Reciprocal number 4. Reciprocal number

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Reciprocal number(reciprocal, reciprocal) to a given number x is the number whose multiplication by x, gives one. Accepted entry: $\backslash frac(1)x$ or $x^(-1)$. Two numbers whose product is equal to one are called mutually inverse. The reciprocal number should not be confused with inverse function. For example, $\backslash frac(1)(\backslash cos(x))$ different from the value of the inverse cosine function - arccosine, which is denoted $\backslash cos^(-1)x$ or $\backslash arccos\; x$.

Inverse to real number

Complex number forms	Number $(z)$	Reverse $\backslash left\; (\backslash frac(1)(z)\; \backslash right)$
Algebraic	$x+iy$	$\backslash frac(x)(x^2+y^2)-i\; \backslash frac(y)(x^2+y^2)$
trigonometric	$r(\backslash cos\backslash varphi+i\; \backslash sin\backslash varphi)$	$\backslash frac(1)(r)(\backslash cos\backslash varphi-i\; \backslash sin\backslash varphi)$
Demonstration	$re^(i\backslash varphi)$	$\backslash frac(1)(r)e^(-i\; \backslash varphi)$

Proof:
For algebraic and trigonometric forms, we use the basic property of a fraction, multiplying the numerator and denominator by the complex conjugate:

• Algebraic form:

$\backslash frac(1)(z)=\; \backslash frac(1)(x+iy)=\; \backslash frac(x-iy)((x+iy)(x-iy))=\; \backslash frac(x-iy)(x^\; 2+y^2)=\; \backslash frac(x)(x^2+y^2)-i\; \backslash frac(y)(x^2+y^2)$

• Trigonometric form:

$\backslash frac(1)(z)\; =\; \backslash frac(1)(r(\backslash cos\backslash varphi+i\; \backslash sin\backslash varphi))\; =\; \backslash frac(1)(r)\; \backslash frac(\backslash cos\backslash varphi-i\; \backslash sin\backslash \; varphi)((\backslash cos\backslash varphi+i\; \backslash sin\backslash varphi)(\backslash cos\backslash varphi-i\; \backslash sin\backslash varphi))\; =\; \backslash frac(1)(r)\; \backslash frac(\backslash cos\backslash varphi-i\; \backslash sin\backslash varphi\; )(\backslash cos^2\backslash varphi+\; \backslash sin^2\backslash varphi)\; =\; \backslash frac(1)(r)(\backslash cos\backslash varphi-i\; \backslash sin\backslash varphi)$

• Indicative form:

$\backslash frac(1)(z)\; =\; \backslash frac(1)(re^(i\; \backslash varphi))\; =\; \backslash frac(1)(r)e^(-i\; \backslash varphi)$

Thus, when finding the inverse of a complex number, it is more convenient to use its exponential form.

Example:

Complex number forms	Number $(z)$	Reverse $\backslash left\; (\backslash frac(1)(z)\; \backslash right)$
Algebraic	$1+i\; \backslash sqrt(3)$	$\backslash frac(1)(4)-\; \backslash frac(\backslash sqrt(3))(4)i$
trigonometric	$2\; \backslash left\; (\backslash cos\backslash frac(\backslash pi)(3)+i\backslash sin\backslash frac(\backslash pi)(3)\; \backslash right)$ or $2\; \backslash left\; (\backslash frac(1)(2)+i\backslash frac(\backslash sqrt(3))(2)\; \backslash right)$	$\backslash frac(1)(2)\; \backslash left\; (\backslash cos\backslash frac(\backslash pi)(3)-i\backslash sin\backslash frac(\backslash pi)(3)\; \backslash right)$ or $\backslash frac(1)(2)\; \backslash left\; (\backslash frac(1)(2)-i\backslash frac(\backslash sqrt(3))(2)\; \backslash right)$
Demonstration	$2\; e^(i\; \backslash frac(\backslash pi)(3))$	$\backslash frac(1)(2)\; e^(-i\; \backslash frac(\backslash pi)(3))$

Inverse to the imaginary unit

$\backslash frac(1)(i)=\backslash frac(1\; \backslash cdot\; i)(i\; \backslash cdot\; i)=\backslash frac(i)(i^2)=\backslash frac(i)(-1)=-i$

Thus, we get

$\backslash frac(1)(i)=-i$ __ or__ $i^(-1)=-i$

Similarly for $-i$: __ $-\; \backslash frac(1)(i)=i$ __ or __ $-i^(-1)=i$

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Notes

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An excerpt characterizing the reciprocal number

