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Who invented multiplication?

The use of multiplication tables can be traced to ancient Sumerian civilizations, some 4,600 years ago. The Egyptians practiced multiplication using hieroglyphic techniques dating back to 1600 B.C. The ancient Chinese multiplied using a series of multiplication tables similar to modern ones, as documented in "Zhou Bi Suan Jing." Ancient Indian societies multiplied using a system similar to the modern lattice method.

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What is an orthonormal matrix?

An orthogonal matrix Q is necessarily square and invertible with inverse Q−1 = QT. As a linear transformation, an orthogonal matrix preserves the dot product of vectors and therefore acts as an isometry of Euclidean space. In other words, it is a unitary transformation.\nThe set of n × n orthogonal matrices forms a group O(n), known as the orthogonal group. The subgroup SO(n), consisting of orthogonal matrices with determinant +1, is called the special orthogonal group, and each of its elements is a special orthogonal matrix. As a linear transformation, every special orthogonal matrix acts ..

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What is the midpoint theorem?

A line segment is defined by two endpoints on a coordinate axis. These endpoints have x and y values on the coordinate plane. The midpoint theorem takes the average of the x coordinates and the average of the y coordinates to produce the midpoint of the line segment. For example, a line segment with the endpoints of (6,4) and (2,2) is calculated as follows: (6+2)/2 = 4, (4+2)/2 = 3. Therefore, the midpoint of this segment is located at (4,3). This theorem can also be used to find a unknown endpoint, provided that the midpoint and one endpoint are already known.\nThis theorem can be proved usin..

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What is the inverse of a 2x2 matrix?

A two-by-two matrix has two rows and two columns, with values a and b in the first row and c and d in the second row. Make a new matrix and place the values of d and negative b in the first row, then place the values of negative c and a in the second row. Multiply this new matrix by the value of 1 over the determinant, which is 1 over the quantity a times d minus b times c. The final result of this series of computations is the inverse of the original matrix.\nUsing software such as Mathematica simplifies the tedious process of calculating the inverse of a matrix, while also reducing the chanc..

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What are characteristics of a right triangle?

If the lengths of two sides of a triangle are known, it is possible to calculate the length of the third side using Pythagoras' theorem. This theorem states that the sum of the square of the hypotenuse is equal to the sum of the squares on the other two sides. This theorem only applies to right triangles.\nWhen a right triangle has sides with lengths that are each an integer, it's called a Pythagorean triangle.

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What is an octagon layout calculator used for?

A regular octagon is a closed, 8-sided, geometrical figure where all sides are of the same length and all internal angles at each vertex are of 135 degrees. The common dimensions of an octagon are side length (s), height (h), main diagonal (d), minor diagonal (c), area (A) and perimeter (p).\nSince all its dimensions are interrelated, they can be calculated with just one of those dimensions as a reference.

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What is the integral of tan(x)?

The integral of tan(x) can be solved by rewriting the equation as the integral of sin(x)/cos(x) dx, and then using the integration technique called substitution. Using substitution, the value u is used in place of sin(x), and the value for the derivative of u, du, is found to be -sin(x) dx. Substituting In for sin(x) and dx gives the equation as the negative integral of u over du, which is equal to ln |u| + C. The problem is completed by substituting back in for u.

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What are internally tangent circles?

Circles are internally tangent if the distance between the centers of the circles is equal to the absolute value of the radius of the first circle minus the radius of the second circle. Internally tangent circles have a common internal tangent. A line that is tangent to a circle exists on the same plane as the circle and intersects the circle at exactly one point. A common internal tangent is a line tangent to two circles that also intersects the line that joins the center of the two circles. A common external tangent is a line tangent to two circles, but it does not intersect the line that jo..

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What are some real-world examples of direct variations?

Direct variation can be expressed as an equation, an algebraic expression or a geometric expression. To illustrate the concept of direct variation as an equation, take the equation y/x = 6. Y varies directly with X, with 6 being the constant in the equation. From the equation, it can be determined that Y is always six times greater than X. The equation y = kx is an example of an algebraic equation that illustrates direct variation. Both Y and X are always multiplied by the same amount. This equation is a linear equation. In addition to direct variations, other relationships between numbers and..

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How are vectors used in everyday life?

A vector is a quantity with a direction and magnitude. Newton's first, second and third laws are all vector relationships that accurately describe the motion of bodies under the influence of an external force. Newton's laws are far-reaching, describing objects from a ball in free fall to a rocket headed to the moon.\nUsing a front door as the reference point, suppose a dog bolts from the front door traveling 3 mph due east. The dog has a velocity vector with a magnitude of 3 mph and an eastbound direction. This allows matching his direction and exceeding his speed with a car. Quarter..

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