only when he proves that the set without the same elements is not equal to the set with the same elements. Here the fun begins.

First of all, the deputies' logic will work: "you can apply it to others, but not to me!" Further, assurances will begin that there are different banknote numbers on banknotes of the same denomination, which means that they cannot be considered identical elements. Well, we count the salary in coins - there are no numbers on the coins. Here the mathematician will frantically recall physics: different coins have different amounts of dirt, the crystal structure and arrangement of atoms for each coin is unique ...

And now I have the most interesting question: where is the boundary beyond which 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 here is not even close.

Look here. We select football stadiums with the same field area. The area of ​​the fields is the same, which means we have 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 right? And here the mathematician-shaman-shuller takes out a trump ace 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 will 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 they are shamans for that, 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 "Sum of Digits of a Number" page. She 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 can do it elementarily.

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's say we 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 to a number graphic symbol. This is not a mathematical operation.

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

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

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

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

From the point of view of mathematics, it does not matter in which number system we write the number. So, in different systems reckoning, 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. FROM a large number 12345 I don’t 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 consider each step under a microscope, we have 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 like finding the area of ​​a rectangle in meters and centimeters would give you completely different results.

Zero in all number systems looks the same and has no sum of digits. This is another argument in favor of the fact that . A question for mathematicians: how is it denoted in mathematics that which is not a number? What, for mathematicians, nothing but numbers exists? For shamans, I can allow this, but for scientists, no. Reality is not just about numbers.

The result obtained should be considered as proof that number systems are units of measurement of 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 action does not depend on the value of the number, the unit of measure used, and on who performs this action.

Ouch! Isn't this the women's restroom?
- Young woman! This is a laboratory for studying the indefinite holiness of souls upon ascension to heaven! Nimbus on top and arrow up. What other toilet?

Female... A halo on top and an arrow down is male.

If you have such a work of design art flashing before your eyes several times a day,

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

Personally, I make an effort on myself to see minus four degrees in a pooping person (one picture) (composition of several pictures: minus sign, number four, degrees designation). And I do not consider this girl a fool who does not know physics. She just has an arc stereotype of perception graphic images. And mathematicians teach us this all the time. Here is an example.

1A is not "minus four degrees" or "one a". This is "pooping man" or the number "twenty-six" in the hexadecimal number system. Those people who constantly work in this number system automatically perceive the number and letter as one graphic symbol.

BYPASS THESE RAKE ALREADY! 🙂

Multiplication and division of fractions.

Attention!
There are additional
material in Special Section 555.
For those who are strong "not very. »
And for those who “very even. "")

This operation is much nicer than addition-subtraction! Because it's easier. I 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:

Everything is extremely simple. And please don't look for a common denominator! Don't need it here...

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

If multiplication or division with integers and fractions is caught, it's okay. As with addition, we make a fraction from a whole number with a unit in the denominator - and 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 form? Yes, very easy! Use division through two points:

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

In the first case (expression on the left):

In the second (expression on the right):

Feel the difference? 4 and 1/9!

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

then divide-multiply in order, left to right!

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

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

That's all the actions with fractions. The thing is quite simple, but gives more than enough errors. Take note of practical advice, and there will be fewer of them (mistakes)!

1. The most important thing when working with fractional expressions is accuracy and attentiveness! These are not common words, not good wishes! This is a severe need! Do all the 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 up when calculating in your head.

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

3. We reduce all fractions to the stop.

4. We reduce multi-level fractional expressions to ordinary ones using division through two points (we follow the order of division!).

Here are the tasks you need to complete. Answers are given after all tasks. Use the materials of this topic and practical advice. Estimate how many examples you could solve correctly. The first time! Without a calculator! And draw the right conclusions.

Remember the correct answer obtained from the second (especially the third) time - does not count! Such is the harsh life.

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

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 a semicolon.

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

And now we draw conclusions. If everything worked out - happy for you! Elementary 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. it solvable Problems.

In Special Section 555 "Fractions" all these (and not only!) examples are analyzed. With detailed explanations of what, why and how. Such an analysis helps a lot with a lack of knowledge and skills!

Yes, and on the second problem there is something there.) Quite practical advice, how to become more attentive. Yes Yes! Advice that can apply each.

In addition to knowledge and attentiveness, a certain automatism is needed for success. Where to get it? I hear a heavy sigh... Yes, only in practice, nowhere else.

You can go to the site 321start.ru for training. There, in the "Try" option, there are 10 examples for everyone to use. With instant verification. For registered users - 34 examples from simple to severe. It's only for fractions.

If you like this site.

By the way, I have a couple more interesting sites for you.)

Here you can practice solving examples and find out your level. Testing with instant verification. Learn with interest!

And here you can get acquainted with functions and derivatives.

