(R. P. of San Luis Obispo, CA.2001-01-23) (M. M. of Gresham, OR.2001-02-11) Why is zero factorial equal to one?
When n is a positive integer, the quantity n! (pronounced "n factorial" or "factorial n") isdefined as the product of the n integers from 1 to n. The major reason why 0! equals 1 is that it'sjust a product of 0 factors. Such an empty product must be equal to 1, just like a sum of zero terms (an empty sum) must be equal to 0. Let's explain:
The product of (n+1) factors is clearly equal to the product of the first n factorsmultiplied by the last one. This is "clear" to everybody when n is 2 or more. To make this work for n = 1, we have to state that a "product" consistingof a single factor is equal to that factor. It follows (for n = 0) that a product of zero factors multiplied byany number x must be equal to said number x. Therefore, the product of zero factors must be equal to 1. (The same reasoning for sums leads to the conclusion that a sum of zero terms is equal to 0,which is less shocking to most people than the corresponding result for empty products.)
Defining n! as a product of n factors (1,2, ... n) when n is nonzero thus implies that the only consistent definition of 0! is 0! = 1.
Another advanced argument is to define factorials in term of the analyticGamma function ( ) whose properties also imply a value 0! = 1.
If you multiply xn by x, you obtain xn+1.So, the product of x0 and x is x [= x1].If x isnonzero, x0 must therefore be equal to 1. Furthermore :
This seems to bother some people offhand (including a fewtextbook authors,who should know better),but x = 0 isnot an exception to the above rule:It's indeed true that 00 = 1 .The most fundamental explanation is that an empty product[the product of no factor(s), which is what a zeroth power is]cannot possibly depend on the value ofany factor sinceyou arenot usingany such factor to form the "product".Thus, the value obtained for the zeroth power of any nonzero xmust also be the correct value when x iszero.In spite of a superficial similarity, this isnot a "continuity argument"(such analytical arguments will not work past the elementary levelwhere exponents areonly integers, because it so happens thatthe two-variable function xcannot be made continuous at the point x = y = 0, as discussed below).Instead, the above argument rests purely on basic logic [set theory and elementary algebra].
I could leave it at that and rest my case, but I know that the abovelogical argumentis often unable to overcome thepsychological reluctance to acceptthe fundamental fact we're discussing here... Although the mathematical case is closed,some people may find it helpful to take a lexicographer's approach and discover thatunity is the value of zero to the power of zero which isimplied in a number offamiliar mathematical contexts. View the following examples only as asupplement to the above fundamental logic,whichillustratesthat the relevant mathematical discourse is always consistent with it(because it's based on it)...
1
1
1
1
2
1
1
3
3
1
1
4
6
4
1
According to thebinomial theorem, (1-1)nis thealternating sum of the coefficientsin a line ofPascal's triangle. The result is zero,except for the top line where it's equal to unity(namely, the only nonzero coefficient in that top line).
The value of the polynomial anxn is a0 at x = 0.
pn is the number of ways to map a set of n elementsinto a set of p elements. There's no such map from a nonempty set to the empty set,but there's one [and only one] from the empty set to itself.
John Baez about the set of the functions from B to A:|AB| = |A| |B|
tr.v.grok·ked,grok·king,groks. Slang. To understand profoundly through intuition or empathy. To assimilate everything about something to the deepest possible extent,becoming as one with the subject of focus. [Strangerin a Strange Land (1961) byRobert AnsonHeinlein ] Do you grok that?
The numerical expression xy is definedonly under one [or both]of the following conditions. (In particular, 0y is not defined unless y is anonnegative integer.)
x is positive (in which case y could be any real orcomplex number).
y is an integer (if y is anegative integer, x must be nonzero).
Lest the reader object that 0yshould be defined asthe limit of xy as x0+,we'll point out that, wheny is the imaginary number i, we obtainexp(i Log(x)),which keeps going around the unit circle,without approachingany limit...
When theexponent (y) in the expressionxy happens to be an integer, thebase(x) can be any number whatsoever(positive, negative or evencomplex)except that it can't be zero when y<0.
