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

    Arithmetic shift divison question

    If I have the following function: a = b * c/255 The following function is apparently equivalent using only shifts: product = b * c; a = (product + (product>>8) + 1)>>8; I am having trouble following how this function works. Since an arithmetic right shift is division by a power of...
  2. D

    Arithmetic and geometric means

    Homework Statement http://img264.imageshack.us/img264/7505/math.png Homework Equations AM = arithmetic mean = (a+b)/2 GM = geometric mean = sqrt(ab) The Attempt at a Solution I'm totally stuck on this, substituting does not help at all.
  3. B

    How Do Nickels Add Up in Arithmetic Sequences?

    a few questions ... 25. Write an equation to show how the amount of money in a jar of nickels is related to the number of nickels in the jar. If the jar contains 40 nickels, how much money is this? (Hint: Define the variables that are used in your equation. Use your equation to to determine...
  4. H

    Binomial Theorem and Modular Arithmetic Proof Check

    Homework Statement \mbox{Prove or give a counterexample: If p is a prime integer, then for all integers x and y, } (x+p)^p \equiv_p x^p+y^p. Homework Equations \equiv_p \mbox{just means (mod p). Can you please check and see if this proof is well-formed?} The Attempt at a Solution...
  5. S

    Validate Peano Arithmetic Approach

    Homework Statement Show that the natural numbers satisfy commutativity of multiplication and distributivity of multiplication over addition. Homework Equations The Attempt at a Solution I'm wondering if there is any potential circularity in this reasoning. I proved distributivity...
  6. D

    Arithmetic operations on sequences

    Homework Statement If the sequence {a_n} n=1 to infinity converges to (a) with a_n >0 show {sqrt(a_n)} converges to sqrt(a) Homework Equations hint: conjigate first The Attempt at a Solution abs[ (a_n-a) / (sqrt(a_n)+sqrt(a) ) ] < epsilon i do not own LATEX, yet.
  7. A

    Can Arithmetic Operations with Infinities Be Undefined?

    Am I correct in assuming that you can make sense of \infty + \infty and \infty + c for any c\in \mathbb{R} (both evaluate to \infty), but that we can make no sense of \infty - \infty? Are there any other arithmetic operations one can perform with infinities that are undefined?
  8. F

    Fundamental Theorem of Arithmetic Problem

    Problem number 4 on the image has me stumped. I understand the problem (obviously not enough) and what its saying, I'm just having trouble putting it into a proof. Can i get a hint to get me started? Thanks http://img40.imageshack.us/i/asdasdjql.jpg/"
  9. I

    Modular Arithmetic - RSA Encryption

    I got curious about RSA Encryption after the software signing key for TI-83s got cracked earlier this week and so I'm reading the wiki article about it[RSA]. I'm curious about a step so I'll fill in everyone and then highlight the step I'm not sure about. Key generation: 1. pick 2 prime...
  10. P

    Arithmetic Progression + System of equations + binomial

    Homework Statement A third degree polynomial has 3 roots that, when arranged in ascending order, form an arithmetic progression in which the sum of the 3 roots equal 9/5. The difference between the square of the greatest root and the smallest root is 24/5 Given that the coefficient of the...
  11. T

    Modular Arithmetic: Solve (21999+31998+51997) Divided by 7

    Homework Statement Okay, so I'm going to find the smallest positive remainder of (21999+31998+51997) divided by seven. Homework Equations The Attempt at a Solution Well, I did like this: 23 is congruent to 1 (mod 7). Therefore, 21999= (23)1999/3 is congruent to 1 (mod 7). 33 is...
  12. B

    Doing binary arithmetic in Windows Calculator and

    I was doing some conversions from binary to decimal and vice versa today using Windows Calculator and I noticed the following, if I multiply B11111111 and B11111111 I get the following: B1111111000000001. Uhhh...great! What's going on here? It looks like it's rolling over the 8 LSB when it...
  13. J

    Weakness in mental arithmetic = weakness in math?

