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Algebra

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Exponents

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Since an exponent on a number indicates multiplication by that same number, an exponent on a negative number is simply the negative number multiplied by itself a certain number of times: (- 4)3 = - 4× -4× - 4 = - 64 (- 4)3 = - 64 is negative because there are 3 negative signs--see Multiplying Negatives. (- 5)2 = - 5× - 5 = 25 (- 5)2 = 25 is positive because there are 2 negative signs.

French Revolution

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A leading cause of social stress in France during the Revolution was its large population. At the beginning of the eighteenth century, France had 20 million people living within its borders, a number equal to nearly 20 percent of the population of non-Russian Europe. Over the course of the century, that number increased by another 8 to 10 million, as epidemic disease and acute food shortages diminished and mortality declined. By contrast, it had increased by only 1 million between 1600 and 1700. Also important, this population was concentrated in the rural countryside: of the nearly 30 million French under Louis XVI, about 80 percent lived in villages of 2,000 or less, with nearly all the rest in fairly small cities (those with fewer than 50,000 inhabitants).

Complex Numbers

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Complex Numbers Introduction If we try to solve x2 = -1, what happens? We extract square roots to get x = +/- (-1. But if we try to evaluate the square root of ?1 on a scientific calculator, we get ERROR! But still, we need a way to define solutions like this so it is defined that i2 = -1 and thus i = ((-1). This means that the solutions of x2 = -1 are x = i and x = -i We refer to such solutions as Complex Solutions. Furthermore, we refer to a number containing the quantity ?i?, where i = (-1, as an imaginary number. This choice of words ?imaginary? is actually not appropriate, since we use the number ?i? in many real-world engineering applications! Using Complex Numbers To Evaluate Square Roots

algebra history

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The history of algebra began in ancient Egypt and Babylon, where people learned to solve linear (ax = b) and quadratic (ax2 + bx = c) equations, as well as indeterminate equations such as x2 + y2 = z2, whereby several unknowns are involved. The ancient Babylonians solved arbitrary quadratic equations by essentially the same procedures taught today. They also could solve some indeterminate equations.

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