In that elementary context, it's clear that algebraic consistency imposesthe zeroth power of any number (positive, negative or complex) to be unity.There's no reason to make an exception for zero that would introduce an arbitraryand needless discontinuity at the origin for the functionf(x) = x0, which is everywhere else equal to 1.As we generalize the notion of exponentiation, this elementary perspective must beretained, unless it is found to be incompatible with the more general framework.
Historically, fractional values of theexponent were introducedin the lateMiddle Ages (c. 1360) by theFrenchmanNicole Oresme (1323-1382). However, the base xmust be positive whenever fractional exponents are not ruled out!
Exponentiation :
xy is unambiguously defined only in two cases for complex numbers:when x is real and positive (it's then equal to exp(y ln(x))and when y is an integer (provided x is nonzero if y is negative). If exponentiation makes any sense at all,it must make sense for the former case of integral exponents[unless a division by zero is involved]and the latter case is needed too because weneed the exp functionand can't possibly discriminate against anypositivebases as soon as we accept the no-so-special base e... Curiously, you can't extend that "domain" of definition (which is not a "domain" in the strict sense of the term) unless you bringRiemann surfaces [or "multi-valued" functions] into the picture,because logarithms cannot becontinuously defined any other wayfor a nonpositive argument...
The nature of the essential discontinuity of the two-variable function f (x,y) = xy about the origin (x = y =0)is probably best grasped by considering the curves where thisquantity is constant, for positive values of x and y near the origin [more precisely,when x is positive and less than 1].
The cartesian equation of such a curve is y =a / ln(1/x) for somenonnegative value of the parameter a;all these curves include the origin in their closure!Along any of them, the value of the function is constant: It's equaltoexp(-a), which can be essentially anything you want between0 (excluded) and 1 (included).This does imply that the two-variable function f doesnot havea limit at the point (0,0), since you have points in any neighborhood of theorigin for which the value of f is as close as you wish fromanychoice of a number in the interval [0,1].
The fact that the function f (x,y) = xy is not continuous at the origin does not make is undefined there... The bottom line is that zero to the power of zero must be defined tobe unity unless we're willing to rephrase many of the theorems we all take for granted(including the binomial theorem mentioned above). No paradox can arise from continuity arguments because sucharguments are simply disallowed
Without any exceptions, when n is a nonnegative integer, xn denotes a product of n factors equal to x. When n is zero, the value of x is thus disregarded and must be irrelevant... The zeroth power of x is defined to be unityin anymonoid,even if the base (x) is notinvertible.
At this point, some people argue that the fact that 0n is zero for any n ought to be an equally valid argumentcontradicting the above... This is just not so! On one hand, fundamental logic does impose that an empty product must be unity (regardless of the values of the factors, since they're not used). On the other hand, a product vanishes when at least oneof its factors is zero. This simply does not apply to a productof zero factors (as there are no factors, none of them is zero).
Furthermore, if we want to keep alive the theorem (valid in anyintegral domain) that a product vanishes if and only if at least one ofits factors vanishes, we see that a product of no factors cannotvanish: Thus, a zeroth power cannot possibly be zero.
(2015-12-28) More surprisingly, zero divides only zero.
Everyone teaches that you can't divide by zero and that 0/0 is undefined...
More precisely, we ought to say that you can't divide a nonzero quantityinto zero (no ordinary number can be produced as a result). On the other hand, any number can be construed as theresult of dividing zero into zero.
In spite of that lack of a definite ratio, can we still say that zero divides zero? Well, yes we can. At least if we consider that the sentence "a dividesb" in a commutative multiplicative realm is strictly equivalent to:
x0, a x = b
We could stop here but it's always best to judge a mathematical definition by thesimplicity of the theorems it allows (as compared to similar theorems usingcompeting definitions). To satisfy this metamathematical imperative, we need at least one desirable fundamental theoremwhich the proposed definition makes easy to state (i.e., without exceptions). Here's one:
Two integers are distinct if and only if there is an integer which only one of them divides.
If zero didn't "divide" zero, the above would only hold for nonzero integers. The following statement also becomes meaningful (first) and true (second):
Zero is the only element which zero divides.
