What is Arithmetic: Definition and 476 Discussions

Arithmetic (from the Greek ἀριθμός arithmos, 'number' and τική [τέχνη], tiké [téchne], 'art' or 'craft') is a branch of mathematics that consists of the study of numbers, especially concerning the properties of the traditional operations on them—addition, subtraction, multiplication, division, exponentiation and extraction of roots. Arithmetic is an elementary part of number theory, and number theory is considered to be one of the top-level divisions of modern mathematics, along with algebra, geometry, and analysis. The terms arithmetic and higher arithmetic were used until the beginning of the 20th century as synonyms for number theory, and are sometimes still used to refer to a wider part of number theory.

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  1. B

    How can mathematical induction be used to prove a modular arithmetic problem?

    I'm trying to prove something in modular arithmetic that I came upon across my studies in comp sci. Consider a set of natural numbers {n_{1},n_{2},n_{3},...n_{k}} Consider two more natural numbers m and p such that (\sum^{k}_{i=1}n_{i} ) \ mod \ m = p Now prove that ((((n_{1} \...
  2. F

    [LOGIC] Proof by Induction in Peano Arithmetic

    I have to do the following using these axioms PA1-7, the others below it are previously proved results I can use too. [Sa] means the successor of a. Base Case: y = S0 x.S0 = S0 → x.0 + x = S0 → 0 + x = S0 → x = S0 & y=S0 Now the induction step is usually y=a to y=Sa, however this does...
  3. S

    Maximizing Sum of Finite Sets: Solving Simple Arithmetic Question

    if {x1 , x2 , ...xi} and {y1,y2,...yi} are finite sets. are two sets of real numbers. Then sum Ʃ xixj +yiyj must be maximum, and i≠j so is there some general condition to solve this problem?
  4. D

    A question regarding arithmetic progressions

    Fairly recently someone started a topic here regarding the conjecture of Erdos about arithmetic progressions, namely that if A is a subset of the natural numbers and the sum of the reciprocals of elements of A diverges, then A contains arbitrarily long arithmetic progressions. I'm looking for...
  5. J

    Erdos conjecture on arithmetic progression

    I read this through wikipedia and some other sources and find it to be unsolved. Erdos offer a prize of $5000 to prove it. A mathematician at UW has looked at it and verify them to be correct. However, i still have some doubt about it because the proof i give is pretty simple. Can anyone take a...
  6. E

    Comp Sci Arithmetic overflow in Fortran 95

    Homework Statement Wrote a code using Fortran 95 to solve for an advection-dispersion equation but at the spatial steps specified at dx = 20 m over a total length, L of 20000 m, I keep getting an arithmetic overflow error. I have run this same program at smaller spatial intervals (dx =...
  7. D

    Pigeonhole Principle and the Arithmetic Mean

    My discrete mathematics book gives the following definition for the pigeonhole principle: If m objects are distributed into k containers where m > k, then one container must have more than \lfloor\frac{m-1}{k}\rfloor objects. It then states as a corollary that the arithmetic mean of a set...
  8. S

    Arithmetic sequence, geometric sequence

    Homework Statement Posted this thread earlier but had mis read the given answer. please disregard older thread as I don't know how to delete it! Write down the condition for the numbers p, q, r to form an arithmetic sequence & geometric progression. Homework Equations \ a_n =...
  9. S

    Pascals Triangle, arithmetic sequence.

    Homework Statement Write down the condition for the numbers p, q, r to form an arithmetic sequence. Homework Equations The Attempt at a Solution Have no idea, but I looked at the answer and they have assigned each letter with a given value (number). How is this possible?
  10. A

    How to do arithmetic with an abacus?

    I've recently been into mental math and I'm learning how to do arithmetic mentally without the need of calculators or computers, I know that abacus could be a great help for me now, so I bought one. I've figured out how to add and subtract numbers, I also have figured out how to do...
  11. D

    Inductive proof in complex arithmetic

    Homework Statement Prove that for any n \in \mathbb{N} and x \in \mathbb{R}, we have \sum_{k = 0}^{n} {\cos{(kx)}} = \frac{1}{2}+ \frac{\cos{(nx)} - \cos{[(n+1)x]}}{2 - 2\cos {x}} Homework Equations None I can think of. The Attempt at a Solution Try induction. The result holds if n = 0...
  12. M

    Show Arithmetic Sequence: V0=4, Vn+1=√Vn2+2n+3

    Homework Statement V0=4 V_{n+1}=\sqrt{V_{n}^{2}+2n+3} Homework Equations Show that Un is an arithmetic sequence. The Attempt at a Solution I counted Vn and i found that it equals: V_{n}=\sqrt{(Vn+2)^{2}+2} what is there to do after this?
  13. S

    What is the fourth term of an arithmetic sequence with specific given terms?

