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5⋅5, or 5 2 (5 squared), can be shown graphically using a square. Each block represents one unit, 1⋅1, and the entire square represents 5⋅5, or the area of the square. In mathematics, a square is the result of multiplying a number by itself. The verb "to square" is used to denote this operation.
It is possible that n · 1 = 1 + 1 + ... + 1 (n times) can be zero. If n is the smallest positive integer such that this occurs, then n is called the characteristic of R . In some rings, n · 1 is never zero for any positive integer n , and those rings are said to have characteristic zero .
The concept of unit circle (the set of all vectors of norm 1) is different in different norms: for the 1-norm, the unit circle is a square oriented as a diamond; for the 2-norm (Euclidean norm), it is the well-known unit circle; while for the infinity norm, it is an axis-aligned square.
Consider the vectors (polynomials) p 1 := 1, p 2 := x + 1, and p 3 := x 2 + x + 1. Is the polynomial x 2 − 1 a linear combination of p 1, p 2, and p 3? To find out, consider an arbitrary linear combination of these vectors and try to see when it equals the desired vector x 2 − 1. Picking arbitrary coefficients a 1, a 2, and a 3, we want
The greatest common divisor (GCD) of integers a and b, at least one of which is nonzero, is the greatest positive integer d such that d is a divisor of both a and b; that is, there are integers e and f such that a = de and b = df, and d is the largest such integer.
For example, if x 2 − x − 1 = 0, y 3 − y − 1 = 0 and z = xy, then eliminating x and y from z − xy = 0 and the polynomials satisfied by x and y using the resultant gives z 6 − 3z 4 − 4z 3 + z 2 + z − 1 = 0, which is irreducible, and is the monic equation satisfied by the product.
However, this function is not injective (and hence not bijective), since, for example, the pre-image of y = 2 is {x = −1, x = 2}. (In fact, the pre-image of this function for every y, −2 ≤ y ≤ 2 has more than one element.) The function g : R → R defined by g(x) = x 2 is not surjective, since there is no real number x such that x 2 = −1.
This coefficient, usually represented by ρ (rho) for the population and r for the sample, assumes values between −1 and 1, where ρ = 1 represents a perfect positive correlation, ρ = −1 represents a perfect negative correlation, and ρ = 0 is no linear correlation. [10]
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