(That's just another way of saying that anything multiplied into zero is zero.)
chormpy (N. N. of New Zealand.2000-10-21) Explain whatcomplex numbers are, in terms an idiot could understand.
Imagine this: We are facing each other in a yard and you're challengingthe very existence of negative numbers,let alone complex numbers:
I ask politely: "Can you take two steps towards me?"
You nod and you do. So nice of you. I thank you.
Then, I ask you to move "minus two" (-2) steps towards me.
You smile, having understood what negative numbers are,and take two stepsback to your original spot.
I smile back: "Can you take an imaginary step towards me?"
You stare and say "Huh?".
However, after some thinking you take a step sideways. Nice job !
In other words, complex numbers are to the plane what real numbers are to the line. They just describe position and motion in the plane the sameway real numbers do on a line. Thus,imaginary simply meanssideways... That viewpoint was devised in 1806 by Jean-Robert Argand (1768-1822).
Adding two complex numbers is easy: The total number of steps taken in the "real" directionis obviously the sum of all steps taken in the real direction. The same applies to theimaginary direction. Each component (real or imaginary) of the sum isthe sum of the corresponding components of the complex addends. (In learned terms, that's a "direct sum".)
Things become only slightly more delicate if you worry about "multiplying" such "numbers"together. However, just think about it this way:
What's the product z of two numbers x and y? Well, it's the number z which is to x what y is to 1. (Isn't it?)
Picture what this means in the complex plane with, say, x=2+i (I move two steps forward and one step to the left). Multiplying any number y by x is like using x as a new "unit" step. In other words, you're now using a new "grid" where each step is of length5(that's the length of x, because of the Pythagorean Theorem),while the whole grid has been rotated about 26.565°, to align x with the "forward" direction. In that new grid, if you go 3 steps forward and one step right (corresponding to the complex number y=3-i) where do you end up? Well, you end up at the point of the plane which, by definition, is the product of x and y.
You could have taken 7old steps forward and one step to the left and would haveended up at the same location. Draw this on paper (just once in your life) and admire the "coincidence" of the tworesults, obtained with or without an intermediate grid.
Why is it that i 2 = -1 ? Well, the left of your left is your back, isn't it?
On 2000-10-22, Chormpy wrote:
Thank you for your answer, Gerard. Although I'm still far from actuallyunderstanding,your answer did clear things up a little. It helped me to understand some of the other examplesand explanations of complex numbers I've found.
The existence of numbers whose squares are negative was first put forth by Gerolamo Cardano (1501-1576) in his Ars Magna (1545). Cardano didn't understand what they meant but found them usefulto present the general solution of the cubic equation that wasrevealedto him in 1539 by Tartaglia(1499-1557) under an oath of secrecy.
The terms real number and imaginary number (nombres réels et nombres imaginaires) were coined by René Descartes (1596-1650) in La Géometrie,an appendix of Discours sur la méthode (1637).
The real linear combinations a +ib of the real unit (+1) and the imaginary unit (i) form the field of complex numbers which is the two-dimensional field obtained from the real line (the field of reals) by the general Cayley-Dickson construction. The reciprocal of a nonzero complex number z is the number which gives unity when multiplied into z. It's given by the following expression:
z-1 = (a +ib )-1 = (aib )(a2 +b2 )-1
That equation expresses the reciprocal of any nonzero complex number in termsof the reciprocal of a nonzero real number.
Fundamental Theorem of Algebra :
Arguably, a full understanding of the complex numbers was reached only when it became clear that they form the algebraic closure of the real numbers. That's what the Fundamental Theorem of Algebra means:
Every nonconstant complexpolynomialhas at least one complex root.
The full theorem (i.e., for polynomials with complex coefficients) was only established byArgand in 1806. (Elsewhere on this site, we give a nice modern proof of that statement.)
Thus, by induction on n > 0, we see thatany complex polynomial P of positive degree n has exactly n complex roots (not necessarily distinct) andcan be written as a product of n linear factors:
P(z) = a
(z-zk) [where a and zk are complex numbers.]
Thinking outside the box :
Let's indulge in some metaphysics about the above introductionof a complex realm whereplanar angles andtwo-dimensionalcurvature live:
As an extra imaginary unit i transforms the real line into the complex plane, so does it transform 3-dimensional space into4-dimensionalspacetime. Time is imaginary length. Length is imaginary time...