    Our 8th grade math counts team met today and I didnt know how to do this problem: The first three terms of an arithmetic sequence are p, 2p+6, and 5p-12. What is the 4th term of this sequence? Please explain how to do this. Arigato!
  14. J

    How to state fundamental theorem of arithmetic in a formal way?

    I think a best informal way to state the theorem is Hardy's: every positive integer (except the number 1) can be represented in exactly one way apart from rearrangement as a product of one or more primes But clearly, this statement does not reveal the structure of the statement in the formal...
  15. T

    Prolog Arithmetic Operations within Function?

    So, I'm trying to learn prolog, and since it's used a lot in the AI community, I thought I would try my hand at implementing a few of the simple "wumpus world" rules. The rule for a "pit" existing at location (2,2) means that a breeze is felt at locations (1,2), (2,1), (2,3), and (3,2). So...
  16. B

    Exploring 3-Digit Floating Point Arithmetic: What is it and How Does it Work?

    what is 3(or i, where i=1,2,3,4...∞) digit arithmetic? is it just working with 3 decimals or 3 significant figures? or is it base 3 arithmetic?
  17. N

    Sequence and series - Arithmetic mean question have been breaking my head

    Homework Statement Two consecutive numbers from 1,2,3...n are removed A.M of remaining numbers is 105/4. Find n and those numbers removed .Homework Equations Answer n = 50 those numbers are 7 and 8 The Attempt at a Solution I solved this question like a few weeks ago but now it escaped my...
  18. Saitama

    Arithmetic Progression question

    Homework Statement If x ε R, the numbers 51+x+51-x, a/2, 25x +25-x form an AP, then 'a' must lie in the interval:- a)[1,5) b)[2,5] c)[5,12] d)[12,∞) Homework Equations Not required The Attempt at a Solution I substituted y=5x. The terms are in AP, so the common difference is...
  19. E

    Modular arithmetic question about functions

    Hello, I'm new to modular arithmetic, but I was wondering - Given that b = R mod(n) where b,R, and n are all integers Is it plausible to consider G(x) = R(x) mod(f(x)) with G,R,f all functions of x? Has this been done or does this not even make sense? If its been done, what is it...
  20. P

    Maximum of arithmetic operations needed

    Hello, I tried to figure out what is the maximum count of arithmetic operation (*,:,+,-) need for gauss elimination and gauss-jordan elimination, but can not get it right. what I get from wikipedia is but I don't understand how to get to this result. Thanks for any help.
  21. N

    Proving the Arithmetic Series Property for x and y When x=! -1, y=! -1, x=! -y

    Homework Statement When x=! -1, y=! -1, x=! -y then x and y are two numbers so that 1/(x+1) + 1/(x+y) + 1/(y+1)... is an arithmetic serie. Show that then also x2 + 1 + y2... must be an arithmetic serie. The Attempt at a Solution I tried to find the differentials between each number in...
  22. B

    Finding two numbers in 9's complement arithmetic

    Homework Statement I have already converted the following numbers 15,765 and -8,773 into 9's complement form. Which gave me the result: 9's Complement of 15,765 = 99,999-84,234 = 84,234 9's Complement of -8,773 = 99,999-(-8,773) = 108,772 Now that I converted the following 5 digit numbers...
  23. A

    Please help on arithmetic mean of continuous distributions.

