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So, instead of proving that all positive integers eventually lead to 1, we can try to prove that 1 leads backwards to all positive integers. For any integer n, n ≡ 1 (mod 2) if and only if 3n + 1 ≡ 4 (mod 6). Equivalently, n − 1 / 3 ≡ 1 (mod 2) if and only if n ≡ 4 (mod 6).
The solution set for the equations x − y = −1 and 3x + y = 9 is the single point (2, 3). A solution of a linear system is an assignment of values to the variables ,, …, such that each of the equations is satisfied. The set of all possible solutions is called the solution set. [5]
If x 2 = 9, one may give the solution as x = ±3. This indicates that the equation has two solutions: x = +3 and x = −3. A common use of this notation is found in the quadratic formula =, which describes the two solutions to the quadratic equation ax 2 + bx + c = 0.
The earliest known use of the × symbol to indicate multiplication appears in an anonymous appendix to the 1618 edition of John Napier's Mirifici Logarithmorum Canonis Descriptio. [3] This appendix has been attributed to William Oughtred , [ 3 ] who used the same symbol in his 1631 algebra text, Clavis Mathematicae , stating:
For example, the polynomial x 2 y 2 + 3x 3 + 4y has degree 4, the same degree as the term x 2 y 2. However, a polynomial in variables x and y, is a polynomial in x with coefficients which are polynomials in y, and also a polynomial in y with coefficients which are polynomials in x. The polynomial
Animation depicting the process of completing the square. (Details, animated GIF version)In elementary algebra, completing the square is a technique for converting a quadratic polynomial of the form + + to the form + for some values of and . [1]
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The graph of the logarithm to base 2 crosses the x axis (horizontal axis) at 1 and passes through the points with coordinates (2, 1), (4, 2), and (8, 3). For example, log 2 (8) = 3, because 2 3 = 8. The graph gets arbitrarily close to the y axis, but does not meet or intersect it.