Numbers are reciprocals of data. Reciprocal numbers, finding the reciprocal of a number

Let's give a definition and give examples of reciprocal numbers. Let's look at how to find the inverse of a natural number and the inverse of a common 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 numbers whose product equals one.

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

The simplest example of reciprocal numbers is two units. Indeed, 1 · 1 = 1, therefore 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 numbers above is equal to one. If this condition is not met, as for example for the numbers 2 and 2 3, then the numbers are not mutually inverse.

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

How to find the inverse of a given number

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

For natural numbers and ordinary fractions finding the reciprocal number is quite simple. One might even say it's obvious. If you find a number that is the inverse of an irrational or complex number, you will have to make a series of calculations.

Let's consider the most common cases of finding the reciprocal number in practice.

The reciprocal of a common fraction

Obviously, the reciprocal of the common fraction a b is the fraction b a. So to find reciprocal fraction number, fraction you just need to turn over. 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 number 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 inverse of any natural number in the same way as finding the inverse of a fraction. It is enough to represent the natural number a in the form of an ordinary fraction a 1. Then its inverse number will be the number 1 a. For natural number 3 its reciprocal is the fraction 1 3, for the number 666 the reciprocal is 1 666, and so on.

Special attention should be paid to one, since it is the only number 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 looks like a b c. To find its inverse number, you need mixed number present in the side improper fraction, and select the reciprocal number for the resulting fraction.

For example, let's find the reciprocal number for 7 2 5. First, let's imagine 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.

Reciprocal of a decimal

A decimal can also be represented as a fraction. Finding the inverse decimal numbers comes down to representing a decimal as a fraction and finding its reciprocal.

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

Let's look at another example.

Example. Finding the reciprocal of a decimal

Let's find the reciprocal number for the periodic decimal fraction 2, (18).

Converting a decimal fraction to an ordinary fraction:

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 the reciprocal number for the fraction 24 11. This number will obviously be 11 24.

For an infinite and non-periodic decimal fraction, the reciprocal number 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 number will be 1 3, 6025635789. . . .

Similarly, for irrational numbers corresponding to non-periodic infinite fractions, the reciprocal numbers are written in the form of fractional expressions.

For example, the reciprocal for π + 3 3 80 will be 80 π + 3 3, and for the number 8 + e 2 + e the reciprocal will be the fraction 1 8 + e 2 + e.

Reciprocal numbers with roots

If the type of two numbers is different from a and 1 a, then it is not always easy to determine whether the numbers are reciprocals. 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 whether the numbers are reciprocals, let's calculate their product.

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

The product is equal to one, which means the numbers are reciprocal.

Let's look at another example.

Example. Reciprocal numbers with roots

Write down the reciprocal of 5 3 + 1.

We can immediately write that the reciprocal number 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

Let's say 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 the number a n is the number a - n . Let's check it out. Indeed: a n · a - n = a n 1 · 1 a n = 1 .

Example. Reciprocal numbers with powers

Let's find the reciprocal number for 5 - 3 + 4.

According to what was written above, the required number is 5 - - 3 + 4 = 5 3 - 4

Reciprocal numbers with logarithms

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

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

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

Example. Reciprocal numbers 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 inverse of a complex number

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

Complex numbers are usually represented in algebraic form z = x + i y. The reciprocal of the given number is a fraction

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

Example. The inverse of a complex number

Let there be a complex number z = 4 + i. Let's find the inverse 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 = 4 - i 4 + i 4 - i = 4 - i 4 2 - i 2 = 4 - i 16 - (- 1) = 4 - i 17 .

In addition to 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 inverse 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

Let's consider examples with the representation of complex numbers in trigonometric and exponential form.

Let's find the inverse number for 2 3 cos π 6 + i · sin π 6 .

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

3 2 cos - π 6 + i sin - π 6

Example. Find the inverse of a complex number

What number will be the reciprocal of 2 · e i · - 2 π 5 .

Answer: 1 2 e i 2 π 5

Sum of reciprocal numbers. Inequality

There is a theorem about the sum of two mutually inverse numbers.

Sum of reciprocal numbers

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

Let us give a 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 will take the form:

a + 1 a 2 ≥ a 1 a a + 1 a ≥ 2

Q.E.D.

Let's give practical example, illustrating this property.

Example. Find the sum of reciprocal numbers

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

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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Material from Wikipedia - the free encyclopedia

Reverse number(reciprocal value, reciprocal value) to a given number x is a 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, $\backslash frac(1)(\backslash cos(x))$ differs from the value of the function inverse to cosine - arc cosine, which is denoted $\backslash cos^(-1)x$ or $\backslash arccos\; x$.

