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. robertphy

    B How to pick some numbers out of 13 integers, by a 4 digits code

    This is the question I read in a Math quizzes book: - you have the first 13 integers from 1 to 13 - someone gives you a code made of at most 4 digits (0 to 9) - you decipher the code and tell him exactly which numbers he wanted from the list each number can be picked only once For example. case...
  2. docnet

    B Does Infinity Times Two Equal Infinity and Other Mathematical Paradoxes?

    We have come to accept that Infinity times two is infinity. In the sense of 'size' we use to think about everyday numbers, the rules of arithmetic with infinities seem like nonsense. For example, consider the computable number $$0.100100100100100....$$ In the decimal expansion, there are...
  3. Feynstein100

    B Arithmetic mean of an infinite number of points?

    So I was thinking about arithmetic, geometric and harmonic means when I had a thought. Let's say we have a curve y = x^2. We want to find the AM of the points on the curve between x=1 and x=2 i.e. y = 1 and y = 4. To make thing easier, we'll start with just the endpoints and keep adding...
  4. P

    Python Floating point arithmetic and Fourier collocation

    >>> from numpy import exp, pi >>> exp(1j*pi) (-1+1.2246467991473532e-16j) The fact that the imaginary part of this is not zero is wrecking a fourier collocation scheme for a nonlinear PDE with periodic boundary conditions: the coefficient corresponding to the Nyquist frequency, which should be...
  5. xDocMath

    Comp Sci Rounding errors in computer arithmetic

    For the first part, I have used a= 2, b = eps/2, c = eps/2 which I believe works and I have tested it in MATLAB however haven't had any luck reproducing the second part in MATLAB with any numbers. Any hints? Thanks
  6. V

    Comp Sci Computer Arithmetic for Double Precision Numbers

    I know that this expression evaluates to 1 when a is equal to 0. Also for when a is equal to 1/n when n is a positive number, but I'm confused about how to go about this in double precision?
  7. A

    I Arithmetic representation of symbols according to certain rules

    Hi, Suppose I am given a sequence of special symbols and I want to produce the next sequence of symbols according to certain rules that transform one symbol into one or more symbols. They are 10 symbols in total, say; A, B, C, D, E, F, G, H, I, J The rules are: A transformed into B and it is...
  8. S

    B What the result of null added to itself?

    Hey,... is that correct to say that "The null part added to itself will always remain the same as itself, namely null."" In arithmetic, 3 * 0 = 0 + 0 + 0 = 0; It is therefore healthy to extrapolate as follows:∞-1 * 0 = 0 + 0 + 0 +... + 0 + 0 + 0 + 0 = 0 ∞ * 0 = 0 + 0 + 0 +... + 0 + 0 + 0...
  9. chwala

    Alternative Methods to Find Arithmetic Sequence's Common Ratio

    My attempt; The terms in the arithmetic sequence are ;##[ a, a+2d,a+5d]##. It follows that; Common ratio ##r=\dfrac{a+2d}{a}=\dfrac{a+5d}{a+2d}## ##⇒ar+2rd=a+2d+3d## ##ar+2rd=ar+3d## ##ar+2rd-ar=3d## ##2rd=3d## ##r=\dfrac{3d}{2d}=\dfrac{3}{2}## The solution given on the textbook is...
  10. M

    Prove the following assertion about modular arithmetic

    Proof: Suppose ## a\equiv b \mod n ##. Then ## n\mid (a-b)\implies a-b=kn ## for some ## k\in\mathbb{Z} ##. Since ## m\mid n ##, it follows that ## n=mp ## for some ## p\in\mathbb{Z} ##. Note that ## a-b=k(mp)\implies a-b=mq ## where ## q=kp ## is an integer. Thus ## m\mid (a-b) ##. Therefore...
  11. S

    Probability of getting arithmetic sequence from 3 octahedron dice

    I try to list all the possible sequences: 1 2 3 1 3 5 1 4 7 2 3 4 2 4 6 2 5 8 3 4 5 3 5 7 4 5 6 4 6 8 5 6 7 6 7 8 I get 12 possible outcomes, so the probability is ##\frac{12 \times 3!}{8^3}=\frac{9}{64}## But the answer key is ##\frac{5}{32}## . Where is my mistake? Thanks
  12. rudransh verma

