Mixed numbers, converting a mixed number to an improper fraction and vice versa. How to make a proper fraction from an improper fraction

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.

The word itself - a fraction means that the number is fractional, it is less than a whole (at least one).

Therefore, it is necessary to extract an integer from the numerator. For example, the number 30/4 is an incorrect fraction, since 30 is greater than 4. So, you just need to divide 30 by 4 and get the number before the decimal point - 7, and then put it in front of the fraction. We multiply 7 by 4 and subtract this number from 30 - we get 2 - it will be in the numerator of the fraction. The result is 7 2/4, we reduce - 7 1/2. In your example, the answer is 2 3/4.

To do this, you need a denominator: the denominator.

The integer that turned out - write in the numerator. The denominator is the one that was. When divided - write down in the whole part.

11:4=2 (3rd remainder).

We get the rule-th fraction: 2 - as many as 34

To convert an improper fraction into a correct one, you need to identify the whole parts and subtract them from the improper fraction. In our case, the improper fraction is 11/4. There will be two (2) whole parts. We subtract them and get the correct fraction: two point three fourths (2 point 3/4).



An improper fraction, in our case, 11/4 must be converted to a correct one, i.e. in this case a mixed fraction. If in a simple way, then the fraction is incorrect, because in addition to the fraction there is also an integer in it. It's like standing in the fridge an unfinished cake, albeit cut, and on the table there are a few pieces left from the second one. When we talk about 11/4, we no longer know about two whole cakes, we see only eleven large pieces. 11 divided by 4 gives 2, and the remainder is 11-8=3. So, 2 whole 3/4, now the fraction is correct, in it the numerator will be smaller than the denominator, but mixed, since the calculation could not do without whole units.

To convert an improper fraction to a correct one, divide the numerator by the denominator. The resulting integer is taken out before the fraction, and the remainder is entered in the numerator. The denominator does not change.

For example: 11/4 is an improper fraction where the numerator is 11 and the denominator is 4.

First, we divide 11 by 4, we get 2 integers and 3 remainder. We take out 2 before the fraction, and write the remainder 3 in the numerator 3/4. Thus, the fraction becomes regular - 2 integers and 3/4.

For an improper fraction, the denominator is less than the numerator, which indicates that this fraction has integer parts that can be distinguished and obtained as a proper fraction with a whole number.

The easiest way to divide the numerator by the denominator. The resulting integer is put to the left of the fraction, and the remainder is written to the numerator, the denominator remains the same.

For example 11/4. We divide 11 by 4 and get 2 and the remainder 3. Two is the number that we put next to the fraction, and we write the three in the numerator of the fraction. Comes out 2 and 3/4.

To answer this simple question, you can solve the same simple problem:

Petya and Valya came to the company of their peers. All together there were 11 of them. Valya had apples with him (but not a lot) and in order to treat everyone Petya cut each into four parts and distributed it. Enough for everyone and even five pieces left.

How many apples did Petya distribute and how many apples are left? How many were there?

Can you write it down mathematically

11 apple slices is in our case 11/4 - we got an improper fraction, since the numerator is greater than the denominator.

To highlight the whole part (convert improper fraction to proper), you need divide the numerator by the denominator, the incomplete quotient (in our case it is 2) is written on the left, the remainder (3) is left in the numerator and the denominator is not touched.

As a result, we get 11/4 = 11:4 = 2 3/4 Peter handed out the apples.

Similarly, 5/4 = 1 1/4 apples left.