So the stories say, and all this is completely unfair, as anyone who wants to delve into the essence of the matter will easily be convinced of.
The Russians did not look for a better position; but, on the contrary, in their retreat they passed many positions that were better than Borodino. They did not stop at any of these positions: both because Kutuzov did not want to accept a position that was not chosen by him, and because the demand for a popular battle had not yet been expressed strongly enough, and because Miloradovich had not yet approached with the militia, and also because other reasons that are innumerable. The fact is that the former positions were stronger and that the Borodino position (the one on which the battle was given) is not only not strong, but for some reason is not at all a position more than any other place in Russian Empire, which, guessing, would indicate with a pin on the map.
The Russians not only did not fortify the position of the Borodino field to the left at a right angle from the road (that is, the place where the battle took place), but never before August 25, 1812 did they think that the battle could take place on this place. This is evidenced, firstly, by the fact that not only on the 25th there were no fortifications in this place, but that, begun on the 25th, they were not completed on the 26th; secondly, the position of the Shevardinsky redoubt serves as proof: the Shevardinsky redoubt, in front of the position on which the battle was taken, does not make any sense. Why was this redoubt fortified stronger than all other points? And why, defending it on the 24th until late at night, were all efforts exhausted and six thousand people lost? To observe the enemy, a Cossack patrol was enough. Thirdly, the proof that the position on which the battle took place was not foreseen and that the Shevardinsky redoubt was not the forward point of this position is that Barclay de Tolly and Bagration until the 25th were convinced that the Shevardinsky redoubt was the left flank of the position and that Kutuzov himself, in his report, written hastily after the battle, calls the Shevardinsky redoubt the left flank of the position. Much later, when reports about the battle of Borodino were written in the open, it was (probably to justify the mistakes of the commander in chief, who had to be infallible) that unfair and strange testimony was invented that the Shevardinsky redoubt served as an advanced post (whereas it was only a fortified point of the left flank) and as if the battle of Borodino was accepted by us in a fortified and pre-selected position, while it took place in a completely unexpected and almost unfortified place.
The matter, obviously, was like this: the position was chosen along the Kolocha River, which crosses the main road not under a straight line, but under acute angle, so that the left flank was in Shevardin, the right flank was near the village of Novy and the center was in Borodino, at the confluence of the Kolocha and Voyna rivers. This position, under the cover of the Kolocha River, for the army, whose goal is to stop the enemy moving along the Smolensk road to Moscow, is obvious to anyone who looks at the Borodino field, forgetting how the battle took place.
Napoleon, leaving on the 24th to Valuev, did not see (as the stories say) the position of the Russians from Utitsa to Borodin (he could not see this position, because it was not there) and did not see the advanced post of the Russian army, but stumbled in the pursuit of the Russian rearguard on the left flank of the position of the Russians, on the Shevardinsky redoubt, and unexpectedly for the Russians transferred troops through Kolocha. And the Russians, not having time to enter into a general battle, retreated with their left wing from the position they intended to take, and took up a new position, which was not foreseen and not fortified. Having crossed to the left side of Kolocha, to the left of the road, Napoleon moved the entire future battle from right to left (from the side of the Russians) and transferred it to the field between Utitsa, Semenovsky and Borodino (in this field, which has nothing more advantageous for the position than any another field in Russia), and on this field the whole battle took place on the 26th. In rough form, the plan for the proposed battle and the battle that took place will be as follows:

