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In Geometry

in automatic control the nyquist theorem is used to determine if a system is stable or not. there is also something called the simplified nyguist theorem that says if the curv…e cuts the "x-axies" to the right of point (-1,0) then the system is stable, otherwise its not. (MORE)

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The hypotenuse leg theorem states that any two right triangles that have a congruent hypotenuse and a corresponding, congruent leg arecongruent triangles.

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A lemma, or a subsidiary math theorem, is a theorem that one proves as an interim stage in proving another theorem. Lemmas can be viewed as scaffolding for the proof. Usually,… they are not that interesting in and of themselves, but there are exceptions. See the related link for examples of lemmas that are famous independently of the main theorems. (MORE)

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Milller's Theorem is used to simplify a circuit for circuit analysis. Instead of one impedance, which connectes two non-grounded nodes, Miller's Theorem allows this impedance …to be broken down into two parallel impedances. One impedance can be seen as Z/(1-A) and the other impedance can be simplified to Z/(1-(1/A)). In this case, Z was the value of the original impedance, and A is the gain of the amplifier being analyzed. (MORE)

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The Pythagorean theorem can be done this way. a²+b²=c² lets say that you have a triangle with three sides, but you are only given two. Their values are 3 and 4. Now you… have to fill in the values with a=3 and b=4 (doesn't matter which order you put it in) 3²+4²=c² c is still unknown so we have to do the next step. 3² is 9 and 4² is 16. knowing this, we have to do this next: 9+16=c² 9+16 is 25. 25=c² now you must get rid of the ². you do this by using the square root. don't ask me why you square root, that's just how the Pythagorean theorem works. √25=√c² the square root gets rid of the c squared so its just the square root of 25. 5=c triangle sides: 3,4,5 The process can also be reversed. a²=c²-b² or b²=c²-a² P.S: Please recommend using button below, thank you. (MORE)

As it turns out, these stars actually regret the roles responsible for launching their careers into unforgettable stardom. After you read our explanations, perhaps you'll unde…rstand why. (MORE)

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When it comes to basic facts, what you don't know can hurt you, or at the very least surprise you.… (MORE)

As the saying goes, you can't teach an old dog new tricks. Some tricks, however, are so simple that even an old dog or new puppy can learn them. Just practice any of the follo…wing a few times a day with your dog. (MORE)

While traveling with a dog down the open road makes for good memories, doing it properly takes some thought. Help you and your dog enjoy the time you spend journeying together… by keeping these tips in mind. (MORE)

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In Technology

In The Middle Of 19th Century ,an English mathematician George Boole developed rules for manipulations of binary variables, known as Boolean Algebra. This is the basis of all …digital systems like computers, calculators etc. 0and 0=0 0 and 1=0 1 and 0=0 1 and 1=1. He did not develop his Boolean Algebra until he became a professor in Ireland. (Cork I think.) There is a well developed article in wikipedia.org. (MORE)

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In Geometry

Since the 4th century AD, Pythagoras has commonly been given credit for discovering the Pythagorean theorem, a theorem in geometry that states that in a right triangle the… square of the hypotenuse (the longest side, opposite the right angle) is equal to the sum of the squares of the other two sides. . In other words, given a right-triangle with side lengths a and b, and a hypotenuse of length c, then a2 + b2 = c2 Example: Given a right-triangle with side a = 3 and side b = 4, what is the hypotenuse (side c)? Solution: a2 + b2 = c2 32 + 42 = c2 (3 x 3) + (4 x 4) = c2 9 + 16 = c2 25 = c2 c = √25 c = 5. (Note that c can't equal -5 because c is the length of the hypotenuse of the triangle and length must be positive.) The importance of the theorem goes way beyond triangles, in fact the Pythagorean theorem is the basis for the definition of distance between two points in space of any dimension of size 2 or more. (There is a related link to 81 different proofs of this theorem.) Euler Improvement The mathematician George Euler improved the Pythagoras theorem to apply to all triangles using the cosine of the included angle: a2 + b2 -2abcosT= c2 where T is the angle between a and b and cos the goniometric function. (The cosine of 90Â° is 0 which makes this the Pythagoras theorem.) Example IF BC=A=5CM=base of right angle, and AB=B=6CM the perpendicular and AC=C=the hypotenuse. (HYP)2=(BASE)2+(PERP)2 C2=A2+B2 So we have: C2=25+36 C2=61 Now we use the square root property but take the positive square root. So C is approximately equal to 7.81 CM Generalizing the theorem to higher dimensions The Pythagorean Theorem works in higher dimensions too. If you have three legs, each one in a different dimension, and each at right angles to the other two, the hypotenuse joining these three lines has a length which equals sqrt(a2+b2+c2). You can't have four mutually right legs in three dimensions, but you can in four dimensions, in which case h=sqrt(a2+b2+c2+d2) and so on. (MORE)

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The importance is that the sum of a large number of independent random variables is always approximately normally distributed as long as each random variable has the same dist…ribution and that distribution has a finite mean and variance. The point is that it DOES NOT matter what the particular distribution is. So whatever distribution you start with, you always end up with normal. (MORE)

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I guess, because one man had an enquiring mind and somehow discovered the relationship between the squares of the sides of triangles. Also they needed to ensure they had r…ight angles for building and the 3-4-5 right angle triangle was perfect for this. (MORE)

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In Geometry

The hypotenuse angle theorem, also known as the HA theorem, states that 'if the hypotenuse and an acute angle of one right triangle are congruent to the hypotenuse and an …acute angle of another right triangle, then the two triangles are congruent.' (MORE)