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Browse Classroom Capsules and Notes

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Displaying 981 - 990 of 1214

The author finds visually sums of squares and sums of odd squares.
Given positive integers \(n\geq 3\), the author finds a single inequality condition for every \(r\) of them (\(3 \leq r \leq n \)) to be the lengths of sides of a \(r\)-gon.
The author presents a geometric proof of Machin's formula, which expresses \(\frac {\pi}{4}\) as the difference of two inverse tangent functions.
The author proves visually the law of cosines.

The author gives two applications of a method for finding a function \(g\) such that \(f(g(x)) \geq f(x)\).

The author visually expresses the cube of a number as an arithmetic sum.

A visual determination of the area of a right triangle is given using an inscribed circle.

An odd prime \(p\) has \((p-1)/2\) quadratic residues mod \(p\), and for relatively prime \(p\) and \(q\) there are \((p-1)(q-1)/2\) non-representable Frobenius numbers. The author discusses a...

An old and forgotten method for determining whether a quartic polynomial over the rationals is reducible is revived.
The Butterfly Theorem is generalized to include Sidney Kung`s Butterfly Theorem.