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. D

    Fortran What is the correct way to handle a potential divide by zero issue in FORTRAN?

    Good day, I'm working with a FORTRAN program that has the line: EDEN=EDEN+(STVS**2)/(N(J)-1) I would like to check if N(J)-1 is equal to zero. If so make N(J)-1=1.0 The following doesn't work: IF ((N(J)-1).EQ.0.0) THEN N(J)-1=1.0 END IF I'm using gfortran and am getting...
  2. T

    Arithmetic progression. find p.

    Homework Statement johns father gave him a loan of $1080 to buy a car. the loan was to repaid in 12 monthly installments starting with an intial payment of $p in the 1st month. there is no interest charged on the loan but the installments increase by $60/month. a) show that p = 570 and find in...
  3. D

    Fortran What is causing the ARITHMETIC exception in my FORTRAN program?

    Good day. My FORTRAN program is throwing an ARITHMETIC exception on this line of code: SAREA(2,ISN)=SAREA(2,ISN)/IND(13)/1000./6080.2*IND(12)/1000. When I check this using gdb and print the contents of the variable SAREA(2,ISN), the value is 1432. gdb error message: Program received signal...
  4. arun-siara

    A formula based approach to Arithmetic Coding

    I have been doing research on entropy encoding for some time.. I found some interesting relationships between Arithmetic coding and other methods such as Huffman Coding. I made an article to explain them and am presenting here for review: http://siara.cc/arithmetic_coding_new_approach/ I have...
  5. W

    I'm making an Arithmetic Error, Electrostatic force diagrams

    Homework Statement Three charged particles are placed at each of three corners of an equilateral triangle whose sides are of length 2.7 cm . Two of the particles have a negative charge: q1 = -6.0 nC and q2 = -12.0 nC. The remaining particle has a positive charge, q3= 8.0 nC . What is the net...
  6. Math Amateur

    MHB Arithmetic for Quotient Groups - How exaclty does it work

    I have just received some help from Euge regarding the proof of part of the Correspondence Theorem (Lattice Isomorphism Theorem) for groups ... But Euge has made me realize that I do not understand quotient groups well enough ... here is the issue coming from Euge's post ... We are to consider...
  7. B

    Where Can I Find Comprehensive Math Resources for Re-learning?

    Hi All, I am looking for the best way to learn math from elementary algebra (just above arithmetic) to calculus. I am looking for the most *comprehensive* way to do this--that is, I want continuity from subject to subject, for instance, a series of textbooks from a single author on Algebra I...
  8. A

    Intro Math A book on arithmetic that doesn't treat you like a baby

    The state of arithmetic today is disgusting. The textbooks on it are absolutely repelling, the authors treat it like a subject that will be of concern to only babies. They don't show any love, they treat the subject like a dirty rug. It's been two years since I majored in mathematics, since...
  9. osirvics

    Arithmetic Progression: Find Common Difference with Given First Term and Ratio

    Homework Statement The first term of an a.p is -8, the ratio of the 7th term to the 9th term is 5:8. what is the common difference of the progression? Homework EquationsThe Attempt at a Solution I've tried... it confuses me. Can anyone give me some hints or tips...?
  10. C

    Mathematica WolframAlpha miscalculates a simple arithmetic expression

    There is a crucial bug on both WolframAlpha and Mathematica. When we give the input "Floor(sqrt(11.44-10)/0.2)" it gives wrong answer. The actual answer is six (6) but wolframalpha gives five (5). The problem is Wolframalpha calculates 11.44-10 incorrectly. Although the answer is exactly 1.44...
  11. G

    What is the Greatest Common Divisor of Two Polynomials?

    Homework Statement What is the greatest common divisor of ##X^a - 1 ## and ## X^b - 1 ##, ##(a,b) \in \mathbb{N}^\star## ? Homework EquationsThe Attempt at a Solution Assuming that ## a\le b ##, I find by euclidian division of ##b## by ##a## that ## b = an + r \Rightarrow X^b - 1 = (X^a-1)...
  12. G

    Arithmetic Homework: Showing Bezout Theorem

    Homework Statement We are given ##a_1,...,a_n ## in ## \mathbb{N}^\star ##, all mutually prime, and ## a = a_1 \times ... \times a_n ##. Show that for all ##(b_1,...,b_n)\in\mathbb{Z}^n##, there is ##\beta \in \mathbb{Z} ## such that for all ##x \in \mathbb{Z} ## : ## (\forall i = 1 ... n, \...
  13. nomadreid

    Recursive sets as delta^0_1 in arithmetic hierarchy.

