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Mathematics: explore STEM content (page 11 of 23) · openstem
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Mathematics
Algebra, calculus, proofs and statistics on openstem — flashcards, notes, quizzes and plots shared by the community. Study free, or clone anything into your library.
Bring in a clock Usually a curve is y in terms of x. Parametric form writes BOTH coordinates in terms of a third variable, the parameter t. As t changes, the point (x(t), y(t)) moves and traces a path. Tracing a curve Take x = t and y = t².
What it is, and what breaks it A rational expression is a fraction with a polynomial on top and a polynomial underneath: (x + 2)/(x − 5). Because a zero denominator is undefined, it comes with restrictions attached. Simplify by factoring, t
What a σ-algebra is for You can't consistently assign a length to every subset of ℝ. A σ-algebra is the collection of sets you agree to measure, chosen so the operations you need stay inside it. The pair (X, ℱ) is a measurable space. A meas
Mathematics
Mathematics · L5 · Measure & the Lebesgue Integral
A compound event combines two or more simple events, like flipping a coin AND rolling a die. Instead of asking about one outcome, you're asking about a combination of outcomes.
The greatest common divisor: the largest whole number that divides evenly into both a and b. For example, GCD(12, 18) = 6, since 6 is the biggest number that divides both 12 and 18.
Mathematics
Mathematics · L3 · Number Theory: GCD, LCM & the Euclidean Algorithm
The most-used and most-misread number in statistics. Here's what it actually says. Definition The p-value is the probability of a test statistic at least this extreme, assuming H₀ is true. That z is a position on the curve below. Slide the
Mathematics
Mathematics · L4 · p-values and statistical significance: what they really mean
A rational number is any number that can be written as a fraction p/q of two integers, where q is not zero. Its decimal form either terminates (stops) or repeats forever in a pattern.
What does it mean for an ordered pair (x, y) to be a solution to a system of two linear equations?1 / 8
It must satisfy BOTH equations at once — plug the x and y values into each equation and check that both come out true. Graphically, it's the point where the two lines cross.
Three routes to the same destination Most theorems say "if P then Q". There's more than one valid way to get there, and the right choice can turn a painful proof into a three-line one. Direct proof Prove: if n is even, then n² is even. Assu
Independent or dependent? The whole topic hinges on one question: does the first outcome change the odds for the second? P(heads AND a six) = 1/2 × 1/6 = 1/12. The coin has no opinion about the die. Conditional probability Read it as: given
Mathematics
Mathematics · L3 · Independent, Dependent & Conditional Probability
Skip counting means counting forward by jumping the same amount each time, instead of counting by ones. Counting by 2s goes 2, 4, 6, 8 — each jump adds 2.
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