Exercises Warm Up to the Theory of Computation
... Which of the two is always a subset of the other? Give an example (i.e., say what L1 and L2 are) so that the opposite inclusion does not hold. 4. Sipser, Problem 0.1 5. Sipser, Problem 0.2 6. Sipser, Problem 0.8 7. Sipser, Problem 0.9 ------------------------------------8. Prove that the square root ...
... Which of the two is always a subset of the other? Give an example (i.e., say what L1 and L2 are) so that the opposite inclusion does not hold. 4. Sipser, Problem 0.1 5. Sipser, Problem 0.2 6. Sipser, Problem 0.8 7. Sipser, Problem 0.9 ------------------------------------8. Prove that the square root ...
induction problems - Harvard Math Department
... Thanks to Zach Abel ’10, our resident computational geometer, for finding this construction. Where did we use n > 3? With some more care we can even use this construction to prove that P has an “interior”, that is, the fact (which I relegated in class to an application of the Jordan curve theorem) t ...
... Thanks to Zach Abel ’10, our resident computational geometer, for finding this construction. Where did we use n > 3? With some more care we can even use this construction to prove that P has an “interior”, that is, the fact (which I relegated in class to an application of the Jordan curve theorem) t ...
The Quadratic Formula
... distinct real solutions (if it’s positive), one real solution (if it’s zero), or two complex, but not real solutions (if it’s negative – a topic to be discussed in Math 120). ...
... distinct real solutions (if it’s positive), one real solution (if it’s zero), or two complex, but not real solutions (if it’s negative – a topic to be discussed in Math 120). ...
1 Warming up with rational points on the unit circle
... Geometry of the multiplicative structure Observe what happens when two complex numbers are multiplied together in polar coordinates: we have that z1 · z2 = r1 eiθ1 · r2 eiθ2 = (r1 · r2 )ei(θ1 +θ2 ) so that their radii are multiplied and the angles are added to each other. So, in fact, one can simply ...
... Geometry of the multiplicative structure Observe what happens when two complex numbers are multiplied together in polar coordinates: we have that z1 · z2 = r1 eiθ1 · r2 eiθ2 = (r1 · r2 )ei(θ1 +θ2 ) so that their radii are multiplied and the angles are added to each other. So, in fact, one can simply ...
Arithmetic Sequence (1).notebook
... Arithmetic sequences are linear where the domain consists of natural numbers {1, 2, 3,...} and the range is the terms of the sequence arithmetic explicit formula formula that will work to find any term in an arithmetic sequence What do you think is the explicit formula for an arithmetic sequen ...
... Arithmetic sequences are linear where the domain consists of natural numbers {1, 2, 3,...} and the range is the terms of the sequence arithmetic explicit formula formula that will work to find any term in an arithmetic sequence What do you think is the explicit formula for an arithmetic sequen ...
ppt
... • Our conjecture is that the decimal expansion of p/q will terminate when q = 5x * 2y, where x and y are positive integers. Essentially, this means that the expansion will terminate if q is a multiple of 5 or 2, or a combination of multiples of 5 and 2. Any other value of q will cause the decimal ex ...
... • Our conjecture is that the decimal expansion of p/q will terminate when q = 5x * 2y, where x and y are positive integers. Essentially, this means that the expansion will terminate if q is a multiple of 5 or 2, or a combination of multiples of 5 and 2. Any other value of q will cause the decimal ex ...
Mathematics 220 Homework for Week 7 Due March 6 If
... Because m, n and m + 1 are positive, from the above inequality we conclude that m < n < m + 1. But there is no integer which is strictly between m and m + 1. This contradicts the assumption that n is an integer and proves the statement. 5.36 Let a, b ∈ R. Prove that if ab 6= 0, then a 6= 0 by using ...
... Because m, n and m + 1 are positive, from the above inequality we conclude that m < n < m + 1. But there is no integer which is strictly between m and m + 1. This contradicts the assumption that n is an integer and proves the statement. 5.36 Let a, b ∈ R. Prove that if ab 6= 0, then a 6= 0 by using ...
Exam
... AD = AB, and BD intersecting AC. Choose point E so that ΔADE ≅ ΔABC and AE intersects BC . Find the area common to the two triangles. ...
... AD = AB, and BD intersecting AC. Choose point E so that ΔADE ≅ ΔABC and AE intersects BC . Find the area common to the two triangles. ...
Special Products – Blue Level Problems In
... In Exercises 11 - 13, Rick Claims that if you multiply four consecutive integers together and add 1, you always get a perfect square. 11. Show that Rick’s statement is true if the smallest of the integers is 2. 12. Show that Rick’s statement is true if the smallest of the integers is 3. 13. Suppose ...
... In Exercises 11 - 13, Rick Claims that if you multiply four consecutive integers together and add 1, you always get a perfect square. 11. Show that Rick’s statement is true if the smallest of the integers is 2. 12. Show that Rick’s statement is true if the smallest of the integers is 3. 13. Suppose ...
Number theory
Number theory (or arithmetic) is a branch of pure mathematics devoted primarily to the study of the integers. It is sometimes called ""The Queen of Mathematics"" because of its foundational place in the discipline. Number theorists study prime numbers as well as the properties of objects made out of integers (e.g., rational numbers) or defined as generalizations of the integers (e.g., algebraic integers).Integers can be considered either in themselves or as solutions to equations (Diophantine geometry). Questions in number theory are often best understood through the study of analytical objects (e.g., the Riemann zeta function) that encode properties of the integers, primes or other number-theoretic objects in some fashion (analytic number theory). One may also study real numbers in relation to rational numbers, e.g., as approximated by the latter (Diophantine approximation).The older term for number theory is arithmetic. By the early twentieth century, it had been superseded by ""number theory"". (The word ""arithmetic"" is used by the general public to mean ""elementary calculations""; it has also acquired other meanings in mathematical logic, as in Peano arithmetic, and computer science, as in floating point arithmetic.) The use of the term arithmetic for number theory regained some ground in the second half of the 20th century, arguably in part due to French influence. In particular, arithmetical is preferred as an adjective to number-theoretic.