(11+5)/4 = 16/4 = 4 apples brought by Valya

Very often in school curriculum Mathematics children are faced with the problem of how to convert a common fraction to a decimal. In order to convert a common fraction to a decimal, let's first recall what a common fraction and a decimal fraction are. A common fraction is a fraction of the form m/n, where m is the numerator and n is the denominator. Example: 8/13; 6/7 etc. Fractions are divided into regular, improper and mixed numbers. A proper fraction is when the numerator is less than the denominator: m / n, where m 3. An improper fraction can always be represented as a mixed number, namely: 4/3 \u003d 1 and 1/3;

Converting an ordinary fraction to a decimal

Now let's look at how to convert a mixed fraction to a decimal. Any ordinary fraction, whether it is correct or incorrect, can be converted to a decimal. To do this, you need to divide the numerator by the denominator. Example: simple fraction(correct) 1/2. We divide the numerator 1 by the denominator 2, we get 0.5. Take the example of 45/12, it is immediately clear that this is an improper fraction. Here the denominator is less than the numerator. We turn the improper fraction into a decimal: 45: 12 \u003d 3.75.

Convert mixed numbers to decimals

Example: 25/8. First, we turn the mixed number into an improper fraction: 25/8 = 3x8+1/8 = 3 and 1/8; then we divide the numerator equal to 1 by the denominator equal to 8, in a column or on a calculator and get decimal equal to 0.125. The article provides the easiest examples of converting to decimal fractions. Having understood the method of translation into simple examples, you can easily solve the most difficult of them.

In this material, we will analyze such a thing as mixed numbers. We start, as always, with a definition and small examples, then we will explain the connection between mixed numbers and improper fractions. After that, we will learn how to correctly extract the integer part from a fraction and get an integer as a result.

The concept of a mixed number

If we take the sum n + a b , where the value of n can be any natural number, and a b is a proper ordinary fraction, then we can write the same thing without using a plus: n a b . Let's take specific numbers to be clear, so 28 + 5 7 is the same as 28 5 7 . Writing a fraction next to an integer is called a mixed number.

Definition 1

mixed number is a number that is equal to the sum of a natural number n with a proper ordinary fraction a b . In this case, n is whole part number, and a b is its fractional part.

It follows from the definition that any mixed number is equal to what will result from the addition of its integer and fractional parts. Thus, the equality n a b = n + a b will hold.

It can also be written as n + a b = n a b .

What are some examples of mixed numbers? So, 5 1 8 belongs to them, while five is its whole part, and one eighth is fractional. More examples: 1 1 2 , 234 34 53 , 34000 6 25 .

We wrote above that only a proper fraction should be in the fractional part of a mixed number. Sometimes you can find entries like 5 22 3 , 75 7 2 . They are not mixed numbers, because their fractional part is wrong. They need to be understood as the sum of an integer and a fractional part. Such numbers can be reduced to standard mixed numbers by taking the integer part of the improper fraction and adding it to 5 and 75 in these examples, respectively.

Numbers of the form 0 3 14 are also not mixed. The first part of the condition is not met here: the whole part must be represented only natural number, and zero is not.

How are improper fractions and mixed numbers related?

This connection is easiest to trace on a concrete example.

Example 1

Let's take a whole cake and another three quarters of the same. According to the addition rules, we have 1 + 3 4 cakes on the table. This sum can be represented as a mixed number as 1 3 4 cakes. If we take a whole cake and also cut it into four equal parts, then we will have 7 4 cakes on the table. Obviously, the amount did not increase from cutting, and 1 3 4 = 7 4 .

Our example proves that any improper fraction can be represented as a mixed number.

Let's go back to our 7 4 cakes left on the table. Let's put one cake back from its pieces (1 + 3 4). We will again have 1 3 4 .

Answer: 7 4 = 1 3 4 .

We figured out how to convert an improper fraction to a mixed number. If the numerator of an improper fraction contains a number that can be divided by the denominator without a remainder, then you can do this, and then our improper fraction will become a natural number.

Example 2

For example,

8 4 = 2 since 8: 4 = 2 .

How to convert a mixed number to an improper fraction

To successfully solve problems, it is useful to be able to perform the reverse action, that is, to make improper fractions from mixed numbers. In this paragraph, we will analyze how to do it correctly.

