The standard proof is by contradiction.
If they were both rational, then
would also be rational, which implies that (π−e) is an algebraic number (i.e., the root of some nonzero polynomial with rational coefficients).
Since π+e is rational and thus algebraic by hypothesis, it follows that (π−e)+(π+e)=2π is algebraic (being the sum of two algebraic numbers) and hence so is π.
Similarly, we can infer that e is algebraic.
However, both are known to be transcendental — contradiction.
Imagine that a family has two children, one of whom we know to be a boy. What then is the probability that the other child is a boy? The obvious answer is to say that the probability is 1/2—after all, the other child can only be either a boy or a girl, and the chances of a baby being born a boy or a girl are (essentially) equal. In a two-child family, however, there are actually four possible combinations of children: two boys (MM), two girls (FF), an older boy and a younger girl (MF), and an older girl and a younger boy (FM). We already know that one of the children is a boy, meaning we can eliminate the combination FF, but that leaves us with three equally possible combinations of children in which at least one is a boy—namely MM, MF, and FM. This means that the probability that the other child is a boy—MM—must be 1/3, not 1/2.
What kind of supports should be given to a chair while designing the chair?
Supposing a simplistic chair (stool) with four legs,
What kind of supports should we give to the end of the legs in the chair?
If this question was asked in mechanics, the answer would be simply supported.
But in design, we should fix three legs in y direction and one leg in all directions.
That is because the fixed leg acts as reference for the movement of the other three legs. In other words, we are restricting the velocity of the chair in all three directions, ie giving boundary conditions in all three directions.
Then we get the solution as below.
Now let us consider the following cases:
The solution for case 3 is given below
Note that the left leg(as shown in figure 1) is the fourth leg.
The bi-linear equation
with the essential boundary condition of u(0) = U0 and natural boundary condition of
at x = L , and the domain 0 < x <L appears in several forms in engineering.
CABLE(ROPE) UNDER A TRANSVERSE LOAD
u = deflection
q(x)= transverse load (eg: weight of cable)
Q = Axial force
BAR IN TENSION
u = deflection
q(x)= Friction(Traction on the bar)
Q = Axial force
u = Temperature
a= Thermal conductivity(k)
q(x)= Heat generated
Q = Heat flux
ONE DIMENSIONAL LAMINAR INCOMPRESSIBLE FLOW (Gradient(p)=const)
u = velocity
a= viscosity (η)
q(x)= Pressure gradient
Q = Axial stress
FLOW IN POROUS MEDIA IN ONE DIMENSION
u = Fluid head
a= Permeability const
Q = Flow
u = Electrostatic potential
a= dielectric constant
q(x)= charge density
Q = Electric Flux
Hot water seems to freeze faster than cold water, known as the Mpemba effect. The effect was first observed by Aristotle in the 4th century BC.
Theories for the Mpemba effect have included faster evaporation of hot water, therefore reducing the volume left to freeze; formation of a frost layer on cold water, insulating it; and different concentrations of solutes such as carbon dioxide, which is driven off when the water is heated. Unfortunately the effect doesn’t always appear - cold water often does actually freeze faster than hot, as you would expect. But this Mpemba effect occurs regularly, and no one has ever been able to definitively answer why.
Some scientists have found evidence that it is the chemical bonds that hold water together that provide the effect. Each water molecule is composed of one oxygen atom bonded covalently to two hydrogen molecules. The separate water molecules are also bound together by weaker forces generated by hydrogen bonds. These forces occur when a hydrogen atom from one molecule of water sits close to an oxygen atom from another.
They suggest that it is these bonds that cause the Mpemba effect. They propose that when the water molecules are brought into close contact, a natural repulsion between the molecules causes the covalent bonds to stretch and store energy. When the liquid warms up, the hydrogen bonds stretch as the water gets less dense and the molecules move further apart.
The stretching in the hydrogen bonds allows the covalent bonds to relax and shrink somewhat, which causes them to give up their energy. The process of covalent bonds giving up their energy is essentially the same as cooling, and so warm water should in theory cool faster than cold.