Reverse 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	$r(\backslash cos\backslash varphi+i\; \backslash sin\backslash varphi)$	$\backslash frac(1)(r)(\backslash cos\backslash varphi-i\; \backslash sin\backslash varphi)$
Indicative	$re^(i\backslash varphi)$	$\backslash frac(1)(r)e^(-i\; \backslash varphi)$

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:

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

• Demonstrative 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 $\backslash frac(1)(2)\; \backslash left\; (\backslash frac(1)(2)-i\backslash frac(\backslash sqrt(3))(2)\; \backslash right)$
Indicative	$2\; e^(i\; \backslash frac(\backslash pi)(3))$	$\backslash frac(1)(2)\; e^(-i\; \backslash frac(\backslash pi)(3))$

Inverse to 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__ $i^(-1)=-i$

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

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Excerpt characterizing the Reverse Number

This is what the stories say, and all this is completely unfair, as anyone who wants to delve into the essence of the matter can easily see.
The Russians could not find a better position; but, on the contrary, in their retreat they passed through many positions that were better than Borodino. They did not settle on 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 people’s 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 previous positions were stronger and that the Borodino position (the one on which the battle was fought) is not only not strong, but for some reason is not at all a position more than any other place in Russian Empire, which, when guessing, would be indicated with a pin on the map.
The Russians not only did not strengthen the position of the Borodino field to the left at right angles to 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 at this place. This is evidenced, firstly, by the fact that not only on the 25th there were no fortifications at this place, but that, begun on the 25th, they were not finished even on the 26th; secondly, the proof is the position of the Shevardinsky redoubt: the Shevardinsky redoubt, ahead of the position at which the battle was decided, 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, all efforts were exhausted and six thousand people were lost? To observe the enemy, a Cossack patrol was enough. Thirdly, proof that the position in which the battle took place was not foreseen and that the Shevardinsky redoubt was not the forward point of this position is the fact 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 in the heat of the moment after the battle, calls the Shevardinsky redoubt the left flank of the position. Much later, when reports about the Battle of Borodino were being 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 a forward post (while it was only a fortified point of the left flank) and as if battle of Borodino was accepted by us in a fortified and pre-chosen position, whereas it happened in a completely unexpected and almost unfortified place.
The thing, obviously, was like this: the position was chosen along the Kolocha River, which crosses the main road not at a right angle, but at an acute angle, so that the left flank was in Shevardin, the right near the village of Novy and the center in Borodino, at the confluence of the Kolocha and Vo rivers yn. This position, under the cover of the Kolocha River, for an 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, having gone to Valuev on the 24th, did not see (as they say in the stories) the position of the Russians from Utitsa to Borodin (he could not see this position, because it did not exist) and did not see the forward post of the Russian army, but stumbled upon the Russian rearguard in pursuit to the left flank of the Russian position, to the Shevardinsky redoubt, and, unexpectedly for the Russians, transferred troops through Kolocha. And the Russians, not having time to engage in a general battle, retreated with their left wing from the position they intended to occupy and took new position, which was not foreseen and not strengthened. By going to left side Kolochi, to the left of the road, Napoleon moved the entire future battle from right to left (from the Russian side) and transferred it to the field between Utitsa, Semenovsky and Borodin (to this field, which has nothing more advantageous for the position than any other field in Russia ), and on this field the entire 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 an attack on the redoubt immediately in the evening, but had launched an 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 take place as we expected. In this case, we would probably defend the Shevardinsky redoubt, our left flank, even more stubbornly; Napoleon would have been attacked in the center or on the right, and on the 24th a general battle would have taken place 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 begin a general battle on the same evening of the 24th, Borodinsky’s first and main action The battle was lost on the 24th and, obviously, led to the loss of the one fought 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, but the disadvantage of this situation was increased by the fact that the Russian military leaders did not recognize the completely accomplished fact (the loss of 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 during the battle from right to left. Thus, throughout the entire battle, the Russians had against all French army, aimed at our left wing, twice the weaker forces. (Poniatowski’s actions against Utitsa and Uvarov on the French right flank were actions separate from the course of the battle.)
So, the Battle of Borodino did not happen at all as they describe it (trying to hide the mistakes of our military leaders and, as a result, diminishing the glory of the Russian army and people). The Battle of Borodino did not take place in a chosen and fortified position with forces that were somewhat weaker on the Russian side, but the Battle of Borodino, due to the loss of the Shevardinsky redoubt, was accepted by the Russians in an open, almost unfortified area with forces twice as weak against the French, that is, in 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 morning of the 25th, 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 a service was going on and the gospel was being preached, Pierre got out of the carriage and went on foot. Behind him, some cavalry regiment with singers in front was descending onto the mountain. A train of carts with those wounded in yesterday's case was rising towards him. The peasant drivers, shouting at the horses and lashing them with whips, ran from one side to the other. The carts, on which three or four wounded soldiers lay and sat, jumped over the stones thrown in the form of a pavement on a steep slope. The wounded, tied with rags, pale, with pursed lips and frowning brows, holding onto the beds, jumped and pushed in the carts. Everyone looked at Pierre's white hat and green tailcoat with almost naive childish curiosity.