To do this, you need to reproduce the following sequence of actions:

1. To begin with, we present the available mixed number n a b as the sum of the integer and fractional parts. It turns out n + a b

3. After that, we perform an already familiar action - we add two ordinary fractions n 1 and a b. The resulting improper fraction will be equal to the mixed number given in the condition.

Let's analyze this action on a specific example.

Example 3

Write 5 3 7 as an improper fraction.

Solution

We perform the steps of the above algorithm in sequence. Our number 5 3 7 is the sum of the integer and fractional parts, that is, 5 + 3 7. Now let's write the five as 5 1 . We got the sum 5 1 + 3 7 .

The last step is to add fractions with different denominators:

5 1 + 3 7 = 35 7 + 3 7 = 38 7

All solution to short form can be written as 5 3 7 = 5 + 3 7 = 5 1 + 3 7 = 35 7 + 3 7 = 38 7 .

Answer: 5 3 7 = 38 7 .

Thus, with the help of the above chain of actions, we can convert any mixed number n a b into an improper fraction. We have obtained the formula n a b = n b + a b , which we will take to solve further problems.

Example 4

Write 15 2 5 as an improper fraction.

Solution

Take this formula and substitute the desired values ​​into it. We have n = 15 , a = 2 , b = 5 , therefore 15 2 5 = 15 5 + 2 5 = 77 5 .

Answer: 15 2 5 = 77 5 .

We usually don't list the improper fraction as the final answer. It is customary to bring the calculations to the end and replace it with either a natural number (dividing the numerator by the denominator) or a mixed number. As a rule, the first method is used when it is possible to divide the numerator by the denominator without a remainder, and the second - if such an action is impossible.

When we extract the whole part from an improper fraction, we simply replace it with an equal mixed number.

Let's see how exactly this is done.

Definition 2

We present a proof of this assertion.

We need to explain why q r b = a b . To do this, the mixed number q r b must be represented as an improper fraction by following all the steps of the algorithm from the previous paragraph. Since is an incomplete quotient, and r is the remainder of dividing a by b, then the equality a = b · q + r must hold.

So q b + r b = a b so q r b = a b . This is the proof of our assertion. To summarize:

Definition 3

The selection of the integer part from the improper fraction a b is carried out as follows:

1) we divide a by b with a remainder and write the incomplete quotient q and the remainder r separately.

2) Write the results as q r b . This is our mixed number, equal to the original improper fraction.

Example 5

Express 1074 as a mixed number.

Solution

We divide 104 by 7 in a column:

Dividing the numerator a = 118 by the denominator b = 7 gives us the incomplete quotient q = 16 and the remainder r = 6.

As a result, we get that the improper fraction 118 7 is equal to the mixed number q r b = 16 6 7 .

Answer: 118 7 = 16 6 7 .

It remains for us to see how to replace an improper fraction with a natural number (provided that its numerator is divisible by the denominator without a remainder).

To do this, remember what relationship exists between ordinary fractions and division. From this we can derive the equalities: a b = a: b = c . It turns out that the improper fraction a b can be replaced by a natural number c.

Example 6

For example, if the answer turned out to be an improper fraction 27 3, then we can write 9 instead, since 27 3 \u003d 27: 3 \u003d 9.

Answer: 27 3 = 9 .

If you notice a mistake in the text, please highlight it and press Ctrl+Enter

You can convert an improper fraction to a proper one by dividing the numerator of such a fraction by the denominator - this way we get the correct fraction. Otherwise, an improper fraction can be written as a simple decimal number.

An improper fraction is a fraction whose numerator is greater than the denominator. correct - that fraction, in which, accordingly, the numerator is less than the denominator. there is no way to turn an improper fraction into a proper one, but it can be represented as a mixed number consisting of two parts (one part will be an integer, and the other will be just a proper fraction).

for example 5/2=2+1/2 (only a fraction is usually written immediately after an integer without a plus sign)

here you need to divide the numerator of the improper fraction by the denominator. write down the integer part of the division (in our case 2). then the remainder of the division (that is, 1) is written as the numerator of the fraction, which we write next to the two.

