What is the reciprocal of 1 9

What is the reciprocal?

......The reciprocal value is known from fractions. It is the fraction where the numerator and denominator are reversed.
So one can reduce dividing to multiplying.
You divide by a fraction by multiplying by the reciprocal of the fraction.

......Every real number can be written as a fraction with the denominator 1: x = x / 1. Then the reciprocal is 1 / x.
You can assign a reciprocal value (or an inverse number, reciprocal number, inverse number) to every real number up to 0.
The exception is therefore the number zero, because 1/0 is not defined.
The arrow describes the function with k (x) = 1 / x.

Reciprocal of a decimal number
......The reciprocal of a decimal number can easily be determined with the calculator.
Enter the decimal number and press the 1 / x key.

If you press the key many times, the inverse number and decimal number follow one another.
The reciprocal of the reciprocal number is again the number.
 


Make denominators rational
The reciprocal values ​​of roots should also be mentioned.
For example, it is common not to leave the reciprocal of a number like sqrt (2), namely 1 / sqrt (2), but to remove the root in the denominator. You can do this by expanding the fraction with sqrt (2). Then 1 / sqrt (2) = sqrt (2) / 2. This makes the number clearer and leads to the usual representation of an irrational number ..
This calculation is called "making the denominator rational".

Opposite number
The reciprocal value is called "Multiplicative inverse" in English. There is also the term "additive inverse", and that is the opposite number in German. As for the multiplication x * (1 / x) = 1, so for the addition x + (- x) = 0. The number -x is the opposite number. Zero plays the same role in addition as one does in multiplication. In both cases the ones and zeros are "neutral elements". If you combine them with any real number, they do not change the number: x * 1 = x and x + 0 = x.

On this website I have put together a few things about the reciprocal value that I found interesting.

Graphic representations Top

......
The output function is f (x) = x.
The graph is the first bisector.

The graph for the function with reciprocal value k (x) = 1 / x is a right-angled hyperbola.

The reciprocal of the reciprocal value is also shown and that is again the first bisector.
This is because 1 / (1 / x) = 1 * (x / 1) = x. The corresponding functional equation is f (x) = x.

This property is called involutorial, and it occurs in many images.
For example, the point reflection and the axis reflection of figures are involutorial.

 


Derivatives
...The output function is the function with the functional equation k (x) = 1 / x or k (x) = x-1.

The derivatives are
k '(x) = - x-2 = -1 / x2
k '' (x) = 2x-3 = 2 / x3
k '' '(x) = - (2 * 3) x-4 = - (2 * 3) / x4
...

k(n)(x) = [(-1)nn!] x-n = [(-1)- (n + 1)n!] / xn.


Indefinite integral
......The logarithmic function can be differentiated as the inverse function of the exponential function.
Claim: The derivative of the function with g (x) = ln (x) is g '(x) = 1 / x.
Derivation
From y = ln (x) it follows that x = ey
The general rule for the derivative is g '(x) = 1 / f' (y), in this case [ln (x)] '= 1 / (ey) = 1 / x, wzbw ..
The rule g '(x) = 1 / f' (y) applies to the Reciprocal of the differential quotient dx / dy, namely dy / dx = 1 / (dx / dy), and fits this website.
The function f (x) = 1 / x has the antiderivative g (x) = ln (| x |) + C. The dashes take into account that the term 1 / x is also defined for x <0.

Reciprocal of function terms Top
A new function arises from a function with the term f (x) if the reciprocal value r (x) = 1 / f (x) is formed.
Her name is that reciprocal function the output function.


Inverse function
In addition to the reciprocal function, there is also the inverse function for a function. This is a function in which an x-value is uniquely assigned to each y-value. With the function, there is exactly one y-value for each x-value.
The inverse function can easily be confused with the reciprocal function, not only because of the similar name, but also because of the spelling of the function term. It is better not to write 1 / f (x) than f-1(x). The term f-1 namely describes the inverse function of f.
For f-1(x) one should therefore, if at all, [f (x)]-1 write.

Reciprocal function
There follows a series of graphs of the function and the reciprocal function.

f (x) = sin (x), r (x) = csc (x)

f (x) = cos (x), r (x) = sec (x)

f (x) = tan (x), r (x) = cot (x)
The six trigonometric functions are split into three pairs of function and reciprocal function.
These are sines and cosecans, cosines and secans, tangents and cotangents.




f (x) = x (x-1) (x-2) +1, r (x) = 1 / [x (x-1) (x-2) +1]
If one compares the graphs of f (x) and r (x), a number of statements are possible.
The graph opposite shows all relationships in the examples above.

(1) If f (x) = 1, then also r (x) = 1.

(2) If f (x) tends towards infinity, then r (x) tends towards 0.

(3) If f (x1) = 0, then r (x) has at x1 a pole.

(4) Has f (x) at position x2 a minimum, then r (x) has a maximum there.
(4 ') Has f (x) at the point x3 is a maximum, then r (x) has a minimum there.
 


Justifications
The statements (1), (2) and (3) are easy to see.