    B Analysis vs arithmetic approach to solving motion

    https://www.feynmanlectures.caltech.edu/I_10.html https://www.feynmanlectures.caltech.edu/I_09.html Using Mathematical approach we can describe the motion of a falling body whose gravity is 32 m/s^2. Analysis shows that this is simply ##s-s_0=ut+1/2at^2##. Similarly we can describe the motion of...
  13. Leo Liu

    I [Congruence class] Proof of modular arithmetic theorem

    Could someone explain why ##[a][x_0]=[c]\iff ax_0\equiv c\, (mod\, m)##? My instructor said it came from the definition of congruence class. But I am not convinced.
  14. pairofstrings

    B Arithmetic progression, Geometric progression and Harmonic progression

    How do I build functions by using Arithmetic Sequence, Geometric Sequence, Harmonic Sequence? Is it possible to create all the possible function by using these sequences? Thanks!
  15. J

    Foundations Klein's Encyclopedia: Is an English Translation Possible?

    https://en.wikipedia.org/wiki/Klein%27s_Encyclopedia_of_Mathematical_Sciences Originals are in German or French, the Japanese version cut all the historical content :( Do you think that some day we will see this published in English? Size is big, 20k pages, but it cannot be more interesting I...
  16. W

    B Extending the Fundamental Theorem of Arithmetic to the rationals

    The Fundamental Theorem of Arithmetic essentially states that any positive whole number n can be written as: ##n = p_1^{a_1} \cdot p_2^{a_2} \cdot p_3^{a_3} \cdot \dots## where ##p_1##, ##p_2##, ##p_3##, etc. are all the primes, and ##a_1##, ##a_2##, ##a_3##, etc. are non-negative integers...
  17. S

    MHB Is the Sum of n^2 Terms in an Arithmetic Sequence Limited to 1?

    How many different arithmetic sequences have the sum of the first n terms n^2? solution an= 2n-1.Does that mean there is only one arithmetic sequence?
  18. anemone

    MHB Arithmetic Progression: Expressing d in Terms of x,y,z,n

    Let $a_1,a_2,\,\cdots,\,a_{2n}$ be an arithmetic progression of positive real numbers with common difference $d$. Let (1) $a_1^2+a_3^2+\cdots+a_{2n-1}^2=x$ (2) $a_2^2+a_4^2+\cdots+a_{2n}^2=y$ (3) $a_n+a_{n+1}=z$ Express $d$ in terms of $x,\,y,\,z,\,n$.
  19. AN630078

    Solving for nC8,nC9, and nC10 in an Arithmetic Progression

    Well, I am having a little difficulty knowing how to approach finding a solution to this problem. I am aware that in an arithmetic progression the first term is a and there is a constant common difference defined as d=un+1-un Expanding the binomial given...
  20. Purpleshinyrock

    How to Solve Arithmetic Sequence Problems

    Summary:: Sequences, Progressions Hello. I have been Given the following exercise, Let (a1, a2, ... an, ..., a2n) be an arithmetic progression such that the sum of the last n terms is equal to three times the sum of the first n terms. Determine the sum of the first 10 terms as a function of...
  21. AN630078

    Finding a and d from the Sum of an Arithmetic Series

    Question 1; Method 1 If the sum of the first four terms is 139 then S4=139 139=1/2(4)(2a+(4-1)d) 139=2(2a+3d) 139=4a+6d----- [1] The part of this question that is confusing is the "the sum of the next four terms is 115". Would this mean that S8=S4+115=139+115=254? In which case...
  22. e2m2a

    I The Fundamental Theorem of Arithmetic and Rational Numbers

    The fundamental theorem of arithmetic applies to prime factorizations of whole numbers. Can this theorem also correctly be invoked for all rational numbers? For example, if we take the number 3.25, it can be expressed as 13/4. This can be expressed as 13/2 x 1/2. This cannot be broken...
  23. R

    I Modular Arithmetic: Find Multiples, Understand the Reason

    Hi everyone, I can find multiple of number for example 2,3,4 and so on. But is there any reason why those number does work.
  24. R

