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. K

    Engineering Arithmetic Circuits & Full Adders

    Homework Statement An arithmetic circuit has two select lines S1 and S2 and does the following arithmetic operations using a full adder: [function table] Determine the simplified Boolean functions for Xi and Yi for a single stage of the circuit given inputs A & B are n-bit registers...
  2. G

    Bell's Theorem, EPRB, QM, spin-1/2 particles, basic arithmetic: In reply

    .. This thread is intended to focus on arithmetic issues, mainly raised by JesseM, vanesch, DrChinese, ThomasT, in the context of: "What's wrong with this local realistic counter-example to Bell's theorem?" https://www.physicsforums.com/showthread.php?t=475076 : For newcomers to the issues...
  3. X

    How to solve modulo / modular arithmetic ?

    Homework Statement Find some x such that x\equiv8 mod (18) Find the inverse of 12 modulo 41 Solve 2x=7 mod (13) Homework Equations The Attempt at a Solution I'm not sure how you do this. Can someone explain please? I know it's actually easy... but can't figure it out!
  4. X

    Solving Modular Arithmetic Problems: How to Explained

    How do I solve those problems? Like, Find some x such that x\equiv8 mod (18) Find the inverse of 12 modulo 41 Solve 2x=7 mod (13) I know it's easy, but I don't get it. That's the explanation in the book. I'm not getting it. Can somebody please explain this modular arithmetic...
  5. R

    Cryptography - modulo arithmetic

    Finding log3(4) in Z(11).. Is that the same as finding log3(4)(mod11)?
  6. M

    Arithmetic progression question

    Homework Statement Series Q is an arithmetic series such that the sum of its first n even terms is more than the sum of its first n odd terms by 4n. Find the common difference of the series Q. The answer provided is 4. Homework Equations The Attempt at a Solution I have no ideas on this...
  7. M

    Arithmetic progression homework

    Homework Statement Need help with number (9).. Homework Equations The Attempt at a Solution Can anyone give me some hints? Thanks.
  8. M

    Sum of Positive Integers Less Than 150 Not Multiples of 5 or 7

    Homework Statement Find the sum of the positive integers which are less than 150 and are not multiples of 5 or 7. Homework Equations The Attempt at a Solution I tried it... Can anyone give me some hints or tips...?
  9. E

    Can't find error in my modular arithmetic

    Homework Statement Solve the equation, if a solution exists: 3x = 4 (mod 6) Homework Equations If a = c (mod m), b = d (mod m), then: a * b = c * d (mod m) The Attempt at a Solution There is no solution, because 3x will always result in 0 or 3 for all integers x. However, what I...
  10. F

    Need Help: How is arithmetic synthetic ?

    I am having difficulty in understanding how the statement "7 + 5 = 12 " is a synthetic statement ? (I don't know whether this view is commonly accepted or whether opinion is divided on this issue.But I would like to know arguments put by those in favor of it.) I didn't have trouble in...
  11. J

    CPU has built-in circuitry to do simple arithmetic operations

    Hi I'm a layman so please keep your reply as simple as possible. Thanks. They say that a CPU has built-in circuitry to do simple arithmetic operations such as addition, subtraction, etc. and logical operation such as greater than, less than, etc. I don't understand how could some...
  12. T

    Arbitrary-precision Integer Calculator for bignum arithmetic

    Beaconaut APICalc 2 just released on Jan.18, 2011, which is an arbitrary-precision integer calculator for bignum arithmetic, cryptography analysis and number theory research... http://www.beaconaut.com/forums/default.aspx?g=posts&t=13"
  13. P

    Solve Corey's Arithmetic Sequence Homework

    Homework Statement Corey has take a job with an initial salary of $26,000 an annual raises of $1,250/ (a) What will his salary be in the 6th year? (b) How much money in total will Corey have earned after six years? Homework Equations an = a1 + (n - 1) d Sn = n / 2 (a1 + an)...
  14. P

    Solve for x and Find t10 | Arithmetic Sequence

    Homework Statement For the arithmetic sequence (2 - x), (-6 + 2x), (x + 2), solve for x and find t10. Homework Equations an = a1 + (n - 1) d The Attempt at a Solution Would I have to start off like this below::: an = a1 + (n - 1) d d = (-6 + 2x)-(2 - x) = (x + 2)-(-6 + 2x)
  15. I

    Modular Arithmetic: 8^7 ≡ (-5)^7 (mod 13) and (25)^3 ≡ (-1)^3 (mod 13)?