To prove the statements (4) and (4 ') you have to do some calculations.
According to the chain rule r '= (1 / f)' = (f-1) '= - (f-2) f '= -f' / f² and according to the quotient rule r '' = - (f '' f²-2f f '²) / f4= (-f "f + 2f" ²) / f³.

Apply at the point x2 the statements f '(x) = 0 and f' '(x)> 0, then there is a minimum.
This property also has the reciprocal function r (x) except for a sign:
The following applies: r '(x) = -f' (x) / f² (x) = 0 and r '' (x) = [-f '' (x) f (x) + 2f '² (x)] / [f (x)] ³ = [-f "(x) / f (x)] <0.
Result: Has f (x) at position x2 a minimum, then r (x) has a maximum there.


(4 ') Is at the point x3 a maximum, then r (x) assumes a minimum there.
The proof is analogous.

Sum of reciprocal values Top
Harmonious mean
The simplest sum of reciprocal values ​​is 1 / x + 1 / y.
If one forms the arithmetic mean value for two reciprocal values, the result is (1/2) (1 / x + 1 / y) = (1/2) (x + y) / (xy).
This is also the reciprocal of the harmonic mean h = (2xy) / (x + y).


Crossing routes
......You enter a rectangular pot like on the left and mark the lines a and b on the walls. You connect the free end points of the lines with the opposite corners of the pot. The height h 'of the intersection of the two connecting lines above the bottom of the pot is half the harmonic mean h of a and b.
Proof:
Let it be w1+ w2= w.
According to the ray theorem, w applies to the right2: w = h ': b or w2= h'w / b and left w1: w = h ': a or w1= h'w / a.
From this it follows with w1+ w2= w the equation w = h'w / b + h'w / a or 1 = h '(1 / b + 1 / a) or h' = ab / (a ​​+ b) = (1/2) h, wzbw ..

The figure also represents the well-known ladder task of the crossing ladders.

heptagon
......
According to Ptolemy's theorem, the side a and the diagonals d apply1 and d2 of the green square in the heptagon is the formula d1d2= ad1+ ad2 . From this it follows that a = (i.e.1d2) / (d1+ d2) or 1 / a = 1 / d1+ 1 / d2

Harmonic triangle
......One can develop a harmonic triangle of inverses from Pascal's triangle.

The special thing is that in the harmonic triangle each number is the sum of the two underneath lying numbers is.


Series of reciprocals
Harmonic sequence and harmonic series
The sequence of reciprocal values ​​of the natural numbers is called the harmonic sequence.

1, 1/2, 1/3, 1/4, ... (general term 1 / n)
For n towards infinity, the sequence tends towards zero.
The reciprocal values ​​of the natural numbers are called parent fractions in fractions.



The harmonic series emerges from the harmonic sequence. It is the sequence of the partial sums of the harmonic sequence.
1, 1 + 1/2, 1 + 1/2 + 1/3, 1 + 1/2 + 1/3 + 1/4, ... (general term 1 + 1/2 + 1/3 + 1 / 4+ ... + 1 / n)
For n towards infinity, the series tends towards infinity.

Alternating harmonic series
The alternating harmonic series emerges from the harmonic series. The sign of the summands changes successively.
1, 1-1 / 2, 1-1 / 2 + 1/3, 1-1 / 2 + 1 / 3-1 / 4, ... [general term 1-1 / 2 + 1 / 3-1 / 4+ ... + (- 1)n + 1(1 / n)]
For n towards infinity, the series tends towards the limit value ln (2).

Leibniz series
In the alternating harmonic series, the odd numbers are used instead of the natural numbers.
1, 1-1 / 3, 1-1 / 3 + 1/5, 1-1 / 3 + 1 / 5-1 / 7, [general term 1-1 / 3 + 1 / 5-1 / 7 +. .. + (- 1)n + 1(1 / (2n-1))]
For n towards infinity, the series tends towards the limit value (1/4) pi.

Series with the reciprocal values ​​of the square numbers
In the harmonic series, square numbers are used instead of natural numbers.
1, 1 + 1 / 2², 1 + 1 / 2² + 1 / 3², 1 + 1 / 2² + 1 / 3² + 1 / 4², ... (general term 1 + 1 / 2² + 1 / 3² + 1 / 4² + ... + 1 / n²)
For n towards infinity the series tends towards the limit value (1/6) pi².

Kempner series
The Kempner series emerge from the harmonious series. Remove all numbers that contain the number 1 in the denominator.
The row 1 + 1/2 + 1/3 + 1/4 + 1/5 + 1/6 + 1/7 + 1/8, + 1/9 + 1/10 + 1/11 ... then becomes 1 + 1/2 + 1/3 + 1/4 + 1/5 + 1/6 + 1/7 + 1/8, + 1/9 + 1/20 + 1/22 + ...
Because of the 10 digits, there are 10 Kempner rows. The special thing is that all Kempner series are convergent.
Found at http://mathworld.wolfram.com (url below).

Reciprocal values ​​for sizes Top
A quantity generally determines a property of a physical object. It is indicated by a measure and a unit.
Examples of quantities with units are volume (cm³), distance (m), time (hours), speed (km / h), electrical resistance (ohms), ...
The following are examples of quantities for which reciprocal values ​​are also known.


