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... Definition 1 ϕ is valid (or logically true, or true by virtue of logic, or a tautology) if v0 |= ϕ for all interpretations v0 . We typically write this as |= ϕ. Thus,|= (p ∧ q) ⇒ p is valid. In fact, more generally, for any ϕ and ψ, |= (ϕ ∧ ψ() ⇒ ϕ. How do you determine if a formula ϕ is valid? We n ...
... Definition 1 ϕ is valid (or logically true, or true by virtue of logic, or a tautology) if v0 |= ϕ for all interpretations v0 . We typically write this as |= ϕ. Thus,|= (p ∧ q) ⇒ p is valid. In fact, more generally, for any ϕ and ψ, |= (ϕ ∧ ψ() ⇒ ϕ. How do you determine if a formula ϕ is valid? We n ...
focus on problem solving 10
... How do we discover patterns? In many cases, a good way to start is to experiment with the problem at hand. How can we be sure a pattern always holds? One way is to use mathematical induction, but there are other ways. In the example we give here we find a pattern for adding the cubes of the first n ...
... How do we discover patterns? In many cases, a good way to start is to experiment with the problem at hand. How can we be sure a pattern always holds? One way is to use mathematical induction, but there are other ways. In the example we give here we find a pattern for adding the cubes of the first n ...
Quadratic Formula
... We look at b2 – 4ac Why is b2 – 4ac so important? It is under a root, and we know that square roots of negative numbers do not exist in the real domain. So if b2 – 4ac < 0 there will be no real solution 2x2 + 4x + 3 = 0 b2 – 4ac = 16 – 4(6) b2 – 4ac is Negative – so x is non real. If b2 – 4ac is a p ...
... We look at b2 – 4ac Why is b2 – 4ac so important? It is under a root, and we know that square roots of negative numbers do not exist in the real domain. So if b2 – 4ac < 0 there will be no real solution 2x2 + 4x + 3 = 0 b2 – 4ac = 16 – 4(6) b2 – 4ac is Negative – so x is non real. If b2 – 4ac is a p ...
5.6 – Quadratic Equations and Complex Numbers
... Classify and find all roots of a quadratic equation. Graph and perform operations on complex numbers. Standard: 2.5.11.C. Present mathematical procedures and results clearly, systematically, succinctly and correctly. ...
... Classify and find all roots of a quadratic equation. Graph and perform operations on complex numbers. Standard: 2.5.11.C. Present mathematical procedures and results clearly, systematically, succinctly and correctly. ...
MATH 117 The Development of Complex Numbers
... first mathematical treatment of the square root of a negative number. He called such values imaginary numbers, which is a typical deragatory name like “negative” numbers or “irrational” numbers. Bombelli was the first to define the arithmetic operations for these numbers. In particular, ...
... first mathematical treatment of the square root of a negative number. He called such values imaginary numbers, which is a typical deragatory name like “negative” numbers or “irrational” numbers. Bombelli was the first to define the arithmetic operations for these numbers. In particular, ...