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---------------- 1865 - Equations are just another language to learn.
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- The language of mathematics is a shorthand. It was created to make things simpler to understand and to manipulate. Learn the new vocabulary in math and how equations can be transformed into longhand, words and sentences. For example:
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---------------------------- h^2 = a^2 + b^2
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---------------------------- The square of the hypotenuse of a right triangle is equal to the sum of the squares of the other two sides.
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- Equations are designed to simplify ideas and theories One of the simplest ideas was discovered 100 years ago.
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-------------------------- E = m
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------------------------- Energy = mass
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- This equation says that energy and mass is the same thing. To be an equality we need a constant of proportionality depending on the units used. Mass in kilograms. Energy is kilograms * meters^2 / seconds^2.
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-----------------------kg * m^2 / sec^2 = kg ( constant )
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-------------------- The constant must be m^2 / sec^2, which is the square of velocity. The genius of Einstein was to realize the constant velocity was the speed of light ( c ).
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---------------------------- E = m*c^2
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- The equation in the first paragraph involved a simple right triangle. Take 2 sticks connected at an angle, a vertex. Measure 3 feet along one stick and make a mark. Measure 4 feet along the other stick and make a mark. Measure 5 feet between the marks and the angle at the vertex is exactly 90 degrees, perpendicular, a right angle.
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- This is the Pythagorean Theorem.
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--------------------(3)^2 + (4)^2 = (5)^2
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------------------- 9 + 16 = 25
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------------------- a^2 + b^2 = h^2
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-This simple equation helped build the pyramids and was around 1,000 years before Pythagoras , a Babylonian, made it famous. Today it is still the core of geometry, trigonometry , surveying, map making , navigation, even art, and of course our GPS satellite navigation system. The Pythagorean Theorem works in plane geometry. A right angle on the surface of a sphere, the Earth’s surface for example, does not work with this theorem. In that case the 3 angles of the triangle does not add up to 180 degrees.
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- There at least 6 different proofs of the Pythagorean Theorem. My favorite uses geometry. Puts a right triangle inside a trapezoid. Doubles the trapezoid to form a rectangle. One side being the hypotenuse.
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---------------- ½ ab + ½ ab + ½ c^2 = h ( a+b) / 2
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-------------------- h = a+ b
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----------------- c^2 = a^2 + b^2
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- Logarithms are another math invention to make life simpler. Numbers can be multiplied and divided by adding and subtracting their exponents with a common base. The common base can be 10, or 2, or “e”. Reference tables are created that make multiplying large numbers less tedious. Just add up the exponents.
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- Today logarithms are used to model compound interest and exponential biological growth, and radioactive decay.
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- Logarithms are the inverse operations to exponentials. For logarithms to the base 2:
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------------------- y = log 2(x)
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------------------ 64 = 2^6
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------------------Log 2 (64) = 6
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- Binary logarithms are used in computer science Natural logarithms using the base “e”, 2.71828.……. Are used in physics.
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--------------- Log ( x*y) = Log x + Log y
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- Calculus was a math invented to calculate instantaneous rates of change. The inventors were Isaac Newton and Gottfried Leibniz. Calculus is the foundation of many of Nature’s laws. It is the source of differential equations. It is used to measure solids, curves, and areas. It s used to calculate the optimal solutions in medicine, economics, physics, engineering and computer science.
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- A linear function is the equation y = mx +b
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- “m” is the slope or “the rate of change” of y with respect to x at the point of intersection of “a” and “b” coordinates.
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---------------------- m = delta y / delta x = change in y / change in x
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--- Differential----------------- d(x^2) / dx = 2x
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----Integral ----------------------- 2x*dx = x^2 + a constant.
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- The equation for gravity:
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-------------------- F = G * m*M /r^2
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- This equation calculates the force of gravity between two objects ( m and M ). Isaac Newton, Johannes Kepler, and Robert Hooke all had a hand in its creation. Newton’s law descries how gravity works everywhere from the baseball field to the evolution of galaxies. An enhancement to Newton’s law was introduced by Albert Einstein’s Theory of Relativity which is needed in dealing with massive objects or motion approaching the speed of light.
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- Complex Numbers were needed to work with the square roots of negative numbers. The square root of (-2) has no “real” solution. There is no real number multiplied by itself that has the answer (-2). The imaginary solution to the square root of (-4) is (2i) Math using these imaginary numbers is essential in working with electrical systems and a variety of modern data processing algorithms.
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- Euler’s formula for polyhedral describes a numerical relationship for solid shapes. Polyhedral are 3-dimensional versions of polygons. The corners are “vertices”, the connections lines are “ edges”. a cube had 8 vertices, 12 edges, and 6 “faces”. If you add the vertices and faces together ( 8 + 6) and subtract the edges ( 12) you will always get the number 2.