    This is an elementary question that I may blush about later, but for now: given that a recursively enumerable set is a set modeling a Σ01 sentence, and a recursive set is a recursively enumerable set S whose complement ℕ\S is also recursively enumerable. Fine. But then, letting x→ = the...
  14. F

    Arithmetic mean Fermi Dirac & Bose Einstein

    Hi everybody, I was doing one asignment form class, I was tasked to prove that in one system, the arimetic mean of FD and BE distributions is equal to MB's distribution for undishtingable particles. After doing the numbers I found out that it actually was, but I don't know why this happens, can...
  15. Superposed_Cat

    Which is faster? reading from memory or arithmetic?

    Hey all, I was writing this program just now and wondered whether it would be easier for the cpu if I stored a number before I divided it by 2 or if i multiplied by 2 later to get that number back, I know it won't affect speed at all really but was just wondering, Any help appreciated.
  16. C

    MHB Arithmetic Progression: Finding the First Term and Common Difference

    The sum of the first 100 terms of an arithmetic progression is 15050; the first, third and eleventh terms of this progression are three consecutive terms of a geometric progression. Find the first term, a and the non-zero common difference, d, of the arithmetic progression.
  17. PsychonautQQ

    Finding inverses in modular arithmetic

    Hey PF! Is there a systematic way to calculate the inverse of a number in a modular setting (modular setting? is that what I call it? lol). How about 108x == 1 (mod 625), wolfram alpha calculated x = 272, how could I have arrived at this number besides guess and check?
  18. D

    Engineering Complex arithmetic for circuit equation

    Hi, I am trying to find the current of a circuit using mesh analysis so far I have; My voltages V1 = 415 ∠ 90° or 0 + j415 V2 = 415 ∠ 0° or 415 + j0 impedances Z1 = j4 Z2 = j6 My formula is; -V1+Z1*I+Z2*I+V2=0 Which equates to; -0+j415+j4*I+j6*I+415+j0=0 Could...
  19. E

    Prove that for a,b,c > 0, geometric mean <= arithmetic mean

    Homework Statement Let ## a,b,c \in \mathbb{R}^{+} ##. Prove that $$ \sqrt[3]{abc} \leq \frac{a+b+c}{3}. $$ Note: ## a,b,c ## can be expressed as ## a = r^3, b = s^3, c = t^3 ## for ## r,s,t > 0##. Homework Equations ## P(a,b,c): a,b,c \in \mathbb{R}^{+} ## ## Q(a,b,c): \sqrt[3]{abc} \leq...
  20. C

    MHB Modular arithmetic and factorials

    Hi there, I actually have a few questions I came across on my studies. They are (a) Show that if p is odd and x is an integer such that x^2 ≡ 1 mod p^k, then x = ±1 mod p^k (b) Find the solutions of the congruence equation x^2 ≡ 1 mod 2^k (c) What is the remainder of (p − 1)!, when divided by...
  21. FuturePhysicist

    Getting worse at mental arithmetic

    Hello I have a question reguarding human calculation. I'm in high school and I have been called a "math whiz" when it comes to subjects like algebra or calculus. But my school has kids using calculators so much that I haven't been using my head much for calculations. Because of this, I feel...
  22. Suraj M

    Arithmetic progression sum and nth term

    Homework Statement The ratio of sums of 2 AP for n terms each is ## \frac{3n + 8}{7n + 15}## that is $$ {\frac{s_a}{s_b}} = \frac{3n + 8}{7n + 15} $$ find the ratio of their 12th terms. $$ Required= \frac{a₁_a+(n-1)d_a}{a_b + (n-1)d_b}$$Homework Equations Tn = a + (n-1)dThe Attempt at a...
  23. M

    Proofing My "Arithmetic?" Proof

    This is a self-assigned question. Not homework. I may have the right answer, but would like some reviewing. It came to me while reading on a CS topic, this did not come from a math textbook, otherwise it would be slightly more in context. I did not use any formula (so 2 would be empty) and 1...
  24. moriheru

    Embarrassing Mistake in Arithmetic? Check Here for Help!