    PROVE mean (X bar) of a continuous distribution is given by: ∫x.f(x)dx {'a' is the lower limit of integration and 'b' is the upper limit}
  24. J

    Modular Arithmetic Proof with exponents

    Homework Statement Let p be a prime number. Prove: (a+b)^p modp = [(a^p modp) + (b^p modp)]modp Homework Equations modular arithmetic. The Attempt at a Solution I honestly haven't the slightest clue. Would induction be my best bet here? If so, when I suppose the...
  25. J

    Modular Arithmetic Proof: How to Solve with Help | Homework Statement

    Homework Statement Suppose a, b, n are integers with n >/= 2 Prove that: (a + b) mod n = ((a mod n) + (b mod n)) mod n Homework Equations Modular arithmetic rules. The Attempt at a Solution r1 = a(modn) => a = q1n + r1 r2 = bmodn => b = q2n + r2 r1 + r2 = a -...
  26. G

    Understanding the Effects of 2's Complement on Arithmetic Operations

    Why is it that when using 2's complement, the result of arithmetic operations differ by two? 11011001 (-39) + 11100111 (-25) = 11000000 (which is -62 in 2's complement, even though it's supposed to be -64) 00110011 (51) + 11101110 (-16) = 00100001 (33, but it's supposed to be 35)
  27. B

    Boolean Arithmetic Simplification

    Homework Statement I am asked to prove that (\sim x)\vee z = \sim(x\vee y)\vee\sim(y\vee\sim z)\vee\sim(x\vee\sim y)\vee\sim(\sim y\vee\sim z). I've tried using all combinations of DeMoran's rule, the distributive rule to get the y terms together, and the absorption rule to get rid of the...
  28. E

    Modular Arithmetic proofs (multiplication and addition mod n)

    Homework Statement Let n be a fixed positive integer greater than 1. If a (mod n) = a' and b (mod n) = b', prove that (a+b) (mod n) = (a'+b') (mod n) and that (ab) (mod n) = (a'b') (mod n) Homework Equations When a = qn + r a mod n = r The Attempt at a Solution (a'+b') (mod n) = (a...
  29. D

    Calc 3 (parametric equations) My answers won't match/can't find arithmetic error

    two questions. I know I'm doing the work right, but I can't get my answers to match and they are pretty close. I think it's just some arithmetic errors. Help? I've been trying to solve my mistakes forever. I can't find them! Homework Statement Use the parametric equation of an ellipse x =...
  30. J

    Mod. Arithmetic Proof: I don't see flaws in my logic, but it isn't working out.

    1. Homework Statement . 1. Let a and b be constant integers with a \not = 0, and let the mapping f : Z \rightarrow Z be defined by F(x) = ax + b. Determine all values of a such that f is a bijection. Prove that the aforementioned values are the only possible values resulting in a bijection. The...
  31. Y

    Proof of a^{\frac{p-1}{2}}=-1 mod p

    If p is a prime and a an integer coprime with p, why is a^{\frac{p-1}{2}}\equiv -1 mod p ?
  32. J

    Incredibly close to a modular arithmetic proof by minimum counterexample

    As stated in the title, I am trying to prove a statement by minimum counterexample involving modular arithmetic. My problem is producing the contradiction, but I feel so close! (The contradiction is p^m | (1 + p)^{p^{m - 1}} - 1) Homework Statement Let p be an odd prime and let n be a...
  33. Q

    Looking for series of books(maths) arithmetic to calculus(or higher)

    Hi, I am trying to learn maths at home and was wondering if anyone knew of any series(or group of books that you would recommend) of books with the structure: learn arithmetic practice arithmetic learn algebra practice algebra and so on up until calculus or higher I would prefer if the...
  34. K

    How do primes come out of Peano arithmetic?

    Let (N, s(n), 0) be a Peano space. That is, N=\{1,2,3,\dots \} is a set in which http://en.wikipedia.org/wiki/Peano_arithmetic" can be used. We can then define: 0=\varnothing, 1=\{0\}, 2=\{0,1\},\dots \implies n=\{0,1,2,\dots ,n-2,n-1\} s(a)=a\cup \{a\}\implies s(a)=a+1 From here we...
  35. N

    Solving Arithmetic Progressions

    Homework Statement The sum of the first 8 terms of an AP is 56, and the 6th term is 4 times the sum of the 2nd and the 3rd. Find the first term and the common difference Homework Equations The Attempt at a Solution 8th term = 56 6th term = 4x2nd+3rd
  36. B

    Dirichlet's Theorem on Arithmetic Progressions

    Hello, I'm wondering if this is true, or if anyone has seen this before: Let q, t be coprime integers. Then there exist infinitely many primes r such that 1. q is primitive root modulo r and 2. r = q + kt, for some k > 0.If we take away 1, this becomes Dirichlet's Thm...
  37. O