If Napoleon had not left on the evening of the 24th for Kolocha and had not ordered to attack the redoubt immediately in the evening, but had begun the attack the next day in the morning, then no one would have doubted that the Shevardinsky redoubt was the left flank of our position; and the battle would have taken place as we expected it to. In that case, we would probably have defended the Shevardino redoubt, our left flank, even more stubbornly; they would attack Napoleon in the center or on the right, and on the 24th there would be a general battle in the position that was fortified and foreseen. But since the attack on our left flank took place in the evening, following the retreat of our rearguard, that is, immediately after the battle of Gridneva, and since the Russian military leaders did not want or did not have time to start a general battle on the same 24th evening, the first and main action of Borodinsky the battle was lost on the 24th and, obviously, led to the loss of the one that was given on the 26th.
After the loss of the Shevardinsky redoubt, by the morning of the 25th we found ourselves without a position on the left flank and were forced to bend back our left wing and hastily strengthen it anywhere.
But not only did the Russian troops stand only under the protection of weak, unfinished fortifications on August 26, the disadvantage of this situation was further increased by the fact that the Russian military leaders, not fully recognizing the accomplished fact (the loss of a position on the left flank and the transfer of the entire future battlefield from right to left ), remained in their extended position from the village of Novy to Utitsa and, as a result, had to move their troops from right to left during the battle. Thus, during the entire battle, the Russians had against all French army, aimed at our left wing, twice the weakest forces. (The actions of Poniatowski against Utitsa and Uvarov on the right flank of the French constituted actions separate from the course of the battle.)
So, the battle of Borodino did not happen at all as (trying to hide the mistakes of our military leaders and, as a result, belittling the glory of the Russian army and people) describe it. The battle of Borodino did not take place on a chosen and fortified position with only the weakest forces on the part of the Russians, and the battle of Borodino, due to the loss of the Shevardinsky redoubt, was taken by the Russians in an open, almost unfortified area with twice the weakest forces against the French, that is, under such conditions, in which it was not only unthinkable to fight for ten hours and make the battle indecisive, but it was unthinkable to keep the army from complete defeat and flight for three hours.

On the 25th in the morning Pierre left Mozhaisk. On the descent from the huge steep and crooked mountain leading out of the city, past the cathedral standing on the mountain to the right, in which there was a service and the gospel, Pierre got out of the carriage and went on foot. Behind him descended on the mountain some kind of cavalry regiment with peselniks in front. A train of carts with the wounded in yesterday's deed was rising towards him. The peasant drivers, shouting at the horses and whipping them with whips, ran from one side to the other. The carts, on which three and four wounded soldiers lay and sat, jumped over the stones thrown in the form of a pavement on a steep slope. The wounded, bound in rags, pale, with pursed lips and frowning eyebrows, holding on to the beds, jumped and jostled in the carts. Almost with naive childish curiosity, everyone looked at white hat and Pierre's green tailcoat.

Reverse - or reciprocal - numbers are called a pair of numbers that, when multiplied, give 1. In itself general view numbers are reversed. Characteristic special case reciprocal numbers - a pair. The inverses are, say, the numbers ; .

How to find the reciprocal

Rule: you need to divide 1 (one) by the given number.

Example #1.

The number 8 is given. Its inverse is 1:8 or (the second option is preferable, because such a notation is mathematically more correct).

When looking for the reciprocal of common fraction, then dividing it by 1 is not very convenient, because recording becomes cumbersome. In this case, it is much easier to do otherwise: the fraction is simply turned over, swapping the numerator and denominator. If a correct fraction is given, then after turning it over, an improper fraction is obtained, i.e. one from which a whole part can be extracted. To do this or not, you need to decide on a case-by-case basis. So, if you then have to perform some actions with the resulting inverted fraction (for example, multiplication or division), then you should not select the whole part. If the resulting fraction is the final result, then perhaps the selection of the integer part is desirable.

Example #2.

Given a fraction. Reverse to it:.

If you want to find the reciprocal of decimal fraction, then you should use the first rule (dividing 1 by a number). In this situation, you can act in one of 2 ways. The first is to simply divide 1 by this number into a column. The second is to form a fraction from 1 in the numerator and a decimal in the denominator, and then multiply the numerator and denominator by 10, 100, or another number consisting of 1 and as many zeros as necessary to get rid of the decimal point in the denominator. The result will be an ordinary fraction, which is the result. If necessary, you may need to shorten it, extract an integer part from it, or convert it to decimal form.

Example #3.

The number given is 0.82. Its reciprocal is: . Now let's reduce the fraction and select the integer part: .

How to check if two numbers are reciprocals

The principle of verification is based on the definition of reciprocals. That is, in order to make sure that the numbers are inverse to each other, you need to multiply them. If the result is one, then the numbers are mutually inverse.

Example number 4.

Given the numbers 0.125 and 8. Are they reciprocals?

Examination. It is necessary to find the product of 0.125 and 8. For clarity, we present these numbers as ordinary fractions: (let's reduce the 1st fraction by 125). Conclusion: the numbers 0.125 and 8 are inverse.

Properties of reciprocals

Property #1

The reciprocal exists for any number other than 0.

This limitation is due to the fact that you cannot divide by 0, and when determining the reciprocal of zero, it will just have to be moved to the denominator, i.e. actually divide by it.

Property #2

The sum of a pair of reciprocal numbers is never less than 2.

Mathematically, this property can be expressed by the inequality: .