    I have a simple question: Why does 8^7 ≡ (-5)^7 (mod 13) and (25)^3 ≡ (-1)^3 (mod 13)? In essence I want to show that 8^7 + 5^7 = 13^7, so that both sides of the equation ≡ 0 (mod 13) and therefore 8^7 ≡ (-5)^7 (mod 13). I know that in a field of characteristic p>0, (x + y)^p = x^p +...
  16. S

    What is the Limit of the Arithmetic Mean as n Approaches Infinity?

    Homework Statement prove: lim x_n = L. Then \lim_{n\to\infty}\frac{x_1+\cdots+x_n}{n}=L Homework Equations The Attempt at a Solution i don't know abolutely. i tried definition \left|\frac{x_1+\cdots+x_n}{n}-L\right|=\frac{1}{n}\left|(x_1-L)+\cdots+(x_n-L)\right|
  17. T

    Arithmetic and geometric progression

    If the fourth, seventh and sixteenth terms of an AP are in geometric progression, the first six terms of the AP have a sum of 12, find the common difference of the AP and the common ratio of the GP. I've been assuming that the fourth, seventh and sixteenth terms of the AP are the fourth...
  18. T

    What are the three terms in an A.P. with a sum of 36 and a product of 1428?

    This isn't a homework question, it's in a textbook I have and I'm a bit stumped. I know there's something relatively simple I'm missing so any help would be much appreciated (working too). Three consecutive terms of an A.P. have a sum of 36 and a product of 1428. Find the three terms.
  19. M

    Problems with inverses in arithmetic in ring z

    Homework Statement Calculate 7*11 + 9*11^-1 in the group Z20 Homework Equations The Attempt at a Solution 77+ (9*1/11) in group Z20 77 + 9/11 17 +11x= 20mod+9 My solution was 12, this makes 149 on both sides when you multiply the mod times 7. I am doing independent study...
  20. M

    Solve Arithmetic in Group Z7: Calculating 6-3*5 = 5 or 2? Find Out Here!

    Homework Statement Make four calculations in the group Z7 -------------------------------------------------------------------------------- First, calculate in Z7 6 - 3*5 = Homework Equations The Attempt at a Solution 6-15=6-1=5 =2 The program I am using...
  21. M

    Prove this inequality : Geometric Mean and Arithmetic Mean

    Homework Statement let r_{1}, r_{2}, ... , r_{n} be strictly positive numbers. Suppose an investment of one dollar at the beginning of the year k grows to 1+r_{k} at the end of year k (so that r_{k} is the "return on investment" in year k). Then the value of an investment of one dollar at...
  22. S

    Arithmetic progression of prime numbers

    what is the maximum number of terms can a arithmetic progression of only prime numbers have?
  23. C

    Number Theory with modular arithmetic

    Homework Statement When a is odd, show \frac{a^2-1}{8} is an integer. Then prove by induction n \geq 2 that for all odd numbers a_1,a_2,...,a_n, \frac{(a_1a_2...a_n)^2 - 1}{8} \equiv \frac{a^2_1 - 1}{8} + \frac{a^2_2 - 1}{8} + ... + \frac{a^2_n - 1}{8} \ mod \ 2 Homework Equations The Attempt...
  24. D

    How to Compute the Sum of Arithmetic Series?

    Compute the following sums and give your answers in terms of n. (a) 1 + 2 + 3 + ... + n (b) 2 + 4 + 6 + ... + 2n (c) 1 + 3 + 5 + ... + (2n + 1) I actually have the answers but there seem to be an error... For (a) 1 + 2 + 3 + ... + n, My answer given is n(n+1)/2 For(b) 2 + 4 + 6...
  25. N

    Arithmetic progression/algebra problem

    Homework Statement The three number a, b, c, whose sum is 15, are successive terms of an geometric sequence, and b, a and c are successive terms of an arithmetic sequence. Find the values of a, b and c. Edit: I wrote the question wrong. It now reads correctly. I also fixed my latex code...
  26. I

    Comp Sci Help evaluating boolean number and arithmetic operators C++

    Homework Statement !( ((count<10) || (x<y)) && (count >=0) ) where count is equal to 0Homework Equations i don't think any equations here are necessary except maybe the precedence lawThe Attempt at a Solution they combined 'and' and 'or' which confused the heck out of me. How do i figure out...
  27. N

    Can Arithmetic Alone Prove This Inequality?