    Modular Arithmetic: Solving Equations with 22x^2 = 11 mod 13

    I am thinking of taking modular of 13 to both sides of equation. So it will become 22x^2 is 11 mod 13. And try all the values from x equal to zero to x equal to 11. Is their easier way to solve it
  25. hackedagainanda

    B How to avoid arithmetic and sign errors

    I find most of the errors I make are related to arithmetic or using the wrong sign. (Note: This applies to mainly algebra and trigonometry for me.) I've tried writing slower and neater, and have been structuring my operations linearly on the paper step by step. Do any of you have some tried...
  26. S

    I A matter of style? Algebra, Arithmetic, Variables

    When people graduate and have their degrees in engineering or physics or mathematics or what they may have done, some of these people will use some mathematics, very often which is some-what complicated (or not) arithmetic. Why will some people choose to strictly avoid using variables in the...
  27. G

    MHB Arithmetic string equal to the geometric string, count x

    Let x1, ..., x25 be such positive integers that x1⋅x2⋅ ... ⋅x25 = x1 + x2 + ... + x25. What is the maximum possible value of the largest of numbers x1, x2, ..., x25?
  28. S

    B A proper comprehensive rule for arithmetic operation precendence

    Here is a "problem" that is "befudding" folks on the internet. To the less mathematically mature, it might be vexing, but to us mathematically mature, we should have a clear comprehensive meme so that stupid things like this don't happen...
  29. F

    B Why is the distributive law correct in algebra, like in arithmetic?

    When they give reason for multiplication the negative numbers leading to positive number, they base on distribute law.But why the distribute law in algebra is correct like in arithmetic?(e.g why -5(8-6)=-5.8+-5.-6?).In abstract algebra they use distribute law as axiom.But in elementary algebra...
  30. C

    B How do I calculate power plant capacity loss over 25 years?

    Hello, I have a simple question and am hoping someone can help. I have a power plant that loses 0.5% production capacity per year for 25 years. When working with the plant capacity in terms of percentages, year 0 is defined as 100% capacity. For year 1 and each subsequent years, is it correct...
  31. nomadreid

    N mod m =1 for 1<m<7, N mod 7=0. Find N. (N=301, but how?)

    First, I know the answer: 301. I thought (despite the injunction that the problem is to be done without number theory) that the Chinese Remainder Theorem might be of help (if I would use a subset which contained only relatively primes), but that didn't get very far. I also tried to spot a...
  32. chwala

    Reverse substitution to find the inverse of modular arithmetic

    ##132,289≡1973* 67 + 98## ##1973≡98*20+13## ##98≡13*7+7## ##13≡7*1+6## ##7≡6*1+1## now in reverse my attempt is as follows, ##1≡7-6## ## 1≡7-(13-7)## ##1≡2*7-(1973-20*132,289+1340*1973)## ##1≡2*7-(1341*1973-20*132,289## which is correct but my interest is in finding the inverse of 1973 help?
  33. R

    Find percentage of red gumballs in this gumball machine

    A gumball machine contains 232323 green gumballs, 525252 red gumballs, 343434blue gumballs, 616161 yellow gumballs, and 303030 pink gumballs What percentage are red ,do you add up the elements and put it over the total?
  34. chwala

    Modular Arith: How is Remainder 2?

    ok this is a bit confusing to me, long since i did this things...-10/3=quotient -3+ remainder -1. How is the remainder 2?
  35. E

    MHB Isomorphism of logic, arithmetic, and set theory

    Has anybody ever heard of this? I learned about it in a discrete math class in grad school, and I've never heard of it anywhere else !? For example, logical disjunction (OR) and set-theoretic UNION are isomorphic in this sense: 0 OR 0 = 0. {0} UNION {0} = {0}. Similarly, logical AND & set...
  36. A

    MHB Robinson Arithmetic (Q) Proofs

    In Peter Smith's Godel book, 2 conditions are proven of several that make Q "order adequate" O2: For any n, Q ⊢ ∀x ({x=0 v x=1 v...v x=n} → x≤n) 03: For any n, Q ⊢ ∀x (x≤ n → {x=0 v x=1 v...v x=n}) O3 is proved by induction. O2 is not. It would appear as if induction would be required in...
  37. S

