A number is at least

    • [PDF File]The Pigeonhole Principle - Stanford University

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      Theorem: For any natural number n, there is a nonzero multiple of n whose digits are all 0s and 1s. Proof: For any k in the range 0 ≤ ∈ ℕ k ≤ n, consider S k defined as Now, consider the remainders of the S k 's modulo n.Since there are n + 1 S k 's and n remainders modulo n, by the pigeonhole principle there must be at least two S k

    • [PDF File]Upper and lower bounds, sup and inf

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      Upper and lower bounds: A real number is called an upper bound for Sif x for all x2S. The set Sis said to be bounded above if it has an upper bound. In analogous fashion, one de nes a lower bound, and one calls a set that has a lower bound bounded below.

    • [PDF File]MATH 2513{002 The Least Principle

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      a non-empty subset of N. By the Least Principle there is a least such denominator, q say. This means that there exist positive integers p;q 2N so that p 2 = p q and q is the least denominator among the fractions (). Squaring both sides of the equation p 2 = p q gives 2 = p2 q2 and multiplying across by q2 gives 2q2 = p2

    • [PDF File]PigeonHolePrinciple

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      • Any number in the set S, can be represented as 2kx i, where xi is in X. Solution • There are 100 numbers in X. So, any 101 integers chosen will have at least two numbers who has the same xi. • Let the numbers be 2kx i and 2 lx i (l ≥k). We see the smaller number divides the larger number.

    • [PDF File]The Real Numbers and the Integers

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      • A real number a is said to be positive if a > 0. The set of all positive real numbers is denoted by R+, and the set of all positive integers by Z+. • A real number a is said to be negative if a < 0. • A real number a is said to be nonnegative if a ≥ 0. • A real number a is said to be nonpositive if a ≤ 0.

    • [PDF File]Chapter 3 Pseudo-random numbers generators

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      number generators (at least some of them) including properties that should produce a good random number generator. Obviously, we want a large period, but there are more subtle issues. 3. Fast (and not a lot of memory)Most Monte Carlo simulations require a huge number of random numbers. You may want to generate a large number of samples, and

    • [PDF File]Least squares reminder - Cornell University

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      The condition number therefore is a relative distance to singularity, which is why I keep saying ill-conditioned problems are \close to singular." The SVD and rank-de cient least squares If we substitute A= U VT in the least squares residual norm formula, we can \factor out" Ujust as we pulled out the Qfactor in QR decomposition:

    • [PDF File]The Pigeonhole Principle - York University

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      The number of boxes is k and the number of objects is k+1 or more. By the pigeonhole principle, at least one of these boxes contains two or more elements x of the domain. At least two elements of the domain are assigned to the same element in codomain. So, f cannot be one-to-one.

    • [PDF File]PART I. THE REAL NUMBERS - UH

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      A number u ∈ R is called the supremum (least upper bound) of S, denoted by supS, if it satisfies the conditions 1. s ≤ u for all s ∈ S . 2. If v is an upper bound for S , then u ≤ v. THEOREM 3. Let S ⊆ R be bounded above, and let u = supS. Then, given any positive number , there is an element s ∈ S such that u−

    • [PDF File]Translating English Words Into Algebraic Expressions

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      A number increased by 13 x + 13 5 less than 10 10 - 5 A number decreased by 7 x - 7 Difference between x and 3 x - 3 Difference between 3 and x 3 - x Twice a number 2x Ten percent of x 0.10x Ten times x 10x Quotient of x and 3 x/3 Quotient of 3 and x 3/x Five is three more than a number 5 = x + 3 The product of 2 times a number is 10

    • [PDF File]Rules for Significant Figures (sig figs, s.f.)

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      calculated value will have the same number of digits to the right of the decimal point as that of the least precise quantity. In practice, find the quantity with the fewest digits to the right of the decimal point. In the example below, this would be 11.1 (this is the least precise quantity).

    • [PDF File]TRANSLATING KEY WORDS AND PHRASES INTO ALGEBRAIC EXPRESSIONS

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      imply at least two parts – use parentheses when a sum or difference is multiplied. For example, the phrase "the sum of three times a number and five" translates to "3x + 5," while the phrase " three times the sum of a number and five"

    • [PDF File]Poisson distribution examples

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      number of events (such as the number of telephone calls at a business , number of customers in waiting lines, number of defects in a given surface area, airplane arriva ls, or ... Find the probability that the company receives at least seven death claims on a randomly selected day. P(x ≥ 7) = 1 - P(x ≤ 6) = 0.393697 . Ex. 4.

    • [PDF File]Math Definitions: Introduction to Numbers

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      The result when a number is added to itself. x is a multiple of y if x/y is an integer. 27 is a multiple of 3 (since 27/3 = 9, which is an integer) Least Common Multiple The smallest number that is a multiple of each number. The least common multiple of 25 and 10 is 50 Prime . A positive integer that is divisible by exactly two positive numbers, 1

    • [PDF File]Number Theory - Stanford University

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      This algorithm does not require factorizing numbers, and is fast. We obtain a crude bound for the number of steps required by observing that if we divide a by b to get a = bq+r, and r > b=2, then in the next step we get a remainder r0 b=2. Thus every two steps, the numbers shrink by at least one bit. 3.1Extended Euclidean Algorithm

    • [PDF File]An Introduction to Partial Least Squares Regression

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      perimental PLS procedure for performing partial least squares and related modeling techniques. How Does PLS Work? In principle, MLR can be used with very many factors. However, if the number of factors gets too large (for example, greater than the number of observations), you are likely to get a model that fits the sampled

    • [PDF File]2.3 Bounds of sets of real numbers - Ohio State University

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      A number B is called the least upper bound (or supremum) of the set S if: 1) B is an upper bound: any x ∈ S satisfies x ≤ B, and 2) B is the smallest upper bound. In other words, any smaller number is not an upper bound: if t < B then there is x ∈ S with t < x Notation: B = supS = sup x∈S x Upper bounds of S may, or may not belong to S.

    • [PDF File]Hands On: Compare and Order Whole Numbers

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      Order 1,390,674 and 998,390 and 985,722 from least to greatest. Line up the digits and find the greatest place where they differ. 1,390,674 998,390 985,722 The only number with a digit in the millions place is 1,390,674. It is the greatest number. The first place where the other two numbers differ is the ten thousands place: 8 < 9, so 985,722 ...

    • [PDF File]Fraction Competency Packet

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      (Also called Least Common Multiple, LCM) Mixed Number: A whole number and a fraction. (It implies addition of wholes and parts; that is, 3 5 7 is read "three and five sevenths".) Multiple: (Similar to the "times table.") A multiple of a given number is equal to the given

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