    Homework Statement (!) Assuming only arithmetic (not the quadratic formula or calculus), prove that \left\{x \in\Re: x^2-2x-3 < 0\right\} = \left\{x \in\Re: -1 < x < 3\right\} Homework Equations \left\{x \in\Re: x^2-2x-3 < 0\right\} = \left\{x \in\Re: -1 < x < 3\right\}...
  28. E

    Q4 - Arithmetic and Geometric series

    Homework Statement Let a1, a2, a3 denote the first three terms of a geometrical sequence, for which a1 + a2 + a3 = 26. a1 + 3, a2 + 4, a3 - 3 are the first three terms of an arithmetical sequence. Find the first term and the common quotient (ratio) of the geometrical sequence...
  29. MathematicalPhysicist

    Arithmetic quantum gravity and cosmology

    So basically I am about to embark on my Msc in the pure maths department. Now I wish to conduct research on the subject of arithmetic quantum chaos and its applications in number theory and in gravity and cosmology. In my university I found someone who is an expert in its applications to...
  30. H

    What Are the Fundamental Rules and Procedures of Modular Arithmetic?

    I looked at a few modular arithmetic websites and I'm a neophyte when it comes to the legal operations/rules, syntax, and procedures of modular arithmetic. No GRE guide I've seen talks about modular arithmetic (not even the official one!). Following are five relevant questions from my GRE...
  31. J

    Fundamental Theorem of Arithmetic

    Homework Statement Theorem. (Fundamental Theorem of Arithmetic) Ever positive integer n has a prime factorization, which is unique except for reordering of the factors. Homework Equations 6.8 Definition. A prime factorization of n expresses n as a product of powers of distinct primes; the...
  32. H

    Arithmetic Progression: Finding the Sum of Terms with Given Conditions

    Homework Statement the first two terms in an arithmetic progression are 5 and 9. The last term in the progression is the only term which is greater than 200. Find the sum of all the terms in the progression Homework Equations The Attempt at a Solution I want to ask : what is the...
  33. L

    How get the significant difference from arithmetic means and SD?

    The raw data are lost, I have only final outputs (arithmetic means and standard deviations) for equally long measurement series. How I can compare results (calculate results) for the purpose to see significant statistic difference (significant difference, p<0.05)? Thank you in advance.
  34. M

    How many hours does it take to fill a 16m x 7m x 7m tank with water?

    Homework Statement water fills a tank at a rate of 150 litres during the first hour, 350 litres during the second hour, 550 litres during the 3rd hour and so on. find the number of hours neccesary to fill a rectangular tank 16m x 7m x7mHomework Equations l=a+(n-1)d S= n/2 (a+l) where: l =...
  35. E

    Is the Calculation of Arithmetic Means Between Two Numbers Always Intuitive?

    Homework Statement A question gives the problem find the two arithmetic means between 4 and 19. The answer is 9 and 14. Homework Equations (a1+a2+a3+an)/n The Attempt at a Solution Logic would dictate that the arithmetic mean would be adding 4 and 19 then dividing by two. Leaving...
  36. N

    Modular arithmetic in matrices

    This is the question: What must fulfill a matrix to be invertible in module Zn? Demonstrate. Z refers to integers. I really appreciate that someone could help me with this because i couldn't find strong information about it. I think that considering A as a matrix... the det(A) must be coprime...
  37. H

    Sequence of integers and arithmetic progressions

    given any two numbers k,j what is the largest sequence of integers such that the sum of any k consecutive terms is negative and the sum of any j consecutive terms is positive and how may we find a subset containing k of the first n numbers such that out of all subsets with k elements, this...
  38. P

    Can a Basis Be Proven with Divisibility?

