DavidQD

Thank You Aziz. The inspiration comes from studying Heureka’s works of art.

The cyanide needs increased to 6 molecules. (It is not “poisonous” enough to balance the equation).

$$\underbrace {\ 10K_4Fe(CN)_6 } _ {Potassium ferrocyanide }\ +\underbrace{122KMnO_4}_{Potassium\ permanganate}\ +\ \ \underbrace{199H_2SO_4}_{Sulphuric\ acid}\\\\
\Rightarrow \ \underbrace{162KHSO_4}_{Potassium\ bisulphate}\ +\ \ \underbrace {5Fe_2(SO_4)_3}_{ferric \ sulphate}\ \ \ +\ \underbrace {122MnSO_4}_{Manganese(II) sulphate}\ +\ \ \underbrace {60HNO_3}_{Nitric\ acid}\ \\\\
\ +\ \underbrace {60CO_2}_{Carbon\ dioxide}\ +\ \ \underbrace {188H_2O}_{water} \\$$

~~D~~

It is like swimming in an ocean of near infinite complexity. Even so, I do recognize the “water,” such as the plus and minus signs.

I suspect you do too, and much more.

~~D~~

Thanks Melody.

This was an attempt to demonstrate that because an integer squared (x^2) (where all such numbers have an odd number of positive divisors) is equal to another number squared (y^2), plus an additional factor (5) that the value of (x) and/or (y) cannot be integers. This means it is equal to a number that has an even number of divisors, or it does not have an even number of divisors to begin with. Because of this contradiction, the number cannot be an integer. Extending on this was an attempt to explain why an irrational number had characteristics of both even and odd numbers.

The explanation I presented is incomplete. This has been rolling around in my head for years, and it seemed functional, until an examination in a proper light. On closer inspection, it seems it is an incomplete amalgamation of proofs by contradiction and a hint of a proof by infinite decent thrown in for good measure (Fermat would be appalled). Mostly, it is a succession of incongruous comparisons that are not made compatible mathematically. Very lame. This is worse than mixing up metaphors. So, thank you Melody, I now see the darkness at the end of the tunnel.

 Your proof is to the point clear. CPhill’s, detailed and elegant, proof clearly demonstrates the paradox of (even and odd) parity.

Somewhere, there is a direct proof (instead of proofs by contradiction) of irrational numbers. Probably the best place to look is in Andrew John Wiles’ proof of Fermat’s Last Theorem. These maths are far outside of my skill set, and always will be. Even so, after dozens of readings, I have gleaned insights from them.

To find a direct proof for irrational numbers could lead to maths that close the gaps on Cantor’s Alephs of infinites. Considering there are more irrational numbers than integers, the value of this proof may be enormous. Adding to this, Wiles’ proof includes proofs of the Modularity Theorem for semistable elliptic curves, with these kinds of proofs, all three of Georg Cantor’s infinites may converge. Wiles’ proof seems like an excellent place to start.

Noting the post on Fermat, there is one question that remains about Fermat’s Last Theorem: Did Fermat really have the remarkable proof he wrote of in the margin of his book? Considering it took advanced, late twentieth century mathematics to solve it, I seriously doubt it. However, it is an intriguing thought that seventeenth century math may hold a solution.

This forum is great. The mathematicians who visit and especially the ones who live here are truly extraordinary.

Thank you Melody and CPhill!

~~D~~

Composite numbers cannot have both an even and odd number of prime factors because all composite numbers have a unique prime factorization.

That is true. This solution assumes that it is, and proves it is not a composite number.

This is impossible because they are the same number and must have the same number of factors.

The “x” and “Y” values have the same number of factors, the equation does not. They cannot because the total factors are odd. Only at an infinite, diminishing point where the “constant” --the additional factor, converges does the equation balance. That value, in this case, is the SQR( 5).

I do not know which side has the even or odd, only that both exist. The total factors are odd, because adding all the factors, except for the 5 is the same as multiplying by 2, making the total an even amount, then add the one additional factor (the 5), to the count, makes it odd.

Therefore there are no integers x and y that meet this criterion.

That is essentially correct.

This is conjecture, but the mathematics that communicate this may be related to the maths that relate why the set of all integers is equal in number to the set of all even (or odd) integers.