Rule 1

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

Rule 2

To multiply a fraction by a fraction:

1. find the product of the numerators and the product of the denominators of these fractions

2. Write the first product as the numerator, and the second as the denominator.

Rule 3

In order to multiply mixed numbers, you need to write them as improper fractions, and then use the rule for multiplying fractions.

Rule 4

To divide one fraction by another, you need to multiply the dividend by the reciprocal of the divisor.

Example 1

Calculate

Example 2

Calculate

Example 3

Calculate

Example 4

Calculate

Maths. Other materials

Raising a number to a rational power. (

Raising a number to a natural power. (

Generalized interval method for solving algebraic inequalities (Author Kolchanov A.V.)

Method of replacement of factors in solving algebraic inequalities (Author Kolchanov A.V.)

Signs of divisibility (Lungu Alena)

Test yourself on the topic ‘Multiplication and division of ordinary fractions’

Multiplication of fractions

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

Multiplying a fraction by a fraction

This is the simplest case, in which you need to use the following fraction multiplication rules.

To multiply a fraction by a fraction, necessary:

Before multiplying numerators and denominators, check if the fractions can be reduced. Reducing fractions in calculations will greatly facilitate your calculations.

Multiplying a fraction by a natural number

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

If the result of multiplication is an improper fraction, 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 convert them into improper fractions and then multiply according to the rule for multiplying ordinary fractions.

Another way to multiply a fraction by a natural number

Sometimes in calculations it is more convenient to use a different method of multiplying an ordinary fraction by a 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 can be seen from the example, it is more convenient to use this version of the rule if the denominator of the fraction is divisible without a remainder by a natural number.

Division of a fraction by a number

What is the fastest way to divide a fraction by a number? Let's analyze the theory, draw a conclusion and use examples to see how the division of a fraction by a number can be performed according to a new short rule.

Usually, the division of a fraction by a number is performed according to the rule of division of fractions. The first number (fraction) is multiplied by the reciprocal of the second. Since the second number is an integer, its reciprocal is a fraction, the numerator of which is equal to one, and the denominator is the given number. Schematically, dividing a fraction by a natural number looks like this:

From this we conclude:

To divide a fraction by a number, multiply the denominator by that number and leave the numerator the same. The rule can be formulated even more briefly:

When you divide a fraction by a number, the number goes to the denominator.

Divide a fraction by a number:

To divide a fraction by a number, we rewrite the numerator unchanged, and multiply the denominator by this number. We reduce 6 and 3 by 3.

When dividing a fraction by a number, we rewrite the numerator and multiply the denominator by that number. We reduce 16 and 24 by 8.

When dividing a fraction by a number, the number goes to the denominator, so we leave the numerator the same, and multiply the denominator by the divisor. We reduce 21 and 35 by 7.

Multiplication and division of fractions

Last time we learned how to add and subtract fractions (see the lesson "Adding and subtracting fractions"). The most difficult moment in those actions was bringing fractions to a common denominator.

Now it's time to deal with multiplication and division. Good news is that these operations are even simpler than addition and subtraction. To begin with, consider the simplest case, when there are two positive fractions without a distinguished integer part.

To multiply two fractions, you need to multiply their numerators and denominators separately. The first number will be the numerator of the new fraction, and the second will be the denominator.

To divide two fractions, you need to multiply the first fraction by the "inverted" second.

From the definition it follows that the division of fractions is reduced to multiplication. To flip a fraction, just swap the numerator and denominator. Therefore, the entire lesson we will consider mainly multiplication.

As a result of multiplication, a reduced fraction can arise (and often does arise) - of course, it must be reduced. If, after all the reductions, the fraction turned out to be incorrect, the whole part should be distinguished in it. But what exactly will not happen with multiplication is reduction to a common denominator: no crosswise methods, maximum factors and least common multiples.

A task. Find the value of the expression:

By definition we have:

Multiplication of fractions with an integer part and negative fractions

If there is an integer part in the fractions, they must be converted to improper ones - and only then multiplied according to the schemes outlined above.

If there is a minus in the numerator of a fraction, in the denominator or in front of it, it can be taken out of the limits of multiplication or removed altogether according to the following rules:

1. Plus times minus gives minus;
2. Two negatives make an affirmative.

Until now, these rules have only been encountered when adding and subtracting negative fractions, when it was required to get rid of the whole part. For a product, they can be generalized in order to “burn” several minuses at once:

3. We cross out the minuses in pairs until they completely disappear. In an extreme case, one minus can survive - the one that did not find a match;
4. If there are no minuses left, the operation is completed - you can start multiplying. If the last minus is not crossed out, since it did not find a pair, we take it out of the limits of multiplication. You get a negative fraction.

