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----------------------------- 2390 - MATH - The Greeks Invented Numbers
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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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- The Pythagoreans also discovered “prime numbers”. These are numbers that can not 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, an infinite number, are all 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.
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- The products of two prime numbers can never be a perfect square. Other than 2,3,5,7 all other prime numbers end in 1,3,7,9. And, there are an infinite number of these.
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- 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 can not be expressed as a ratio of two integers. Square root of 2, pi, are “irrational numbers”.
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- 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 2^0.5, 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.
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- Where a and b are the smallest positive integers that can satisfy this equation 2^0.5 = a/b.
- a and b can not 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.
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- But, since 2*a^2 = b^2. Then b^2/2 = a^2. Thus, a^2 must be even, and therefore b is even. We had previously determined that b must be odd. +b and 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 unto 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 that 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.
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- 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.
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- 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 and 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.
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- 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.
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- 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. 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.
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- 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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- June 2, 2019 649
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