Reciprocal of a quantity
Oscillation duration and frequency
A characteristic variable of the oscillation is the oscillation period T. The reciprocal value, the frequency f, is also in use. Both quantities are linked by the equation T * f = 1 or f = 1 / T. It makes sense to use the period of oscillation for slow oscillations.
> For example, a thread pendulum with a length of 1 m has the oscillation period T = 0.68 s.
> A tuning fork with the oscillation period T = 0.00227272727 s is better identified with the reciprocal value, namely f = 440 1 / s = 440 Hz.

I am familiar with two other pairs of values, namely the electrical resistance R and the electrical conductance G = 1 / R or, in the case of glasses, the focal length f and the refractive power D = 1 / f, measured in diopters.

Proportional sizes
Two quantities A and B are proportional if their quotient A / B is constant.
There are examples where the reciprocal of the proportionality factor also has a meaning.
Gasoline consumption
The distance traveled by a car and the amount of gasoline used are proportional on average. The proportionality factor is the fuel consumption.
It is customary for us to state the gasoline consumption of a car, e.g. 8.5 l. More precisely this is the quotient 8.5 l / 100 km. This means that the car consumes an average of 8.5 liters of petrol for 100 km.
In Great Britain or the USA, gasoline consumption is given by the reciprocal value. You give the distance in miles that one gallon can travel.
For a comparison with 8.5 l / 100 km you have to know that 1km = 0.6214 miles and 1 gallon = 3.7854 liters or 1 liter = 0.2642 gallons.
Then the reciprocal value (100 km) / (8.5 l) = (100 * 0.6214 miles) / (8.5 * 0.2642 gallons) = (27 miles) / (1 gallon) = 27 mpg.

To add to the confusion: In Great Britain, gasoline consumption is also given in the unit mpg, but 1 gallon = 4.546 liters rather than 1 gallon = 3.7854 liters. A petrol consumption of 8.5 l would then correspond to the larger value of 32.10 mpg.

In English, the name gasoline consumption is actually wrong because it is the reciprocal of gasoline consumption. One also speaks of mileage and i.e. mileage or runtime.


Conversion rate
There are two ways of specifying the conversion rate between the euro and pound sterling,
1 euro = 0.780 pounds or 1 pound = 1.283 euros (October 2, 2014).
Obviously it is a question of a number and its inverse number. Which information is preferred depends on the point of view. We Germans prefer the first entry.
speed
The speed v indicates a uniform movement. It is the proportionality factor v = s / t to the proportionality s ~ t. Where s is the distance traveled and t is the time that has passed.
In principle, any reciprocal value of a proportionality factor can be formed. The only question is whether it makes sense and whether it is also used "in everyday life". Here is an example.

If the reciprocal value t / s is formed, it is a measure of the "slowness", because with a constant path the quotient is greater the more time passes. - One could perhaps express the movement of a snail in terms of t / s, for example in terms of the time it takes for one meter. However, I found, for example, a speed specification on the corresponding Wikipedia page: The snail moves at an average of 7cm / min at the "snail's pace". By choosing suitable units, the dimension figure is single-digit and therefore clear.
(The above path of 1m leads to a time of 14.3 min.)


I found that when Americans drive long distances, most of the time they report distances in hours, also because there are speed limits on the highways. That is more appropriate to her than miles. - We Germans prefer kilometers. We only give hours when we want to demonstrate how fast we were again or how long we were stuck in a traffic jam.

Anti-proportional sizes
Two quantities A and B are inversely proportional if their product AB is constant. The graph is a right-angled hyperbola.
There is an analogy to number and inverse number. For them the product is 1 and the graph is the special hyperbola f (x) = 1 / x.

Sum of reciprocal values
In school physics there are equations with sums of reciprocal values ​​of quantities. A selection follows.
Equivalent resistance of resistors connected in parallel
......If two resistors are connected in parallel, they have the same resistance as the "equivalent resistance" R = (R1R.2) / (R1+ R2).

Replacement capacitance of capacitors connected in series
... ...If two capacitors are connected in series, they have the same capacitance as the "replacement capacitor":
C = (C1C.2) / (C1+ C2).

Substitute inductance of coils connected in parallel
......If two coils are connected in parallel, they have the same inductivity as the "replacement coil":
L = (L1L.2) / (L1+ L2).

Lens formula
......For thin, convex lenses, the lens formula 1 / f = 1 / g + 1 / b applies.

Lens system
..... Two thin converging lenses with focal lengths f1 and f2 that are close to each other act like a converging lens with the focal length f = (f1f2) / (f1+ f2) or expressed by the refractive power D = D1+ D2.

Reciprocal value on the internet Top

German

maths [at]
A small task

Wikipedia
Reciprocal value, reciprocal proportionality, harmonic sequence, harmonic series,
Involution (math), inverse element, group (math)

English

Eric W. Weisstein (MathWorld)
Harmonic Series, Kemper series

Wikipedia
Multiplicative inverse, Inverse proportionality, Harmonic series (mathematics),
Involution (mathematics), Inverse element, Group (mathematics)


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© October 2014 Jürgen Köller

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