Property #3

Multiplying a number by two reciprocal numbers is equivalent to multiplying by one. Let's express this property mathematically: .

Example number 5.

Find the value of the expression: 3.4 0.125 8. Since the numbers 0.125 and 8 are reciprocals (see Example #4), there is no need to multiply 3.4 by 0.125 and then by 8. So the answer here is 3.4.

Content:

Reciprocals are needed when solving all types of algebraic equations. For example, if you need to divide one fractional number by another, you multiply the first number by the reciprocal of the second. In addition, reciprocals are used when finding the equation of a straight line.

Steps

1 Finding the reciprocal of a fraction or integer

1. 1 Find the reciprocal of a fractional number by flipping it."Reciprocal number" is defined very simply. To calculate it, simply calculate the value of the expression "1 ÷ (original number)." For a fractional number, the reciprocal is another fractional number that can be calculated simply by "reversing" the fraction (by swapping the numerator and denominator).
   • For example, the reciprocal of 3/4 is 4 / 3 .
2. 2 Write the reciprocal of a whole number as a fraction. And in this case, the reciprocal is calculated as 1 ÷ (original number). For a whole number, write the reciprocal as a fraction, no need to do any calculations and write it as a decimal.
   • For example, the reciprocal of 2 is 1 ÷ 2 = 1 / 2 .

2 Finding the reciprocal of a mixed fraction

1. 1 What " mixed fraction". A mixed fraction is a number written as a whole number and a simple fraction, for example, 2 4 / 5. Finding the reciprocal of a mixed fraction is done in two steps, described below.
2. 2 Write the mixed fraction as improper fraction. Of course, you remember that the unit can be written as (number) / (same number), and fractions with the same denominator (number under the line) can be added to each other. Here is how it can be done for the fraction 2 4 / 5:
   • 2 4 / 5
   • = 1 + 1 + 4 / 5
   • = 5 / 5 + 5 / 5 + 4 / 5
   • = (5+5+4) / 5
   • = 14 / 5 .
3. 3 Flip the fraction. When a mixed fraction is written as an improper fraction, we can easily find the reciprocal by simply swapping the numerator and denominator.
   • For the example above, the reciprocal would be 14 / 5 - 5 / 14 .

3 Finding the reciprocal of a decimal

1. 1 If possible, express the decimal as a fraction. You need to know that many decimals can be easily converted to simple fractions. For example, 0.5 = 1/2 and 0.25 = 1/4. When you write a number as a simple fraction, you can easily find the reciprocal simply by flipping the fraction.
   • For example, the reciprocal of 0.5 is 2 / 1 = 2.
2. 2 Solve the problem using division. If you can't write a decimal as a fraction, calculate the reciprocal by solving the problem by dividing: 1 ÷ (decimal). You can use a calculator to solve it, or skip to the next step if you want to calculate the value manually.
   • For example, the reciprocal of 0.4 is calculated as 1 ÷ 0.4.
3. 3 Change the expression to work with integers. The first step in decimal division is moving the positional point until all numbers in the expression are integers. Because you move the positional comma the same number of places in both the dividend and the divisor, you get the correct answer.
4. 4 For example, you take the expression 1 ÷ 0.4 and write it as 10 ÷ 4. In this case, you've moved the comma one place to the right, which is the same as multiplying each number by ten.
5. 5 Solve the problem by dividing the numbers by a column. Using division by a column, you can calculate the reciprocal of a number. If you divide 10 by 4, you should get 2.5, which is the reciprocal of 0.4.

• The value of a negative reciprocal will be the reciprocal of the number multiplied by -1. For example, the negative reciprocal of 3/4 is -4/3.
• The reciprocal of a number is sometimes referred to as the "reciprocal" or "reciprocal".
• The number 1 is its own reciprocal because 1 ÷ 1 = 1.
• Zero has no reciprocal because the expression 1 ÷ 0 has no solutions.

A pair of numbers whose product is equal to one is called mutually inverse.

Examples: 5 and 1/5, -6/7 and -7/6, and

For any number a not equal to zero, there is an inverse 1/a.

The reciprocal of zero is infinity.