    I am actually quite sure I have it right but my book would then be wrong so I have posted my problem here to see if I didn't do a embarassing mistake in arithmetics. Homework Statement "Show that the energy equivalent of the length 10-18cm of a large extra dimension is roughly 20TeV" -Zwiebach...
  25. A

    Finding Units Modular Arithmetic

    Homework Statement I am required to find the units of ℤ8. Homework Equations I have that ##\bar{a}## = [a]n = { a + kn, k ∈ ℤ } ##u## ∈ ℤn is a unit if ##u## divides ##\bar{1}##. The Attempt at a Solution I'm not sure how to go about this. My lecturer wrote out the multiplication table...
  26. J

    Why is the fundamental theorem of arithmetic special?

    Why is it significant enough to be fundamental? Some people say that it is fundamental because it establishes the importance of primes as the building blocks of positive integers, but I could just as easily 'build up' the positive integers just by simply iterating +1's starting from 0.
  27. anemone

    MHB Roots of $g'(x)$ in AP: Proving the Theory

    The roots of a fourth degree polynomial $g(x)=0$ are in an AP (arithmetic progression). Prove that the roots of $g'(x)=0$ must also form an AP.
  28. H

    Cardinal Arithmetic: Finite Set X, #X+1

    Homework Statement Let X be a finite set and let x be an object which is not an element of X. Then X U {x} is finite and #(X U {x}) = #(X) + 1 The Attempt at a Solution Let X be a finite set such that X has cardinality n, denoted by #X. Suppose that ## x \notin X##, then the set X U {x} has...
  29. T

    MHB Solving Bitwise Arithmetic: What Trick Makes it Easy?

    1234567 ^ 7 ^ ~~1234567 5 & (12345678 ^ ~~ 12345678) My prof said there is an easy way to solve these as there is a trick to it. Does anyone know what trick is being referred to?
  30. T

    MHB Learn the Basics of Bitwise Arithmetic: A&B = 12 (Integer Variables A=60, B=13)

    "Assume integer variable A holds 60 and variable B holds 13 then:A&B will give 12" Why is this?
  31. nomadreid

    The advantage of modular arithmetic, e.g. cyclic groups?

    In starting to look into the mathematical side of encryption , I note the heavy dependence upon modular arithmetic. What is the advantage is this? For example, why are finite cyclic groups and rings preferable? Note: I know zilch about programming; I am approaching it from the mathematical side.
  32. D

    MHB Finding terms in arithmetic progressions

    3). A company is to distribute \$36,000 in bonuses to its top five sales people. The fifth salesperson on the list will receive \$6,000 and the difference in bonus money between successively ranked salespeople is to be constant. find the bonus for each salesperson. 4). Find the third term of...
  33. D

    MHB Sums of arithmetic progressions

    1). Find the fourth term of the sequence of partial sums for the given sequence. {5+ 3\2 n} 2). A bicycle rider coasts downhill, traveling 7 feet the first second. In each succeeding second, the rider travels 6 feet farther than in the preceding second. If the rider reaches the bottom of the...
  34. D

    MHB Understanding Arithmetic Sequences: Tips and Strategies

    Need help on these questions I haven't been introduced yet. Please look at image I inserted in.
  35. R

    Question about arithmetic progressions

    Homework Statement Of a 4 digit positive integer, the four digits form an Arithmetic progression from left to right. How many such 4 digit integers exist? 2. The attempt at a solution If d = 1, the integers are 1234, 2345, …, 6789. These 6 integers and their reverses satisfy the given...
  36. L

    What is a good arithmetic textbook?

    I am looking for an arithmetic book to use before going on to pre-algebra. However I don't want a book that says "28 + 53, add 8 + 3, carry the 1, etc). I'm looking for a book that has a better approach. For example, instead of that approach, saying "28 + 53 = 30 + 51 = 81" - something like...
  37. J

    Using Modular Arithmetic to Solve Congruence Equations

    Homework Statement Determine whether there is a positive integer k so that the congruence is satisfied. 2k ≡ 1 (mod 11)Homework Equations gcd(2k,11) = 1The Attempt at a Solution Well, I know the answer is false. Because of Fermat's Little Theorem, 2k ≡ 2 (mod 11) But I'm not satisfied with...
  38. PcumP_Ravenclaw

    Proof of fundamental theorem of arithmetic

    Dear all, Please help me understand the proof by induction for only one way of expressing the product of primes up to the order of the factors. Please see the two attachments from the textbook "alan F beardon, algebra and geometry" A is a set of all natural numbers excluding 1 and 0?? r and s...
  39. I

