Jackson an’ Johnson / Murphy an’ Bronson /
One by one dey come / An’ one by one to dreamland dey go.
[From Carmen Jones. Lyrics: Oscar Hammerstein]
Euclid’s Theorem I-47, also known as Pythagoras’ Theorem, in Oliver Byrne’s colourful text, The Elements of Euclid.
Two young high-school students from New Orleans, Ne’Kiya Jackson and Calcea Johnson, recently presented a new proof of the Pythagorean theorem at a meeting of the American Mathematical Society in Georgia. It has been widely believed that no proof based on trigonometry was possible but we now know that to be false. Continue reading ‘The Waffle Cone and a new Proof of Pythagoras’ Theorem’
. Right: partial density of set 
or choris ena, so let us call a set of “oneless numbers” a chorisenic set. 
of a real variable, we may ask what is the value of the function for a particular value 
. Maybe this is an easy question or maybe it is not.
of a number to take values from 1 to 9.
occurs more than 
of the time while 
is found in less than 
of cases (see Figure above). Specifically, the probability distribution is
in reduced form (
and 
coprime), a Ford circle is a circle with center at 
and radius 
. There is a Ford circle associated with every rational number. Every Ford circle is tangent to the horizontal axis and each two Ford circles are either tangent or disjoint from each other.
. Another example is the wave equation derived by Scottish mathematical physicist James Clerk Maxwell. Our modern world would not exist without the knowledge encapsulated in Maxwell’s equations. 
and 
. There is no point on the real line corresponding to the “North Pole” in 
and 
. This immediately yields the well-known sequence
is its modulus or absolute value 
. We measure the “distance” between two real numbers by means of the absolute value of their difference. This gives the Euclidean metric 
and, using it, we can define the usual topology on the real numbers 
and asked for the next number in the sequence. Arguments were given indicating why any number might be regarded as the next number.
. The eighth is a tiny bit less than this.
. Most people would agree that the next number in the sequence is 
. We will show how the number 
could be the “correct” answer. 
to the Weierstrass CND function.![{[a, b]}](https://s0.wp.com/latex.php?latex=%7B%5Ba%2C+b%5D%7D&bg=ffffff&fg=000000&s=0&c=20201002)
, is continuous on that interval. However, the converse is not necessarily true: the continuity of a function at a point or on an interval does not guarantee that it is differentiable at the point or on the interval.
in the plane 
and the sphere 
in three-space 
, we can consider hyper-spheres in higher dimensional spaces. In particular, we will consider the 3-sphere which can be embedded in 
but can also be envisaged as a non-Euclidean manifold in 
are convenient. Here 
is the longitude and 
the latitude. In Figure 1 we show the momentum equations as presented in the monograph of Lorenz (1967):
![{I := [0,1]}](https://s0.wp.com/latex.php?latex=%7BI+%3A%3D+%5B0%2C1%5D%7D&bg=ffffff&fg=000000&s=0&c=20201002)
into the unit square ![{Q:=[0,1]\times[0,1]}](https://s0.wp.com/latex.php?latex=%7BQ%3A%3D%5B0%2C1%5D%5Ctimes%5B0%2C1%5D%7D&bg=ffffff&fg=000000&s=0&c=20201002)
. Georg Cantor found a one-to-one map from 
onto 
, showing that the one-dimensional interval and the two-dimensional square have the same cardinality. Cantor’s map was not continuous, but Giuseppe Peano found a continuous surjection from 
, or its polar coordinates 
. In space, we may specify the location by giving three numbers 
.
) has been demonstrated. There is essentially no doubt about its validity, but no proof has been found.
. Each line corresponds to an ordinal 
where 
and 
are natural numbers [image Wikimedia Commons].
in the triangle (Brothers, 2012(a),(b)). This is indeed surprising. The number 
(in reduced form):
stands for “Neither Odd Nor Even”.
are dense in the real numbers 
is 
. We cannot list them in ascending order, because there is no least rational number greater than 
.
ThatsMaths 