- 2095 - Math through the Ages. - Pythagoras, 580 - 500 B.C., was a Greek mathematician credited with
developing the theory of functions, the significance of numbers, and the
Pythagorean theorem for right triangles, the square of the hypotenuse = the sum
of the squares of the other two sides.
None of his works survived, only his lore. This started the inventions, or discoveries, in
the world of math.
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----------------------------- - 2095 - Math through the Ages.
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- The Greeks
invented the integers, 0,1,2,3,……
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- They later
discovered negative numbers, -1,-2,-3,….
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- This happened
when someone tried to subtract 4 from 3.
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- The
Pythagoreans invented “rational numbers” by creating a ratio of two integers,
¾, 9/28, 1/16, 1/137, …
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- Pythagoras, 580 - 500 B.C., was a Greek
mathematician credited with developing the theory of functions, the
significance of numbers, and the Pythagorean theorem for right triangles, the
square of the hypotenuse = the sum of the squares of the other two sides. None of his works survived, only his lore.
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- The
Pythagoreans also discovered “prime numbers”.
These are numbers that cannot be evenly divided by another integer other
than itself, 2,3,5,7,11,13,17,…. Two is
the only prime number that is even, the rest, and infinite number, are
odd.
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- The prime
numbers are the building blocks out of which all integers are built by
multiplication. Factoring is the process
of finding all the smallest prime numbers that multiplied together get to that
number. The products of two prime
numbers can never be a perfect square.
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- Other than
2,3,5,7 all other prime numbers end in 1,3,7,9.
And, there are an infinite number of these p[rime numbers.. Two of the largest prime numbers are: 391,581*2^216,193, this number has 65,087
digits. 2^756,839 - 1 is an even larger
number with 227,832 digits, which would fill 32 pages. In 1985 a prime number
was discovered by putting 1,031 ones in a row.
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- One day
rational numbers were found to be incomplete.
Some numbers were found that cannot be expressed as a ratio of two
integers. Square root of 2, pi, are
“irrational numbers”. The square root of
2 came up when they tried to calculate the length of a diagonal of a unity
square. You can find the proof for
square root of 2 being an irrational number on the internet, see Euclid.
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- To prove
2^.5, the square root of two, is irrational start with the assumption that it
is rational and can be expressed as the ratio of two integers, a/b. Where a and b are the smallest positive
integers that can satisfy this equation 2^.5 = a/b. +a and b cannot both be even since if they
were, we could divide each by 2 and have a smaller a and b to work with.
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- If we square both sides of the equation: 2 = a^2/b^2.
Or, 2*a^2 = b^2. From this we can
deduce that b^2 is even. If b is even
then a must be odd since a and b cannot both be even. Since b^2 is even then b must be even. If b is even then b^2 must be divisible by
4. Therefore, b^2/2 is an even
integer. But, since 2*a^2 = b^2. Then b^2/2 = a^2. Thus, a^2 must be even, and therefore b is
even.
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- We had
previously determined that b must be odd.
+b and a cannot be both even and odd, so we conclude that no such
integer exists, which invalidates our premise that square root of 2 is
rational, and therefore it is irrational according to Euclid. His logic is so eloquent.
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- The 360
degree circle was introduced by Hypsicles, another Greek, in 180 B.C. The plus and minus signs were not introduced
until 1489 as a translation for surplus and deficit. Greater and less than signs came along in
1631. The divide sign -:- was not used
until 1659.
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- Together, all
these numbers form a continuous line from zero to positive infinity and from
zero to negative infinity. They are
called “ real numbers”. They have some
really unique properties. Take any
3-digit number in which the first digit is larger than the last digit. Reverse the number and subtract the smaller
number from the larger one. Reverse the
result and add this number to the result.
The answer is always 1,089.
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- Someone came
across the problem, x^2 = -9 and discovered that there is no real number that
will solve this problem. They had to
invent “imaginary numbers” that contained the square root of -1 in order
to have a solution. Now, the answer to the problem is two numbers
+3*(square root -1) and -3*(square root -1).
Since the (square root -1) was hard to write ever time, they just called
it “*i” for imaginary.
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- Imaginary
numbers in effect created a whole new set of numbers of the form z = a + bi,
where a and b are real numbers and *i is imaginary. This set of new numbers is called “complex
numbers”. The magnitude of the complex
number z = a+bi is the square root of a^2 + b^2, using the Pythagorean theorem,
since a+bi forms a right triangle.
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- Complex numbers are used to define a vector, which
has both magnitude and direction.
Magnitude at an angle can always be solved using trigonometry or the
Pythagorean theorem. Complex numbers
were first used in 1572. They became a
very useful tool in electrical engineering.
Voltages and currents all have phase which is the same as an angle. Circuits have to be designed using complex
numbers. I used a slide rule in my
day. It was fast and accurate to 3
significant figures.
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- Imaginary
numbers create a 2 dimensional numbering system. The imaginary numbers are along one axis and
perpendicular to it is the real numbers axis.
Both extending to infinities.
This axis structure creates a complex plane of numbers. Complex numbers are vectors on this plane
with each point defined with a magnitude and an angle, or with two coordinates,
x and y. Every complex number on this
plane has the form a + bi, or z(angle).
The magnitude at an angle are called polar coordinates in the complex
plane.
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- Imaginary
numbers are a misnomer because they really do exist in real life. It is not just a mathematical trick used to
solve a problem. In quantum mechanics
there are wave functions that really are complex valued functions in space-time. The magnitude of the wave function is 2*pi*
the frequency of the wave.
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- The energy of the wave function is wave magnitude * Planck’s constant / 2*pi. 2 pi shows up in a log of these formulas
because it is one time around a circle, one circumference. 2*pi also represents one complete cycle of a
sine wave, and all continuous waves can be constructed using some combination
of sine waves. Planck’s Constant is the
Constant of Action for one complete cycle of a wave. It equals 6.625*10^-34 kilogram-meter^2/second.
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- Numbers and
math have been unusually helpful in describing and understanding our
Universe. Our maker must have been good
at mathematics
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------------------------- Sunday, May 27, 2018
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