Nobody has yet figured out (convincingly) what it would mean to move sideways in time. Human time remains confined to a single dimension.
JonBall (2002-10-22) ) to solve z5 = 1. How do you express the 5 fifth roots of unity in terms of = ½(1+5) ?
Using the fact that cos(2/5) =½(1) and the relation , it's not difficult to show that the 5 fifth roots of unity are:
1, ½ [ i ], and ½ [ i ].
The 10 tenth roots of unity include the above and their 5opposites...
ciderspider (Mark Barnes, UK.2000-11-04) Does the equation x= have an infinite number of complex solutions?
No, it does not. The function 2z can only be defined asexp(ln(2) z). Just like the exp function itself, it'ssingle-valued over the entire complex plane. There's nothing to "solve", the value of x is simplysome real number: 8.824977827...
One possible source of confusion is the use of the numericalconstantln(2) in the above definition... Since the extension of theln function to complex argumentsis indeed multivalued, why not take any of the "other" values ofln(2)and go on from there?
If you take "another" value of ln(2)(say:ln(2) + i )to define your own base-2 exponential,you simply get another single-valued function which is different from everybody else's. You could define infinitely many such functions, but so what? The values of two such functions at the same point ( or any other point) are simply different.
Likewise, thesquare root functionis an introductory example of a function which, like the logarithm function, does notpresent a problem for (positive) real numbers, but which cannot be generalizedto acontinuous function over the whole complex plane. As explained in thenext article,a continuous generalization of the square root function involves anentirely newdomain of definition (called aRiemann surface). For the square root function, the Riemann surface consists of the origin andnonzero points identified as (r,)where r is the [positive] distance to the origin and the "angle" is understood "modulo 4",so that (r,) and (r,)identify twodistinct points with different square roots(which are opposite of each other). Loosely speaking this surface is composed of twosheets and you end upback to the same point if you go around the origin aneven number of times.
In the case of the logarithm function,the Riemann surface has infinitely manysheets;you may visualize it as a flattenedhelicoidwhose nonzero points are identified as above by a couple (r,) except that different values of the real number will always identify different points of the surface. What this means, in concrete terms, is that whenever you use a logarithmyou mustabsolutely refrain from adding an arbitrary multiple of to the "angle" of the argument. This is allowed in the complex plane, but prohibited on the relevant Riemann surfaceover which thecontinuous logarithm function is defined.
Do think about Riemann surfaces and you are safe under the umbrella of mathematical rigor.Forget about this fundamental point and you are bound to produce a numberof false proofs, not always for a recreational purpose...
The Obsolete Formula of Roger Cotes (1712):
Before all this became clear,Roger Cotes(1682-1716) came up with the following formula, which is only true up to somenumber of angular "turns":
ln ( cos + i sin ) = i [ modulo 2i ]
The above uses the modern "natural" measurement of angles (in radians)which is due to Cotes himself ! Cotes died at the age of 34 andIsaac Newton (1643-1727) said of him:
"If he had lived, we might have known something. "
That formula is only of historical interest now. It's been superseded by the following celebrated formula due toLeonhard Euler (1707-1783) whichcould be construed as removing all ambiguities bytaking the exponentials of both sides in the above. However, Euler's formula is best derived directly and it's much simpler in theory and in practice,as it involves only unambiguous (i.e., "single-valued") functions:
Euler's Formula (c. 1740)
cos + i sin = exp (i )
The wonderful special case is often heralded as Euler's equation :
-1 = e i usually popularized in the form e i + 1 = 0
An immediate consequence of Euler's formula is De Moivre's formula :
( cos + i sin ) n = cos n + i sin n
Historically, this predates Euler's formula by twenty years or more. It was written in this form in 1722, by Abraham de Moivre (1667-1754) who may have known about it as early as 1707. It can also be proved by induction on n (the negative case is easily deduced from the positive one). Of course, n must be an integer (since you can't raise to any other kind of exponent anything but a positive real number).
silenteuphony (2003-07-20) May the square root function ( ) be generalized to negative numbers?