    Halfway through an undergraduate course in engineering, I'm now planning to review math fundamentals from pre-algebra, algebra, geometry to trigonometry and finally calculus because, as you may know, having a solid foundation in math is vital for any engineering course, and I've always been weak...
  14. T

    How Do You Solve These Challenging Arithmetic Sequence Problems?

    Arithmetic Sequences - PLEASE HELP! I would really appreciate any help to figure out the following 4 questions: 1) The Sum of the first two terms of an arithmetic progression is 18 and the sum of the first four terms is 52. Find the sum of the First eight terms 2) An arithmetic Progression...
  15. G

    Proving Algebraic Equivalence: GCDs & Modular Arithmetic

    Homework Statement 1. Prove that if ca=cb (mod m) and gcd(c,m)=1, then a=b (mod m) 2. Prove that if a=b (mod m), then gcd(a,m)=gcd(b,m)Homework Equations The Attempt at a Solution I can't figure out how to get started on these, especially the last one. Is it just a matter of expanding definitions?
  16. S

    How Do Cardinal Number Exponents Distribute Over Multiplication?

    Homework Statement prove that (a x b)^{}c = (a^{}c x b^{}c where a,b,c are any cardinal numbers Homework Equations The Attempt at a Solution i know that they should first be interpreted as sets A,B,C but what functions should I use.
  17. K

    Arithmetic Series: Find 1st 3 Terms & 20th Term

    Homework Statement The nth term of an arithmetic series is 1/2(3-n). What are the first three terms and the 20th term? Homework Equations nth term = a+(n-1)d The Attempt at a Solution I have made various attempts but cannot seem to work out how this can be done without a...
  18. E

    Find q in F = qVe + (Pe - Pa) * Ae with this simple formula

    I have this formula: F = qVe + (Pe - Pa) * Ae; I want to get q by its self. This what I did to get q by its self. F = qVe + (Pe - Pa) * Ae \frac{F - (P_e - P_a)}{(A_e)} = \frac{(V_e * q)(A_e)}{(A_e)} [( F - (Pe - Pa)) ÷ Ae] ÷ Ve = q This is how I got q by itself in order to solve for...
  19. F

    Proving the Difference of Sums in an Arithmetic Progression

    Homework Statement An arithmetic progression has n terms and a common difference of d. Prove that the difference between the sum of the last k terms and the sum of the first k terms is | (n-k)kd |. Homework Equations \begin{array}{l} {S_n} = \frac{n}{2}\left[ {2{a_1} + \left( {n - 1}...
  20. D

    Finding the nth Term of an Arithmetic Sequence

    The sum of the first n terms in a certain arithmetic sequence is given by Sn = 3n2 - n. Show that the nth term of the sequence is given by an = 6n - 4. so far i have done: Sn = (n / 2) (a1 + an) = 3n2 - n i solved for a1 + an = 6n - 2 i also have an = a1 + d(n-1). i don't know what do...
  21. D

    Help with arithmetic sequence problem

    The zeros of the polynomial f(x) = x^3 - 33x^2 + 354x + k form an arithmetic sequence. What is the value of k? so i let the zeros = a, b, and c. then i did b - a = c - b since it's an arithmetic sequence and they have common differences. so now i have a + c = 2b. i don't know what to do from...
  22. A

    Number of Arithmetic Operations to solve (x^T)(A^-1)x

    Homework Statement Let A be a nxn real symmetric positive definite matrix and x not equal to 0 a real nx1 vector. Show how to computre xTA-1x in n3/3 + O(n2) arithmetic operations. Homework Equations The Attempt at a Solution Some things I think I do know: If A is real spd, so is...
  23. K

    Ten prime numbers describing an arithmetic sequence

    Ten distinct prime numbers, each less than 3000, when arranged in increasing order of magnitude describe an arithmetic sequence. What are these ten prime numbers?
  24. D

    What is the Sum of the First 110 Terms in This Arithmetic Progression?

    If the sum of the first 10 terms and the sum of the first 100 terms of a given arithmetic progression are 100 and 10, respectively, what is the sum of first 110 terms? S(10) = (10/2) (a1 + a10) = 100 S(100) = (100/2) (a1 + a100) = 10 a1 + a10 = 20 a1 + a100 = 0.20 a100 = a1 + 99d a10...
  25. D

    How Can You Prove Zero Product in Modular Arithmetic for Composite Numbers?