We translate all fractions into improper ones, and then we take out the minuses outside the limits of multiplication. What remains is multiplied according to the usual rules. We get:

Let me remind you once again that the minus that comes before a fraction with a highlighted integer part refers specifically to the entire fraction, and not just to its integer part (this applies to the last two examples).

Also pay attention to negative numbers: When multiplied, they are enclosed in parentheses. This is done in order to separate the minuses from the multiplication signs and make the whole notation more accurate.

Reducing fractions on the fly

Multiplication is a very laborious operation. The numbers here are quite large, and to simplify the task, you can try to reduce the fraction even more before multiplication. Indeed, in essence, the numerators and denominators of fractions are ordinary factors, and, therefore, they can be reduced using the basic property of a fraction. Take a look at the examples:

In all examples, the numbers that have been reduced and what is left of them are marked in red.

Please note: in the first case, the multipliers were reduced completely. Units remained in their place, which, generally speaking, can be omitted. In the second example, it was not possible to achieve a complete reduction, but the total amount of calculations still decreased.

However, in no case do not use this technique when adding and subtracting fractions! Yes, sometimes there are similar numbers that you just want to reduce. Here, look:

You can't do that!

The error occurs due to the fact that when adding a fraction, the sum appears in the numerator of a fraction, and not the product of numbers. Therefore, it is impossible to apply the main property of a fraction, since this property deals specifically with the multiplication of numbers.

There is simply no other reason to reduce fractions, so the correct solution to the previous problem looks like this:

As you can see, the correct answer turned out to be not so beautiful. In general, be careful.

Division of fractions.

Division of a fraction by a natural number.

Examples of dividing a fraction by a natural number

Division of a natural number by a fraction.

Examples of dividing a natural number by a fraction

Division of ordinary fractions.

Examples of division of ordinary fractions

Division of mixed numbers.

To divide one mixed number by another, you need:
• convert mixed fractions to improper;
• multiply the first fraction by the reciprocal of the second;
• reduce the resulting fraction;
• If you get an improper fraction, convert the improper fraction to a mixed one.

Examples of dividing mixed numbers

1 1 2: 2 2 3 = 1 2 + 1 2: 2 3 + 2 3 = 3 2: 8 3 = 3 2 3 8 = 3 3 2 8 = 9 16

2 1 7: 3 5 = 2 7 + 1 7: 3 5 = 15 7: 3 5 = 15 7 5 3 = 15 5 7 3 = 5 5 7 = 25 7 = 7 3 + 4 7 = 3 4 7

Any obscene comments will be removed and their authors blacklisted!

Welcome to OnlineMSchool.
My name is Dovzhik Mikhail Viktorovich. I am the owner and author of this site, I wrote all the theoretical material, and also developed online exercises and calculators that you can use to study mathematics.

Fractions. Multiplication and division of fractions.

Multiplying a fraction by a fraction.

To multiply ordinary fractions, it is necessary to multiply the numerator by the numerator (we get the numerator of the product) and the denominator by the denominator (we get the denominator of the product).

Fraction multiplication formula:





Before proceeding with the multiplication of numerators and denominators, it is necessary to check for the possibility of reducing the fraction. If you manage to reduce the fraction, then it will be easier for you to continue to make calculations.

Note! There is no need to look for a common denominator!!

Division of an ordinary fraction by a fraction.

The division of an ordinary fraction by a fraction is as follows: turn over the second fraction (i.e. change the numerator and denominator in places) and after that the fractions are multiplied.

The formula for dividing ordinary fractions:





Multiplying a fraction by a natural number.

Note! When multiplying a fraction by a natural number, the numerator of the fraction is multiplied by our natural number, and the denominator of the fraction remains the same. If the result of the product is an improper fraction, then be sure to select the whole part by turning the improper fraction into a mixed one.



Division of fractions involving a natural number.

It's not as scary as it seems. As in the case of addition, we convert an integer into a fraction with a unit in the denominator. For example:



Multiplication of mixed fractions.

Rules for multiplying fractions (mixed):

• convert mixed fractions to improper;
• multiply the numerators and denominators of fractions;
• we reduce the fraction;
• if we get an improper fraction, then we convert the improper 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 for multiplying ordinary fractions.



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

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

Note! To multiply a fraction by a natural number, it is necessary to 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 a fraction is divided without a remainder by a natural number.

Multilevel fractions.

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

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



Note! When dividing fractions, the order of division is very important. Be careful, it's 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, concentratedly and clearly. It is better to write down a few extra lines in a draft than to get confused in the calculations in your head.

2. In tasks with different types of fractions, go to the type of ordinary fractions.

3. We reduce all fractions until it is no longer possible to reduce.

4. We bring multi-level fractional expressions into ordinary ones, using division through 2 points.

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