The short answer isno. There are popular implementations (on somehandheld calculators and elsewhere) which providepointwise solutions to quadratic equations,but they don't qualify as proper mathematical generalizationsof the square root function.
Such generalizations would invalidate familiar properties established in the realmof nonnegative real numbers, where the square root of a number x is unambiguouslydefined as the nonnegative number whose square is equal to x. Among the casualties would be one of our most trusted relations:
u v = (uv)
Indeed, if a definition of (-1) could be given which wasconsistent with this relation, wewould have: (-1) (-1) =1,so that the square of (-1) would be 1instead of (-1)...
Thereis a number whose square is -1,namely the imaginary number i [note that its opposite-i would do just as well]. However, it's abusive to denote it (-1) for a number of reasons, including the one given above.
Unfortunately, this has not stopped a number ofotherwise distinguished authors from doing so,in order to bypass a more proper introduction to whatimaginaryandcomplex numbers really are. (Seeabove for my own attempt at such an introduction.)
What about the "square root" of a complex number?
If we insist on defining a square root(sqrt or )as a single-valued function over the complex plane,the best we can do is acceptdiscontinuity(jumping from y to -y) on some kind of curve going from the origin to infinity(e.g., one half of a straight line).
We like to call this kind of line a cliff (since ajump discontinuity occurs when the argument crosses such a line). The square-root function can't be defined over the entire complexplane without creating acliff.
This annoying issue was cleverly resolved by Bernhard Riemann (1826-1866)who stated essentially that the "correct"domain ofsqrt wasnotthe complex plane itself, but (roughly) two copies of it,properly interconnected topologically. Each such "copy" (loosely speaking) is called aRiemann sheet and the whole thingis theRiemann surface for thesqrt function.
This surface may be rigorously described as consisting of the origin, together withthe set of ordered pairs (r,)where r [the distance to the origin] is positiveand is a real "angle"modulo 4(whereas a similar definition of the ordinarysingle sheet complex planewould specify that is "modulo 2"). The beauty of this approachis thatsqrt is defined and continous everywhere on itstwo-sheet domain(itsrange is the ordinarysingle sheet complex plane).
The "two-sheet" Riemann surface for the square root function is totally different fromthe set of complex numbers. Loosely speaking, you end up on the same point only if you travel anevennumber of times around the origin. If you wish, you may identify a point on the surface using a notation like(r,) where is between 0 and 4,although it's probably better to make no such restriction and state that the second numberis understood "modulo 4" (as stated above)so that the point (r,)is identical to (r,+4k)for any integer k...
Points on the two-sheet Riemann surface have square roots that are ordinary complex numbers;the square root of the point (r,)is defined as the complex number(r) exp(i/2). Therefore, (r,) and (r,)have two different square roots that are opposite of each other.
Multiplication is well-defined: The product of u = (a,)and v = (b,) isuv = (ab,),where is understood modulo 4. This is how we maintain the validity of properties likeuv =(uv).
Unfortunately, no simple "addition" is defined on this Riemann surface.
A nonzero complex number is associated withtwo distinct points of the Riemann surface, which have different square roots(opposite of each other), sothe "nice" definition of square roots over the Riemann surface does not resolve thesign ambiguity for ordinary complex numbers. One deep explanation for the impossibility of defining a continuousgeneralization of the square root function over complex numbers is that the relevantRiemann surface and the complex plane aren't homeomorphic(i.e., there's nobicontinuous one-to-one correspondence between the two things).
If you choose to define on the domain of complex numbers rather than on the proper Riemann surface,your "square root" function cannot be continuous and the "square root" of a productis not necessarily equal to the product of the "square roots" of its factors. There's no way around this...
A cardboard model of the Riemann surface for thesqrt function is easy to make butnot to describe (the surface goes through itself along one line). The one I made years ago (pictured at right) used to sit on a shelf next to my desk, as aconstant reminder of the above fact in the realm of complex variables.
One 3D embodiment of that Riemann surface is a self-intersecting pseudo-helicoid; a 3D surface whose parametric cartesian equations are:
x = r cos y = r sin z = r cos /2
The parameter goes from 0 to 4.