Reciprocal - or mutually reciprocal - numbers are a pair of numbers that, when multiplied, give 1. In fact general view the reciprocals are numbers. A characteristic special case of reciprocal numbers is a pair. The inverses are, say, numbers; .

How to find the reciprocal of a number

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

Example No. 1.

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

When looking for the reciprocal number for a common fraction, dividing it by 1 is not very convenient, because the recording is cumbersome. In this case, it is much easier to do things differently: the fraction is simply turned over, swapping the numerator and denominator. If given proper fraction, then after turning over the resulting fraction is improper, i.e. one from which a whole part can be isolated. Whether to do this or not must be decided 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 isolating the whole part is desirable.

Example No. 2.

Given a fraction. Reverse to it: .

If you need to find the reciprocal of a decimal fraction, you should use the first rule (dividing 1 by the number). In this situation, you can act in one of 2 ways. The first is to simply divide 1 by that number into a column. The second is to form a fraction from a 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 a 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, select an entire part from it, or convert it to decimal form.

Example No. 3.

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

How to check if two numbers are reciprocals

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

Example No. 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, let's present these numbers in the form of ordinary fractions: (reduce the 1st fraction by 125). Conclusion: the numbers 0.125 and 8 are reciprocals.

Properties of reciprocal numbers

Property No. 1

A reciprocal exists for any number except 0.

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

Property No. 2

The sum of a pair of reciprocal numbers is always no less than 2.

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

Property No. 3

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

Example No. 5.

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

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.

Reverse fractions- these are two fractions whose product is equal to 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:

A number whose product by a given number equals one. Two such numbers are called reciprocals. These are, for example, 5 and 1/5, 2/3 and 3/2, etc... Big Encyclopedic Dictionary

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A number whose product by a given number equals 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 by a given number is equal to ... ... encyclopedic Dictionary

A number whose product with a given number equals one. Two such numbers are called reciprocals. These are, for example, 5 and a, not equal to zero, the opposite exists... Great Soviet Encyclopedia

A number whose product by a given number is equal to one. Two such numbers are called. mutually inverse. These are, for example, 5 and 1/5. 2/3 and 3/2 etc... Natural science. encyclopedic Dictionary

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Content:

Reciprocal numbers are needed for solving all types algebraic equations. For example, if you need to split one a fractional number to another, you multiply the first number by the reciprocal of the second. In addition, reciprocal numbers 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 fraction by reversing 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 of a fraction is another fractional number that can be calculated simply by “reversing” the fraction (switching the places of the numerator and denominator).
   • For example, the reciprocal of the fraction 3/4 is 4 / 3 .
2. 2 Write the reciprocal of a whole number as a fraction. And in this case, the reciprocal number is calculated as 1 ÷ (the original number). For an integer, write the reciprocal as common fraction, there is no need to perform calculations and write it as a decimal fraction.
   • For example, the reciprocal of 2 is 1 ÷ 2 = 1 / 2 .

2 Finding the reciprocal of a mixed fraction

1. 1 What is a "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 carried out in two steps, described below.
2. 2 Write the mixed fraction as an improper fraction. You, of course, remember that a unit can be written as (number)/(same number), and fractions with the same denominator (the number under the line) can be added to each other. Here's how to do it 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 Reverse the fraction. When a mixed fraction is written as an improper fraction, we can easily find the reciprocal simply by swapping the numerator and denominator.
   • For the example above, the reciprocal number would be 14 / 5 - 5 / 14 .

3 Finding the reciprocal of a decimal fraction

1. 1 If possible, express the decimal as a fraction. You need to know that many decimals can be easily converted to fractions. For example, 0.5 = 1/2, and 0.25 = 1/4. Once you have written a number as a simple fraction, you can easily find its reciprocal simply by flipping the fraction over.
   • For example, the reciprocal of 0.5 is 2 / 1 = 2.
2. 2 Solve the problem using division. If you cannot write a decimal as a fraction, calculate the reciprocal by solving the problem by division: 1 ÷ (decimal). You can use a calculator to solve this or go 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 dividing a decimal is to move the decimal point until all the numbers in the expression are integers. Because you move the decimal place the same number of places in both the dividend and 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 have moved the decimal place one place to the right, which is the same as multiplying each number by ten.
5. 5 Solve the problem by dividing the numbers into a column. Using long division you can calculate the reciprocal 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 number will be equal to the reciprocal number multiplied by -1. For example, the negative reciprocal of 3/4 is - 4/3.
• The reciprocal of a number is sometimes called 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.