# Tangential magnetic force experiment with circular coil

If magnetic force is to respect Newton’s third law, there should be a recoil force on the vertical current which is Ft. This force is tangent to the current I1 and called tangential magnetic force. Some physicists claim that tangential magnetic force exists, this claim is supported by some experiments such as the rail gun recoil force shown by Peter Graneau and Ampère’s hairpin experiment, see Lars Johansson’s paper. But these experiments did not convince the main stream physicists and tangential magnetic force is rejected. I have carried out an experiment to show tangential magnetic force acting on a circular coil.

PDF Tangential magnetic force experiment with circular coil http://pengkuanem.blogspot.com/2017/06/tangential-magnetic-force-experiment.html

# Length-contraction-magnetic-force between arbitrary currents

In ≪Relativistic length contraction and magnetic force≫ I have explained the mechanism of creation of magnetic force from Coulomb force and relativistic length contraction. For facilitating the understanding of this mechanism I used parallel current elements because the lengths are contracted in the direction of the currents. But real currents are rarely parallel, for example, dIa and dIb of the two circuits in Figure 1. For correctly applying length contraction on currents in any direction, we will consider conductor wires in their volume and apply length contraction on volume elements of the wires.

PDF Length-contraction-magnetic-force between arbitrary currents http://pengkuanem.blogspot.com/2017/05/length-contraction-magnetic-force.html

# Relativistic length contraction and magnetic force

Magnetism is intimately related to special relativity. Maxwell’s equations are invariant under a Lorentz transformation; the electromagnetic wave equation gives the speed of light c. Many have explained magnetic force as a consequence of relativistic length contraction, for example Richard Feynman in page 13-8 of his ≪The Feynman Lectures on Physics, Volume II≫ and Steve Adams in page 266 in his ≪Relativity: An Introduction to Spacetime Physic≫. If magnetic force is really created by relativistic length contraction, we should be able to derive the expression for magnetic force from the length contraction formula. And indeed we can, as I will show below.

PDF Relativistic length contraction and magnetic force http://pengkuanem.blogspot.com/2017/04/relativistic-length-contraction-and.html

# Lists of binary sequences and uncountability

Creation of binary lists, discussion about the power set of ℕ, the diagonal argument, Cantor’s first proof and uncountability. Binary system is kind of magic because it can express natural numbers, real numbers and subsets of natural numbers. Below, we will create lists of binary sequences to study the uncountability of the power set of ℕ and real numbers.

1.Infinite list of binary sequences
2.About the Power set of ℕ
3.Frame of Natural Infinity
4.List of numbers smaller than 1
a.Creation of the numbers
b.Denseness of R..
c.Completeness of R..
d.Real numbers in [0,1[
7.Conclusion

PDF Lists of binary sequences and uncountability
http://pengkuanonmaths.blogspot.com/2016/11/lists-of-binary-sequences-and.html

# Continuity and uncountability

Discussion about continuity of line, how continuity is related to uncountability and the continuum hypothesis.
The real line is made of real numbers which are points. Points are discrete objects, but lines are continuous objects. How does continuity arise out of discreteness when points make line? The idea of uncountability solves this problem. Rational numbers are countable, the line they make contains holes. Real numbers are uncountable, the line they make is continuous. So, continuity must be created by the uncountability of the points of a continuous line. One can imagine that uncountable points are so numerous on the real line that real numbers are squeezed together.

Georg Cantor called the set of real numbers continuum, so he probably thought of creating continuity with discreteness when inventing uncountability. But, what does continuity really mean?

PDF Continuity and uncountability
http://pengkuanonmaths.blogspot.com/2016/09/continuity-and-uncountability.html

# Cardinality of the set of decimal numbers

Cardinalities of the set of decimal numbers and ℝ are discussed using denominator lines and rational plane. On the rational plane, a vertical line is referred by its abscissa M. Because the points of a vertical line represent the quotients i/M which have the same denominator M, the vertical line at abscissa M is called denominator line of M.

PDF Cardinality of the set of decimal numbers http://pengkuanonmaths.blogspot.com/2016/03/cardinality-of-set-of-decimal-numbers.html

# Prime numbers and irrational numbers

The relation between prime numbers and irrational numbers are discussed using prime line and pre-irrationality. A rational number is the quotient of 2 whole numbers i and j, coordinates of a points (j, i) in the plane of 2 dimensional natural numbers shown in Figure 1. Each points (j, i) represents a rational number whose value is i/j that equals the slope of the straight line connecting the point (j,i) to the origin (0,0).

PDF Prime numbers and irrational numbers

http://pengkuanonmaths.blogspot.com/2016/02/prime-numbers-and-irrational-numbers.html

# On Cantor’s first proof of uncountability

Discussion about Cantor’s first proof using the next-interval-function, potential and actual infinity. Cantor’s first proof of the uncountability of real numbers is the first rigorous demonstration of the notion of uncountability. Countable sets can be put into a list indexed with natural numbers. If a set cannot be listed, then, it has more members than the set of natural numbers and is uncountable. Cantor’s first proof is a proof by contradiction. First, he supposes that all real numbers are listed in any order by the list X=(x1, x2, x3 …). Then, a real number out of this list is found by using a series of intervals, contradicting that X lists all real numbers.

PDF On Cantor’s first proof of uncountability
http://pengkuanonmaths.blogspot.com/2016/02/on-cantors-first-proof-of-uncountability.html

# On the uncountability of the power set of ℕ

This article discusses the uncountability of the power set of ℕ proven by using the out-indexes subset contradiction. Cantor’s theorem proves that the power set of ℕ is uncountable. This is a proof by contradiction. Suppose that the power set of ℕ is countable. This allows us to put all subsets of ℕ in a list. The contradiction will come from the indexes.