(M. M. of Salem, MA.2000-10-11) Two numbers have a product of 19551 and a sum of 280.Without determining the numbers, find their difference.
If P, S and D are the product, sum and difference of the two numbers, then:
S2 - D2 = 4P
Therefore, in this case, D2 is 2802 - 419551 = 196. The difference D between the two numbers is thus14. (Don't object that it could also be-14.)
You may want to prove the relation S2 - D2 = 4P by noticing that:
FlyingHellfish (Atlanta, GA.2002-10-08) Find the value of D = x4+y4+z4,given the relations at right, in particular when A=1, B=2, and C=3.
Introducing theelementary symmetric functions,U = x+y+z, V = xy+yz+zx, and W = xyz. , we have: A = U, B = U2-2V, and C = U3-3UV+3W. Conversely, U = A, V = (A2-B) / 2, and W = (A3-3AB+2C) / 6.
Since D = U4-4U2V+4UW+2V2, we have D = (A4-6A2B+8AC+3B2) / 6. For the particular case A=1, B=2, C=3, this means D = 25/6.
The quantities x, y and z are the 3 zeroes of X3-UX2+VX-W (in the numerical case above, two of these are complex numbers). Any symmetrical polynomial of such roots is also a polynomial in U,V,W, and its value may thus be obtained without solving the cubic equation. This remark may be generalized to any number of variables...
The Elementary Symmetric Functions: (A. Girard, 1629)
For m variables, the nthelementary symmetric function (n)is defined via:
0 = 1 1 =
Xi2 =
XiXj3 =
XiXjXk etc.
At first (1629) Albert Girard (1595-1932) called n the "nth fraction" of those m variables. Note that the definition of n as thesum of all products consisting of n factors taken from the m variables remains valid for n = 0 (since there's only one product of zero factors and it's equal to 1 ).
If n > m > 0 then n = 0 (no products of n distinct variables to sum up).
The variables X1, X2, ... Xm are clearly the roots of the polynomial [in x]:
(Xi x) =
m-n (-x)n
A polynomial in m variables which remains unchanged under any permutation of the variablesis calledsymmetrical. Any such polynomial can be expressed as a polynomial ofthe aboveelementary symmetric functions. Such is the case, in particular,for the sum of the p-th powers of all the variables (Sp ):
S = S = S = S =
This last relation gave us D = S4 in the above case of 3 variables(). To extend the list in a systematic way, we observe that the followingresults (known asNewton Identities orNewton-Girard Formulas)hold for any m:
0 = mm +
m-n (-1)n Sn
This is true for m variables,because each is a root of theabove polynomial(the right-hand side is thus obtained by summing m zero values of that polynomial). This holds for less than m variables(the result for m variables holds if some of them are set to zero)and also for more than m variables,because of symmetry and degree considerations which we won't go into... For example:
S = SSSS which yields the expression: S = [7 terms]
Such expressions of power-sums in terms of theelementary symmetric polynomialsare known asGirard-Waring expansions (published in 1629 byAlbert Girard and between 1762 and 1782 byEdwardWaring).
That expansion may be given in the form of adeterminant too:
k! k =
S1 S2 S3 S4 ... ... Sk
1 S1 S2 S3 ... ... Sk-1
0 2 S1 S2 ... ... Sk-2
0 0 3 S1 ... ... Sk-3
0 0 0 0 ... k-1 S1
(S. M. of Bagdad, KY.2000-10-18) Find 6 numbers in continued proportion. Their sum is 14 and the sum of their squares is 133.
Three or more numbers are said to be in "continued proportion"when the ratio of one term to the previous one is a constant R.This is now more commonly called a "geometric progression" of (common) ratio R.