~~D~~

The equation is set up as attempt to balance, with a constant, the ratio of two hypothetical integers. The constant extends, in diminishing value, into infinity. In other words, no integers exist that define this ratio.

It is the middle ground of light and shadow.

~~D~~

Test this by assuming √5 is rational. If it is rational then it can be expressed in lowest terms as x/y.

$$\ Let \ (\frac{x}{y})^2 = 5 \\\
\hspace{20pt}\ x^2 / y^2 \ = 5 \\\
\hspace{20pt}\ x^2 = 5y^2 \\$$

Here the factor 5y², on the right, has an additional prime. One side of the equation has an odd number and the other has an even number. Composite numbers cannot have both an even and odd number of prime factors because all composite numbers have a unique prime factorization.

These two factors “compete” for balance into infinitesimal infinity.

This is an irrational number, a surd. As CPhill might say, something that cannot be heard or uttered.

~~D~~

 . . . what about 6 , 7 and 8 . . .

Yes, they are correct. I double checked them.

~~D~~

Q-1) The answer is (A). Remember to subtract the length (1cm from the top and 1 cm from the bottom) already included in the other measurements.

Q-2 through Q-5 ) Answers are correct.

For Q-23 . . . . .

Your answer, (B), is (probably) correct. If it didn’t say “always” and/or have (D) “Any value” as an alternative, then I might be more certain.

This is an ambiguous question. (These are often found on physics tests).

For most equations friction work is (usually) signed as negative. Applied forces, are usually positive.

Technically, friction opposes motion and, hence, does negative work. However, friction is not a vector, so the negative sign does not tell us that it's moving backward.

Friction removes energy from the system. That is why it is (usually) signed as negative. To overcome this, energy needs to be (continuously) added to the system.

Nota bene: Another consideration is friction can be (0) zero. If this is the case, then it is neither positive nor negative. This really is an ambiguous question

~~D~~

P.S. Now, that I have reread the question, it is clear it refers to the previous question (Q-20). Your answer, (B), is correct.

It does help to read the whole question.

(Q-11)

This is a potential energy calculation. The energy is the same whether this is occurs in 4 seconds or 4 minutes. (Note: there is a difference in power because of this).

Here a mass is moved 0.8 meters vertically against a force of 9.81m/s². Potential Energy = Mass x (Force of Gravity per Kg) x vertical distance 20Kg * (9.81m/s²) * 0.8M

$$E = m \times g \times h = 20 \textrm{ kg} \times 9.81 \frac{\textrm{N}}{\textrm{kg}} \times h$$

$$E = 20 \textrm{ kg} \times 9.81 \frac{\textrm{N}}{\textrm{kg}} \times 0.8m \ =\ 156.96J \\$$

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(Q-21)

The answer is A.

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(Q-22) Which shown arrows represents correctly the electric field at point p.

The answer is A. Electric fields are conventionally shown as moving to a negative charge.

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(Q-30)

Magnetic field = Permeability * Turn density * Current

Current = (Magnetic field) / (Permeability * Turn density)

The relative permeability here is assumed to be (1). (This is used as a multiplier of absolute permeability).  

Turn density = The number of “loops” of wire uniformly distributed over a length of 1 meter.

110turns/0.5Meters = 220turns/meter

Magnetic field in milliTesla’s : 2.5mT = 0.002500 T

The magnetic field inside a solenoid is proportional to the applied current and the number of turns per unit length. The field inside is considered constant and independent of the diameter of the solenoid.

The magnetic field only depends on the current (I = amps), the number of turns per unit length (N/L), and the permeability of the core (mu₀). 

B= mu₀(N/L) * I mu₀ is  Permeability in a vacuum (or air)

I=B/(mu₀(N/L)

$$I = \frac{B}{mu_0 \frac {N}{L}}$$

$$I = \frac{B}{220 *\mu_0} \ = \ 9.04 A\\$$

Where

$$\\ \mu_0 = \ 4\pi \* 10^-7} \ H M^-^1$$

Yields Tesla measure of the ratio between enclosed magnetic field and the current.

 0.002500 T corresponds to 9.04 Amps. (Answer D).

~~D~~