    Inverse matrix word problem, matrix arithmetic

    Homework Statement Hello! Please, take a look at the attached picture - there is a quote of the exercise and below is my attempt to make a matrix. Is my matrix correct? I have tried many times to convert it to inverse one, but I can't figure out how to do it - I keep getting "inconvenient"...
  40. A

    Binomial series with coeficients in arithmetic progression

    Homework Statement The binomial expansion of (1+x)^n, n is a positive integer, may be written in the form (1+x)^{n} = 1+c_{1}x+c_{2}x^{2}+c_{3}x^{3}+...c_{r}x^{r}+... Show that , if c_{s-1}, c_{s} and c_{s+1} are in arithmetic progression then (n-2s)^{2} =n+2 Homework Equations The Attempt...
  41. bananaman13

    How to Solve Problems in Arithmetic Series Using Formulas?

    I don't understand on how to get the answer of a certain problem in arithmetic series. formula/s: Arithmetic means An=A1+ (n-1)d Arithmetic series Sn=n/2 (A1+An) I understand on how to get the terms to these but when finding for: Find A1 if, S10= -20 , d= 4 & Find S12 if, 2+5+8+... I...
  42. A

    Problem about arithmetic progression

    Hi, can't solve following prob: Let a, b and c be real numbers. Given that a^2, b^2 and c^2 are in arithmetic progression show that 1 / (b + c), 1 / (c + a) and 1 / (a + b) are also in arithmetic progression. From assumptions: b^2 = a^2 + nk and c^2 = b^2 + mk where k is some real number...
  43. Q

    Binomial theorem and modular arithmetic

    Homework Statement From an old exam: Show that \begin{equation*} \sum_{0 \leq 2k \leq n} \binom{n}{2k}2^k = 0 (3) \text{ iff } n = 2 (4). \end{equation*} By ##a = b (k)## I mean that ##a## is congruent to ##b## modulo ##k##. Homework Equations Binomial theorem: ## (a + b)^m =...
  44. D

    Modular Arithmetic: Exponentiation & Division Algorithm

    Homework Statement Hi, I am currently working on modular arithmetic and recently I have been investigating on the effects of exponentiation on the system.It will be helpful if someone can share some ideas on the following problem. According to the division algorithm: a= nQ+r, where a is the...
  45. B

    MHB Some formulas of Arithmetic progression/series

    An arithmetic progression (AP) or arithmetic sequence is a sequence of numbers such that the difference between the consecutive terms is constant. Example: 2,4,6,8,10….. Arithmetic Series : The sum of the numbers in a finite arithmetic progression is called as Arithmetic series. Example...
  46. M

    Use 3-digit arithmetic with no pivoting?

    Homework Statement Use 3-digit arithmetic with no pivoting to solve the following system: 10-3x-y=1, x+y=0. Homework Equations I know that 10-3=0.001. The Attempt at a Solution The answer for this problem is (0, -1). Here's the work: 1.001 0 1 1 I've...
  47. M

    Use exact arithmetic to solve the following system?

    Homework Statement Use exact arithmetic to solve the following system: 10-3x-y=1, x+y=0. Homework Equations I know that 10-3=0.001. The Attempt at a Solution Here's the work: y=-x 10-3x+x=1 x(10-3+1)=1 x=1/(0.001+1)=1/1.001 y=-1/1.001 since y=-x The answer for this...
  48. D

    Modular Arithmetic: Explained and Proven

    Please see the attached,which was quoted from the following website: http://en.wikipedia.org/wiki/Modular_arithmetic It said that the multiplicative property is only applicable if n is an integer.On the contrary,the addition property can be applied to all real numbers. I don't quite...
  49. T

    Fortran Fortran programming help arrays and arithmetic

    Hi folks, To begin with, I have no past programming experience and have just begun to teach myself programming in FORTRAN 95 and I've hit a wall. I'd be very grateful for any assistance here. I have around 150 text files with three columns of data (I have attached one as an example, and...
  50. D

    How Do You Solve for b in an Arithmetic Sequence Involving 1/a, 1/b, and 1/c?

    Homework Statement * "/" means divided by * 1/a , 1/b , 1/c are consecutive terms in an AS, where a,b,c ε R\0. (whatever that means haha) express b in terms of a and c. give your answer in its simplest form. *thats all it says* Homework Equations there are none :) The...
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