2.15 A metric space is called separable if it contains a countable
... such that (z, y) ⊂ G. Let b = sup{y : (x, y) ⊂ G} and a = inf{z : (z, x) ⊂ G}. Then −∞ ≤ a < x < b ≤ ∞. Put Ix = (a, b). It is clear that Ix is a segment. We claim that b ∈ / G. In fact, there is nothing to prove if b = ∞. If b is finite, and b ∈ G, then there is some δ > 0 such that (b − δ, b + δ) ...
... such that (z, y) ⊂ G. Let b = sup{y : (x, y) ⊂ G} and a = inf{z : (z, x) ⊂ G}. Then −∞ ≤ a < x < b ≤ ∞. Put Ix = (a, b). It is clear that Ix is a segment. We claim that b ∈ / G. In fact, there is nothing to prove if b = ∞. If b is finite, and b ∈ G, then there is some δ > 0 such that (b − δ, b + δ) ...
9.2 The Pythagorean Theorem
... root. Note: There are no negative square roots until you get to Algebra II and introduced to “imaginary numbers.” ...
... root. Note: There are no negative square roots until you get to Algebra II and introduced to “imaginary numbers.” ...
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... v 7→ qv of U thus gives rise to a permutation of the real sphere. It turns out that that permutation is a rotation. Its axis is the line through (0, 0, 0) and (a, b, c), and the angle through which it rotates the sphere is θ. If rotations F and G correspond to quaternions q and r respectively, then ...
... v 7→ qv of U thus gives rise to a permutation of the real sphere. It turns out that that permutation is a rotation. Its axis is the line through (0, 0, 0) and (a, b, c), and the angle through which it rotates the sphere is θ. If rotations F and G correspond to quaternions q and r respectively, then ...
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... 1) You are driving to visit a friend in another state who lives 440 miles away. You are driving 55 miles/hour and have already driven 275 miles. What is the equation and how much longer, in hours, must you drive to ...
... 1) You are driving to visit a friend in another state who lives 440 miles away. You are driving 55 miles/hour and have already driven 275 miles. What is the equation and how much longer, in hours, must you drive to ...