    Homework Statement If n is composite, prove that there exist a,b in Zn such that a and be are nonzero, but ab=0 Homework Equations if a is congruent to b mod n, then n divides (a-b) The Attempt at a Solution So this is what i have so far, please let me know if i am on the right...
  26. T

    Solving Modular Arithmetic: x\equiv2 (mod km)

    i might be making it up, but i am confused. can we say: x\equiv2 (mod k) x\equiv2 (mod m) hence x\equiv2 (mod km) by km i mean k multiplied by m. if not, what is the result? or can it be found? thank you in advance.
  27. Loren Booda

    An arithmetic series of primes

    List all of the possible sums of prime number pairs with each element taken once. For instance: 2+3=5, 2+5=7, 3+5=8, 2+7=9, 3+7=10, 5+7=12, 5+11=16, 5+13=18 . . . Can you find significance in this progression? Have you seen this sequence before?
  28. C

    Finding longest arithmetic progressions

    Meta-note: This post includes both computer science and computational number theory. I could have posted it in the Programming forum, the CS forum, the NT forum, or here; I felt that this was the best place, especially since the CS forum does not actually deal with computer science these days...
  29. A

    Teaching Arithmetic Through Physical Manipulation: A Creative Approach

    i want to teach my boy arithmetic. my current 'best' idea, is to have a board with a few columns with nine spaces in each column. headings from right to left would be (units,tens, hundreds etc.). nine small plastic discs showing value of +1 unit. nine discs showing value +10 units. nine...
  30. J

    How to Bound the Limit of Arithmetic Mean from Below

    Homework Statement Prove if that if the limit of a_n = c as n approaches infinity, then the limit of o_n = c as n approaches infinity, where o_n is the arithmetic mean (a_1 + ... + a_n)/n Homework Equations I can't figure out how to bound it from below. The Attempt at a Solution...
  31. S

    Solve Modular Arithmetic Homework: 12^9 mod71

    Homework Statement find: 12^9 mod71 Homework Equations The Attempt at a Solution =12(12^8) mod71 = 12mod71 x 12^8mod71 = 12 x (12^2)^4mod 71 Now I'm stuck. My teacher solved it but i don't understand what he did so can someone explain how to do it in a very basic way...
  32. S

    Solving Modular Arithmetic Homework: ax = 1 (mod m)

    Homework Statement Find x for ax = 1 (mod m) a) a = 15 , m = 31 b) a = 6, m = 93 c) a = 15, m = 20 if possible. (The equal sign above is equivalence in modular arithmetic) Homework Equations The Attempt at a Solution a) gcd(31,5) = gcd( 31 - 2*15, 15) = 1...
  33. L

    How can I solve an arithmetic progression problem involving the sum of terms?

    Homework Statement In an arithmetic progression, the sum of the first 10 terms is the same as the sum of the next 5 terms. Given that the first term is 12, find the sum of the first 15 terms. 2. The only one I could think of is S= n/2 (2a+(n-1)d) 3. I've tried solving it, but failed. I...
  34. P

    Solve Tennis Score Puzzle: Use Modular Arithmetic

    In tennis, the players switch sides after the odd games. one hot sunday the auther found himself on the side where he had begun the match a while earlier. He knew the game score was either 6-2, 4-3 or 6-2, 5-4. which was it? you must use modular arithmetic to solve this
  35. K

    Getting progressive with arithmetic, geometric and harmonic

    Five positive integers P, Q, R, S and T, with P< Q < R <S < T, are such that: (i) P, Q and R (in this order) are in arithmetic progression, and: (ii) Q, R and S (in his order) are in geometric progression, and: (iii) R, S and T (in this order) are in harmonic progression. (I) Determine...
  36. P

    Modular arithmetic (casting out 9's)

    Homework Statement Let m be a positive integer and m' be an integer obtained from m by rearranging its digits. Prove that m-m' is a multiple of 9 Homework Equations Casting out 9's method The Attempt at a Solution So I found that by applying the casting out 9's method on m and m'...
  37. E