From school course we know mathematics. An improper fraction is a fraction whose numerator is greater than its denominator. To convert it to a proper fraction, you need to divide the numerator of such a fraction by its denominator. Everything is very simple, so it will become a correct, or a decimal fraction.

An improper fraction, for example: 9/5, we select its integer part, it will be: 1 4/5 is now a bit like the correct one, only with the integer part being one.

You can also turn it into a decimal fraction in our case it will be 1.8

To solve the problem, you first need to clearly understand for yourself what is a proper fraction and what is an incorrect one.

Let's start with the statement

true not for all numbers on the number line.

numerator is (-10), denominator is (-4)

similar statement

also not always true

numerator is 2, denominator is (-3)

An improper fraction can be written as the sum of an integer and a proper fraction ( mixed fraction) and for this you need:

divide the numerator by the denominator, write the resulting integer in the integer part, the remainder in the numerator, leave the denominator unchanged

in the numerator (-15), in the denominator 2, we take the minus outside the fraction - (15/2), divide 15 by 2, put the integer 7 in the integer part of the fraction, write the remainder of dividing 1 in the numerator, and leave the denominator 2 without changes.

In order to convert an improper fraction to a proper one, you first need to say:

In an improper fraction, the numerator (the top number in the fraction) is greater than or equal to the denominator;

For a proper fraction, the opposite is true.

We will analyze the conversion process using the example of a fraction 260/7:

1) First we divide 260 by 7, we get the number 37.14 ..

2) The number 37 will come before the fraction as an integer

3) Now 37 * 7 = 259

4) From the numerator we subtract the resulting number 260 - 259 \u003d 1 - this number will be in the numerators of our regular fraction.

5) When writing a new fraction, the denominator remains unchanged. In this case, it is 7. The correct fraction will look like this:



Checking the converted fraction:

We multiply the whole number by the denominator and add the numerator 37 * 7 + 1 = 260.

A proper fraction is a fraction whose denominator is greater than the numerator. This suggests that this fraction shows some part of the whole. For example, the fraction 1/2 indicates that we have half, for example, a watermelon, and the fraction 7/9 indicates that we have seven pieces of watermelon cut into 9 parts. Someone ate two.

If the fraction is incorrect, that is, the numerator is greater than the denominator, then it is completely incomprehensible what part of the whole, but cut watermelon, and how many more whole watermelons are available. Therefore, you have to convert the improper fraction to the correct one. in this case, we will get some integer and the remainder - exactly the correct fraction.

To translate, we divide the numerator by the denominator into a column. Example: 7/4. Seven by four gives one and the remainder is 3/4. So we converted the fraction to the correct one - the answer is 1 and 3/4.

Improper fraction called a fraction that has numerator greater than denominator. So a proper fraction is one whose numerator is less than the denominator. To turn an improper fraction into a proper one, you can represent it as a decimal number. For example, 17/8 can be written like this: 2.125. Or write it like this: 2 1/8.

A proper fraction is considered to be one in which the denominator is higher than the numerator. In order to convert an improper fraction into a proper one, you need to divide the numerator of the improper fraction by its denominator, the result will be a number with a remainder.

For example, 4 integers and three elevenths, we multiply 4 by 11 and +3, then we divide by 11, it turns out 44 +3 and divide by 11, and we get the fraction 47/11. An improper fraction is when there is an integer such as 5.10, that is, five integers and 10/100, five we multiply 100 and +10, it turns out 10/500. Also, if for example 6.6, it’s easier here, we multiply 6 by 6 and +6 turns out 12/6, we cut by two, we get six thirds, we cut six thirds by three, we get the first two, two divided by one, we get two. That is, 6.6 = 2.