If A is the first of 6 such terms, their sum is A(R6-1)/(R-1)and the sum of their squaresis A2(R12-1)/(R2-1).(That's assuming that R differs from 1, but it's easy to check that R=1 does not yield anysolution to the problem at hand.) As we're told that the former is 14 and the latter is 133, we may solve this using a pairof relations giving:
The first quantity, namely: 14 (R-1) = A (R6-1)
The ratio of those two: 133 (R+1) = 14 A (R6+1)
Substituting in (2) the value of AR6 obtained from (1) or the other way around, we obtain the relation 9R+4A=47.(Incidentally, this same relation would hold regardless of the length of the continuedproportion.) Either of the above equations then becomes:
9R7 - 47R6 + 47R - 9 = 0
The obvious root R=1 is to be ruled out, as remarked at the outset(so we could freely divide by R-1). Dividing by (R-1), there remains to solve a polynomialequation of degree 6, namely:
9R6 - 38R5 - 38R4 - 38R3 - 38R2 - 38R + 9 = 0
Clearly, if R = K is a solution, then so is R = 1/K. Both of those correspond to the same solutionof the original problem but with the 6 numbers listed "forwards" or "backwards". This calls for the following change of variable:
X = R + 1/R (giving X2 = R2 + 1/R2 + 2 and X3 = R3 + 1/R3 + 3X)
If we have a solution for X, it will only be a matter of solving a quadratic equation torecover a pair of solutions for R.
That's still a mouthful but it's only of the third degree so we could solve it withalgebraic methods! I hate doing this, so I'll just give the three roots approximately(they happen to be all real):
5.41246229893, 0.469896647164, and -1.66013672387.
Now, a solution in X corresponds to a pair of real solutions in R when the equationR2-XR+1=0 has real solutions. This happens only when X2-4 is positive.Therefore only the solution X=5.412... is to be retained if we are only interested in realsolutions. This corresponds to the solution R=5.2209925253737229 (or the inverse of this tolist the numbers backwards) and A=(47-9R)/4.The unique pair of solutions is thus composed of the following 6 numbers listed eitheras below or in reverse order (last digits not guaranteed):
Now, you may check (I did!) that the sum of the above is indeed 14 and the sum of theirsquares is indeed 133.
tenorboy (Todd A. Moore.2002-05-19) In an alley way, a 12 ft ladder leans against a building on one side;its bottom is on the ground against another vertical building across the alley. Similarly, a 10 ft ladder leans in the other direction across the alley... The ladders intersect 4 ft off the ground. What's the width of the alley?
Let x be the width of the alley and ux the horizontal distance from the bottom ofthe 12-ft ladder to the plumb line at the intersection;(1-u)x is the corresponding quantity for the second ladder. Remark that the 4-foot plumb line is equal to u times the top height of the first ladderand (1-u) times the top height of the second one(because of the two pairs of similar triangles involved). In other words:
4 = u (122-x2) and 4 = (1-u) (102-x2)
Eliminating u, we obtain:
1 / (122-x2) + 1 / (102-x2) = 1/4
We may use this equation directly to find the solution numerically,with ludicrous precision:x = 7.2575891083169677047316337322... ft.
Alternately, we may obtain a polynomial equation,by eliminating the above two radicals: First put one radical by itself on one side of the equation;squaring both sides will then eliminate that first radical. Isolating the remaining radical on one side and squaring again givesa rational expression without radicals. This double squaring gives a quartic [= degree 4] equationin the variable y = x2. Because the equation isonly a quartic,it can be solved algebraically, although everybody (including myself) hates to do so. For the record, here's the quartic:
y4 - 424 y3 + 64912 y2 - 4200448 y + 95420416 = 0
Note that this quartic equation may include roots which do not correspond to solutions ofthe original problem... Indeed the double squaring does introduce just such aspurious solution here(corresponding to x around 9.6668,which is clearly not a solution of our original equation). All told, in this age of computers and nifty scientific calculators,it's probably best to stick with a simple equation (like the one we first gave)rather than insist on some not-so-simple polynomial relation with a few irrelevant roots...
In the 1980's, I saw [a problem just like the above] in the French monthly magazine Science & Vie, staging a smart painter trying to figure out, with pencil and paper,the width of a corridor where two opposing ladders of known lengthscross at a height ofone meter.
The question was: Even if the scene takes place at the top floor of a high-rise building under construction (no lift)wouldn't it be wiser for the painter to fetch the tape measure he left downstairs?
François Robert Milan, Italy
Thanks for the comment, François. I used to be a fan of Science & Vie too.
tr.v.grok·ked,grok·king,groks. Slang.