    Homework Statement Hi everyone. I'm studying a problem and I need to prove that I have a basis. I tryed a proof and to achieve it I need to show that : if k divides a*b and also divides a2 +2*b2 Then k divides both a and b. Homework Equations I'm not sure what I'm asserting is...
  39. E

    Mathematical induction and arithmetic progression

    Homework Statement All the terms of the arithmetic progression u1,u2,u3...,un are positive. Use mathematical induction to prove that, for n>= 2, n is an element of all positive integers, [ 1/ (u1u2) ] + [ 1/ (u2u3) ] + [ 1/ (u3u4) ] + ... + [ 1/ (un-1un) ] = ( n - 1 ) / ( u1un)...
  40. W

    Arithmetic in Mathematical Fields (in particular GF(2^8) )

    Homework Statement * note that I meant fields in the abstract mathematical sense, notphysical (i.e. electric) fields! * Finding the determinant of a matrix in GF(28) I want to know if it is using mod 256 or mod 255 in the field of GF(28) Homework Equations Just math The...
  41. P

    Verifying the Sum of an Arithmetic Progression: Is it -(p+q) or +(p+q)?

    my book says that if sum of p terms of an ARTHMETRIC PROGRESSION is q and sum of q terms is p , then sum of p+q terms will be -(p+q) , but i am getting it as +(p+q), can someone verify it ?
  42. S

    Can we define arithmetic sequences in R^2 using two-dimensional indices?

    Hi all, I was just wondering whether one could define arithmetic sequences in R^2 in a simmilar manner as in R.? Here is what i see as a natural way of doing it, but neither have i read about it, nor heard. \mbox{ Let } x_n \in R^2 \mbox{ be a sequence given as follows : } x_n=a+mb\\...
  43. T

    Fundamental Theorem of Arithmetic

    Homework Statement Using the Fundamental Theorem of Arithmetic, prove that every positive integer can be written uniquely as a power of 2 and an odd number. Homework Equations The Attempt at a Solution Since the FTOA states that any integer can be written as a product of primes...
  44. C

    Consistency of arithmetic Mod N

    "[URL consistency of ordinary arithmetic has not yet been satifactorily settled[/URL]. What is the upper limit for N such that arithmetic modulo N is known to be consistent?
  45. C

    Numerical analysis, floating-point arithmetic

    Hi all, this (probably easy) problem from numerical analysis is giving me trouble. I can't seem to get started and need some poking in the right direction. Homework Statement Consider the following claim: if two floating point numbers x and y with the same sign differ by a factor of at...
  46. P

    Seemingly difficult complex arithmetic problem

    Homework Statement Transmission of a quantum mechanical wave past a one-dimensional square well results in the following expressions relating initial to final wave amplitudes: A= (cos(2ka)-{i\epsilon\over2}sin(2ka))e^{2ia\lambda}F+{i\eta\over2}sin(2ka)G B=...
  47. T

    Arithmetic mean of two unknowns in a system of equations

    Homework Statement When finding the arithmetic mean in a system of equations is there any reason why the method that I am using is wrong? Find the arithmetic mean of x and y in the following set of equations Homework Equations 3x + 5y = 65 and 7x + 14y = 175 The Attempt at...
  48. W

    Predicting Blink Times in an Arithmetic Series

    Need help with the following: Imagine you have 2 lights and a timer. You start the timer, and the first light blinks at 13 seconds and the second light blinks at 15 seconds. At 24 seconds the first light blinks again, and the second light blinks again at 28 seconds. Following this pattern...
  49. B

    Finding the First Term and Common Difference of an Arithmetic Series

    Homework Statement The sum of the first 4 terms in an arithmetic series is -8 and the sum of the first 5 terms is 85. Determine the first term and the common difference. Homework Equations tn = a + (n-1)d Sn = n/2 (2a + (n-1)d) The Attempt at a Solution
  50. F

    Arithmetic Sequence Homework: Find x for Consecutive Terms

    Homework Statement Find x so that x+5, 3x+1, and 4x+1 are consecutive terms of an arithmetic sequence. Not really sure how to do the problem at all. Some assistance would be much appreciated.
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