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------------------------ 8 + 6 -12 = 2
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- This concept is used extensively in the science of topology, which is the geometry of continuous surfaces. This scientific tool is also used to study DNA, and networks like the internet.
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--------------------- e^ix = cos x + i*sin x
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------------------- “e” is the base of natural logarithms
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------------------ “i” is the imaginary unit
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- The Normal Distribution is a bell shaped curve that has defined probabilities at points near the average and extending to the edges. 68% of the values in the distribution are within one “standard deviation” of the mean. 95% of the values are within 2 standard deviations. 99.7% are within 3 deviations.
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- It came from the initial work of Blaise Pascal. The Normal Distribution has become the foundation for modern statistics. The curve mathematically describes purely random occurrences, like flipping a coin, heads or tails. Today it is used to determine whether drugs are sufficiently effective in clinical trials.
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- The Wave Equation is a differential equation that describes the behavior of waves, for example the vibrating guitar string and the resulting sound wave. It works equally well with earthquakes and ocean waves. Oil , gas, mining companies set off explosives and read reflecting data in sound waves to discover geological formations.
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- To describe a sinusoidal wave you need a “ second - order linear partial differential equation”:
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-------------------------- d^2 f(x) / dx^2 = - (w/c) f(x)
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- Fourier Transforms describe patterns of time as a function of frequency. The equation can describe complex wave patterns in heat flow, music, speech, images and signal analysis. Today it is used to compress the JPEG image format in your cell phone and describe the structure of molecules. ( Charles Fourier 1772 - 1837 ).
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- The Navier-Stokes equations are used to describe how fluids work. The left side of the equation is the acceleration of fluid and the right side is the forces that act upon it. It is applying Newton’s 2nd law of motion to fluid motion. The equations are used to model the weather, ocean currents, water flow in a pipe, air flow around a wing. It allows the design of vehicles that are more aerodynamic.
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- Maxwell’s equations are the ones I spent 3 years working on in electrical engineering school. They define the relationship between electric and magnetic fields. Maxwell then realized that light was an electromagnetic wave. Today we owe these equations to the development of radar, television, and modern communications.
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- The 2nd law of Thermodynamics defines how heat and energy dissipate over time. The concept of “ Entropy” is a measure of disorder, or randomness, in the system. The sum of entropies must continually increase over time. The simplest example is that of heat flow from hot to cold until equilibrium is reached. The 2nd law expresses the irreversibility of this process. Thermodynamics is an essential tool in the understanding of chemistry and engines.
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- Einstein’s Theory of Relativity says energy and matter are 2 forms of the same thing, E = mc^2. “c^2” is the speed of light squared. “c” is a universal speed limit such that space / time must change to keep “c” constant in all forms of motion.
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- Schrodinger’s equation. In 1924 Louis de Broglie determined the dual nature of matter being both particles and waves. Erin Schrodinger (1867 - 1961) in 1924 defined the math that describes how particles exist over a range of probabilities. This today is the modern Quantum Mechanics that matches all observations and predictions in physics. The equation is used in nuclear power, semi-conductor based computers, lasers, and other quantum phenomena.
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- Shannon’s Information equation estimates the amount of data in a piece of code by the probabilities of its components. It measure the information content of a message or anything represented symbolically. The Shannon Entropy is how much the message can be compressed without losing content. It is used in source coding given a noisy communications channel.
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- Chaos Theory equations are used in logistic models of population growth. Chaos is a normal consequence of differential equations. Processes evolve based on their initial conditions. Weather is the best known example of a small change in initial conditions resulting in a completely different weather system days later. (popularly known as the butterfly effect). It is also used to model earthquakes.
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- The Black-Scholes equations were created for the multi-trillion dollar financial derivatives market. Improper use of the formula created a financial crisis. Variants of the model are still in use to price derivatives today.
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- And , we still have not gotten to F = ma. Force = mass * acceleration. An object in orbit at a constant speed is accelerating. Velocity is speed AND direction. A circular orbit is constantly changing directions. And, changing velocity is acceleration. The centripetal force is exactly balanced by the pull of gravity.
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----------------------- F = G*m*M / r^2 = m*a = m*v^2 / r
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----------------------- So if you can measure the orbital speed and the distance from the larger mass , “M“ , you can calculate the mass inside the orbit.
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- Or, e^i*pi + 1 = 0, An “ irrational number” raised by the power of and “ imaginary number” plus a normal number 1 = zero. Who ever thought of that?
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- Leonard Euler. ( 1707 - 1783)
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- or, Heisenberg’s Uncertainty principle. Equations are a simplification to help us understand complex things. Simplification helps us know more and more about less and less until we know everything about nothing.
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----- 707-536-3272 ---------------- Thursday, April 28, 2016 -----
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