Inverse fractions- these are two fractions, the product of which is 1. For example, 3/7 and 7/3; 5/8 and 8/5 etc.

see also

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See what "Reverse number" is in other dictionaries:

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A number whose product times a given number is equal to one. Two such numbers are called reciprocals. These are, for example, 5 and 1/5, 2/3 and 3/2, etc. * * * REVERSE NUMBER REVERSE NUMBER, a number whose product times a given number is ... encyclopedic Dictionary

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We give a definition and give examples of reciprocal numbers. Consider how to find the reciprocal of a natural number and the reciprocal of an ordinary fraction. In addition, we write down and prove an inequality that reflects the property of the sum of reciprocal numbers.

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Reciprocal numbers. Definition

Definition. Reciprocal numbers

Reciprocal numbers are those numbers whose product gives one.

If a · b = 1 , then we can say that the number a is the reciprocal of the number b , just as the number b is the reciprocal of the number a .

The simplest example of reciprocal numbers is two ones. Indeed, 1 1 = 1, so a = 1 and b = 1 are mutually inverse numbers. Another example is the numbers 3 and 1 3 , - 2 3 and - 3 2 , 6 13 and 13 6 , log 3 17 and log 17 3 . The product of any pair of the above numbers is equal to one. If this condition is not met, as for example with the numbers 2 and 2 3 , then the numbers are not mutually inverse.

The definition of reciprocal numbers is valid for any numbers - natural, integer, real and complex.

How to find the reciprocal of a given number

Let's consider the general case. If the original number is equal to a , then its reciprocal number will be written as 1 a , or a - 1 . Indeed, a · 1 a = a · a - 1 = 1 .

For natural numbers and common fractions, finding the reciprocal is fairly easy. One might even say it's obvious. In the case of finding a number that is the inverse of an irrational or complex number, a number of calculations will have to be made.

Consider the most common cases in practice of finding the reciprocal.

The reciprocal of a common fraction

Obviously, the reciprocal of the common fraction a b is the fraction b a . So to find inverse fraction number, the fraction just needs to be flipped. That is, swap the numerator and denominator.

According to this rule, you can write the reciprocal of any ordinary fraction almost immediately. So, for the fraction 28 57, the reciprocal will be the fraction 57 28, and for the fraction 789 256 - the number 256 789.

The reciprocal of a natural number

You can find the reciprocal of any natural number in the same way as the reciprocal of a fraction. It is enough to represent a natural number a as an ordinary fraction a 1 . Then its reciprocal will be 1 a . For natural number 3 has a reciprocal of 1 3 , for 666 the reciprocal is 1 666 , and so on.

Particular attention should be paid to the unit, since it is singular, whose reciprocal is equal to itself.

There are no other pairs of reciprocal numbers where both components are equal.

The reciprocal of a mixed number

The mixed number is of the form a b c . To find its reciprocal, you need mixed number present an improper fraction in the side, and choose the reciprocal for the resulting fraction.

For example, let's find the reciprocal of 7 2 5 . First, let's represent 7 2 5 as an improper fraction: 7 2 5 = 7 5 + 2 5 = 37 5 .

For the improper fraction 37 5 the reciprocal is 5 37 .

The reciprocal of a decimal

A decimal fraction can also be represented as a common fraction. Finding the reciprocal of a decimal fraction of a number comes down to representing the decimal fraction as a common fraction and finding the reciprocal of it.

For example, there is a fraction 5, 128. Let's find its reciprocal. First, we convert the decimal to a common fraction: 5, 128 = 5 128 1000 = 5 32 250 = 5 16 125 = 641 125. For the resulting fraction, the reciprocal will be the fraction 125641.

Let's consider one more example.

Example. Finding the reciprocal of a decimal

Find the reciprocal of the periodic decimal fraction 2 , (18) .

Convert decimal to ordinary:

2, 18 = 2 + 18 10 - 2 + 18 10 - 4 + . . . = 2 + 18 10 - 2 1 - 10 - 2 = 2 + 18 99 = 2 + 2 11 = 24 11

After the translation, we can easily write down the reciprocal of the fraction 24 11. This number will obviously be 11 24 .

For an infinite and non-recurring decimal fraction, the reciprocal is written as a fraction with a unit in the numerator and the fraction itself in the denominator. For example, for the infinite fraction 3 , 6025635789 . . . the reciprocal will be 1 3 , 6025635789 . . . .

Similarly for irrational numbers corresponding to non-periodic infinite fractions, reciprocals are written as fractional expressions.

For example, the reciprocal of π + 3 3 80 is 80 π + 3 3, and for 8 + e 2 + e the reciprocal is 1 8 + e 2 + e.

Reciprocal numbers with roots

If the form of two numbers is different from a and 1 a , then it is not always easy to determine whether the numbers are mutually inverse. This is especially true for numbers that have a root sign in their notation, since it is usually customary to get rid of the root in the denominator.