    A Modular Arithmetic Proof Problem

    Homework Statement Let a, b, s, t be integers with s, t > 0. What conditions must s, t satisfy if the following statement is true: If a = b (mod s) and a = b (mod t), then a = b (mod st). The attempt at a solution If s | a, s | b, t | a and t | b, then st | a and st | b if and only if...
  38. P

    Is n^2 congruent to 0 or 1 (mod 3) for any integer n?

    proof: n^2 congruent 0 or 1 (mod3) for any integer n
  39. R

    Attempts to define Pi as a definite arithmetic progression?

    Howdy ho. No reason for a welcome around here, it's not about me it's about the Mathematical Anti-Telharsic Harfatum Septomin, eh!? (I hope at least one of you are familiar with that guy) Nonetheless, I've become obsessed with the transcendental property, and thusly therein my familiarization...
  40. C

    Prove "a-c = (b-d)(mod m)" Using Modular Arithmetic

    Homework Statement Show that if a = (b mod m) and c = d(mod m) and m => 2, then a - c = (b - d)(mod m) Homework Equations c = d(mod m) <=> m|(c - d) d = c + xm The Attempt at a Solution I don't know how any equivalences for a = (b mod m), is there a way to get b from a = (b mod...
  41. B

    Thanks,JimCalculating forces between stars in 3D space

    Hello, What is the best way to determine direction, force, and velocity in three dimensional Cartesian coordinates? Explanation: I am writing a program to do some star motion simulations. I will edit in some stars, give them their mass, along with initial position and velocity, and see what...
  42. E

    Arithmetic mean always greater than geometric mean

    Hey, (sin A + sin B + sin C)/3 >= \sqrt[3]{}(sin A*sin B*sin C) I know this is true by Arithmetic mean always greater than geometric mean... but is there any other way of proving this?
  43. K

    Modular Arithmetic & Number Theory

    1) Suppose 2^k + 1 is a prime number. Prove that k has no prime divisors other than 2. (Hint: if k=ab with b odd, consider 2^k + 1 modulo 2^a +1) First of all, I have a little question. k=ab with b odd. Is this always possible for any natural number k? Why? Assuming it's always...
  44. J

    Master Arithmetic with Comprehensive Tables

    Arithmetic tables...?!? :confused: Any help would be v much appreciated!
  45. D

    Understanding Arithmetic Errors in C Programs

    C Program "Arithmetic Error" I'm writing a C program which prompts a user for an input decimal number (or any integer in base 10) as well as the n-base to which he wishes to convert the number to. However, for some reason my program failed to work and displayed "Arithmetic Error (core dumped"...
  46. J

    Arithmetic versus mathematics.

    IMO, I don't really think arithmetic is really mathematics. It's just pure calculations, that's all.
  47. J

    Decidability of Presburger arithmetic and FOL

    I always thought that first order logic with identity was undecidable if it had either a 2-place relation or a 2-place function. Wikipedia seems to confirm what I'd thought: "The set of logical validities in any first-order signature with equality and either: a relation symbol of arity no less...
  48. C

    Parallel discrete logs (continues: modular arithmetic)

    I'm working on a problem that involves calculating many discrete logarithms in GF(p): given n and an odd prime p, either find a k with 2^k\equiv n\pmod p or return "failure" if no such k exists. Now there are many algorithms for computing discrete logarithms, some of which are designed for many...
  49. D

    Problem of distinct integers chosen from the arithmetic progression

    I have a solution to a problem which I am not certain that is complete. (It's a putnam problem so I can't believe I solved it) Would you mind to take a look at it? The problem stated: "Let A be any set of 20 distinct integers chosen from the arithmetic progression 1,4,7,...,100. Prove that...
  50. F

    Arithmetic Series Perfect Square

    Homework Statement I need to find all arithmetic sequences of integers with the property that the sum of the first n terms is a perfect square for all integers n. Homework Equations a_n = nth term of the sequence = a_1 + (n-1)d d = common difference Sum of the first n terms of the...
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