    Arithmetic Sequences and Series

    Homework Statement Just a quick question I was looking to have cross checked… Q. Find un, the nth term of sequence -5, 0, 5, 10,… Homework Equations un = a + (n-1)d The Attempt at a Solution -5 + (n-1)5 -5 + 5n - 5 5n-10 The answer in the book...
  38. K

    Exact Arithmetic using Continued Fractions

    I've recently started development on a continued fraction based exact arithmetic computational package. This is work based on Bill Gosper's HACMEM algorithm and Peter Potts' Mobius transforms with significant modifications. These algorithms have some remarkable properties and can be made much...
  39. V

    Summing up an Arithmetic Progression via Integration?

    Why doesn't the integration of the general term of an A.P. give its sum? Integration sums up finctions, so if I integrate the general term function of an A.P., I should get its sum. Like 2,4,6,8,... T=2+(n-1)2=2n \int T dn=n^2 ..(1)...
  40. S

    How Do You Calculate Workforce Growth and Total Wages Over Time?

    Homework Statement A woman started a business with a workforce of 50 people. Every two weeks the number of people in the workforce increased by 3 people. How many people were there in the workforce after 26 weeks? Each member of the workforce earned $600 per week. What was the total wage bill...
  41. I

    Integration of Modular Arithmetic Functions

    Hello, I have been searching and can't seem to find anything on the topic of integrating modular arithmetic functions. So far I have created an equation for a function in the form of f(x)=mod(x,a): \int mod (x,a) dx=\frac{a(x-mod (x,a))+mod (x,a)^2}{2}+c But, now I am investigating how to...
  42. J

    Arithmetic Question for Finding Derivative using Quotient Rule

    Homework Statement Find dy/dx for the following function: y = (11-cos(x))/(2+cos(x)) Homework Equations Quotient Rule: y'= ((g(x))(f'(x)) - (f(x))(g'(x)))/ (g(x)^2) The Attempt at a Solution I used the quotient rule to come up with this: y'= ((2+cos(x))(sin(x)) -...
  43. X

    Interesting arithmetic sequence

    Given N= 1.2.3 + 2.3.4 + ... + n(n+1)(n+2), prove that 4N + 1 is a square (n is a positive integer)
  44. E

    What Is the Arithmetic Gamma Function \(\gamma_{m}(n)\)?

    In some exercises I've stumbled upon a function which is denoted \gamma_{m}(n) with m,n natural. I've no idea what is the definition of the function and could not infer from the exercises. Searching google yielded nothing, as it kept suggesting me the OTHER Gamma function. Can anyone here help...
  45. H

    Primes and arithmetic progressions.

    The prime number theorem describes the distribution of the prime numbers, in a sense. Are there other prime number theorems corresponding the asymptotic distributions of primes in other arithmetic progressions containing infinitely many primes? I was just wondering.
  46. T

    Question about Dirichlet's theorem on arithmetic progressions

    Greetings. This is my first post, please be gentle! I am a music theorist who uses a lot of math in my investigations of music. I am writing a paper about transposition and multiplication operations by 1, 5, 7 and 11 which have very interesting properties in music. The reason, of course, is...
  47. A

    How Do You Solve These Tricky Arithmetic Sequence Problems?

    Homework Statement Problem 1: Two Arithmetic Sequences are given. a_n = 200,196,192,188,184... [ltaex]b_n = 100,103,106,109,112...[/itex] For integers l,m, find the number of pairs consisting of (l,m) which satifies condition a_l = b_m Problem 2: Three terms, sin(x)...
  48. S

    Introductory Linear Algebra - Modular Arithmetic

    Homework Statement a) For which values of a does ax = 1 have a solution in Z5? b) For which values of a does ax = 1 have a solution in Z6? c) For which values of a and m does ax = 1 have a solution in Zm?Homework Equations None.The Attempt at a Solution Answers: a) All a ≠ 0 b) a = 1, 5 c)...
  49. L

    Can logarithms be applied to Modular arithmetic

    I was just curious. I believe the answer would be no, but I don;t know why
  50. M

    Modular arithmetic and exponents

    I have a homework problem and I use a rule to solve it that seems to be true, at least for small numbers, but I cannot seem to find a clearly stated theorem assuring me that it is true. Here's the problem with my solution: Find the remainder of the division of 2^(36!) by 37. Proof: By...
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