Let's turn to practice.

Let's answer the question: are the numbers 4 - 2 3 and 1 + 3 2 reciprocal.

To find out if the numbers are mutually inverse, we calculate their product.

4 - 2 3 1 + 3 2 = 4 - 2 3 + 2 3 - 3 = 1

The product is equal to one, which means that the numbers are mutually inverse.

Let's consider one more example.

Example. Reciprocal numbers with roots

Write down the reciprocal of 5 3 + 1 .

You can immediately write that the reciprocal is equal to the fraction 1 5 3 + 1. However, as we have already said, it is customary to get rid of the root in the denominator. To do this, multiply the numerator and denominator by 25 3 - 5 3 + 1 . We get:

1 5 3 + 1 = 25 3 - 5 3 + 1 5 3 + 1 25 3 - 5 3 + 1 = 25 3 - 5 3 + 1 5 3 3 + 1 3 = 25 3 - 5 3 + 1 6

Reciprocal numbers with powers

Suppose there is a number equal to some power of the number a . In other words, the number a raised to the power n. The reciprocal of a n is a - n . Let's check it out. Indeed: a n a - n = a n 1 1 a n = 1 .

Example. Reciprocal numbers with powers

Find the reciprocal of 5 - 3 + 4 .

According to the above, the desired number is 5 - - 3 + 4 = 5 3 - 4

Reciprocals with logarithms

For the logarithm of the number a to the base b, the reciprocal is the number equal to the logarithm of the number b to the base a.

log a b and log b a are mutually reciprocal numbers.

Let's check it out. It follows from the properties of the logarithm that log a b = 1 log b a , which means log a b · log b a .

Example. Reciprocals with logarithms

Find the reciprocal of log 3 5 - 2 3 .

The reciprocal of the logarithm of 3 to base 3 5 - 2 is the logarithm of 3 5 - 2 to base 3.

The reciprocal of a complex number

As noted earlier, the definition of reciprocal numbers is valid not only for real numbers, but also for complex ones.

Usually complex numbers are represented in algebraic form z = x + i y . The reciprocal of this will be a fraction

1 x + i y . For convenience, this expression can be shortened by multiplying the numerator and denominator by x - i y .

Example. The reciprocal of a complex number

Let there be a complex number z = 4 + i . Let's find the reciprocal of it.

The reciprocal of z = 4 + i will be equal to 1 4 + i .

Multiply the numerator and denominator by 4 - i and get:

1 4 + i \u003d 4 - i 4 + i 4 - i \u003d 4 - i 4 2 - i 2 \u003d 4 - i 16 - (- 1) \u003d 4 - i 17.

In addition to the algebraic form, a complex number can be represented in trigonometric or exponential form as follows:

z = r cos φ + i sin φ

z = r e i φ

Accordingly, the reciprocal number will look like:

1 r cos (- φ) + i sin (- φ)

Let's make sure of this:

r cos φ + i sin φ 1 r cos (- φ) + i sin (- φ) = r r cos 2 φ + sin 2 φ = 1 r e i φ 1 r e i (- φ) = r r e 0 = 1

Consider examples with the representation of complex numbers in trigonometric and exponential form.

Find the inverse of 2 3 cos π 6 + i · sin π 6 .

Considering that r = 2 3 , φ = π 6 , we write the reciprocal number

3 2 cos - π 6 + i sin - π 6

Example. Find the reciprocal of a complex number

What is the inverse of 2 · e i · - 2 π 5 .

Answer: 1 2 e i 2 π 5

The sum of reciprocal numbers. Inequality

There is a theorem on the sum of two reciprocal numbers.

Sum of mutually reciprocal numbers

The sum of two positive and reciprocal numbers is always greater than or equal to 2.

We present the proof of the theorem. As is known, for any positive numbers a and b the arithmetic mean is greater than or equal to the geometric mean. This can be written as an inequality:

a + b 2 ≥ a b

If instead of the number b we take the inverse of a , the inequality takes the form:

a + 1 a 2 ≥ a 1 a a + 1 a ≥ 2

Q.E.D.

Let's bring practical example illustrating this property.

Example. Find the sum of reciprocal numbers

Let's calculate the sum of the numbers 2 3 and its reciprocal.

2 3 + 3 2 = 4 + 9 6 = 13 6 = 2 1 6

As the theorem says, the resulting number is greater than two.

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