    Proof by Induction: Arithmetic Sum

    Hi, I am self studying induction and came across the following problem. I am stuck on how to proceed (I need to use induction, I know there is a direct proof). My proof attempt is as follows: Let ## P (m) ## be the proposition that: $$ \sum_{i = m + 1}^{n} i = \frac{(n - m)(n + m + 1)}{2} $$...
  38. T

    Weak Induction implies Strong Induction

    Homework Statement [/B] Weak Induction: If (i) ##S(1)## holds, and (ii) for every ##k \geq 1(S(k) \Rightarrow S(k+1)##. Then ##\forall n \geq 1##, ##S(n)## holds. Strong Induction: If (i) ##S(k)## is true and (ii) ##\forall m\geq k [S(k) \land \cdots \land S(m)]\Rightarrow S(m+1)##. Then for...
  39. WMDhamnekar

    MHB Find Common Difference of A.P. Given G.P. & Logarithms

    If a,b, c, are in G.P and $\log_ba, \log_cb,\log_ac$ are in A.P. I want to find the common difference of A.P. Answer: After doing some computations, I stuck here. $\frac{2(\log a+\log r)}{\log a+2\log r}=\frac{2(\log a)^2+3\log r\log a +2(\log r)^2}{(\log a)^2+\log r\log a}$ How to proceed...
  40. Mark44

    Insights What is the relevance of Assembler programming in modern technology?

    Greg Bernhardt submitted a new blog post AVX-512 Assembly Programming: Opmask Registers for Conditional Arithmetic Conclusion Continue reading the Original Blog Post.
  41. Mark44

    Insights AVX-512 Assembly Programming: Opmask Registers for Conditional Arithmetic - Comments

    Greg Bernhardt submitted a new blog post AVX-512 Assembly Programming: Opmask Registers for Conditional Arithmetic Continue reading the Original Blog Post.
  42. R

    A proof for modular arithmetic theorem

    Homework Statement Let a and b be integers, and let m be a positive integer. Then a ≡ b (mod m) if and only if a mod m = b mod m. Homework EquationsThe Attempt at a Solution By definition a ≡ b (mod m) => m| (a-b) mx = a -b => mx + b = a => b = a mod m b = a - mx => b = m(-x) + a => a = b...
  43. A

    MHB Proving (x = y) using Axioms: Basic Arithmetic Proof

    Which axioms (at minimum) would have to be invoked so the following expression holds: (x = y) ----> [(y=x) <---> (y=y)] ? All help appreciated, am
  44. YoungPhysicist

    Big integer arithmetic functions

    NOTE:This is not a homework question! This is just a topic that I like very much,but don’t have the programming ability to do many of them.That’s why I post this thread. C++ is a language without built-in big integer calculation functions,so building ones that can do such job is a great way to...
  45. Entertainment Unit

    Help with an arithmetic proof by induction

    Homework Statement Show that if ##0 \leq a < b##, then $$\frac {b^{n + 1} - a^{n + 1}} {b - a} < (n + 1)b^n$$ Homework Equations None that I'm aware of. The Attempt at a Solution Proof (Induction) 1. Basis Case: Suppose ##n = 1##. It follows that: $$\frac {b^{1 + 1} - a^{1 + 1}} {b - a} < (1 +...
  46. P

    MHB Calculation of probability with arithmetic mean of the sum of random variables

    Calculation of probability with arithmetic mean of random variables There are 4 people, each of whom has one deck of cards with 500 cards that are numbered from 1 to 500 with no duplicates. Each person draws a card from his deck and I would like to calculate the probability of the event that...
  47. S

    Solving for the Sum of an Arithmetic Progression | m>n | AP Homework

    Homework Statement In an AP, sum of first n terms is equal to m and sum of first m terms is equal to n. Then, find the sum of first (m-n) terms in terms of m and n, assuming m>n. Homework Equations Sum of an AP: n/2 * {2a+ (n-1)d} The Attempt at a Solution We get two equations: m= n/2 * {2a+...
  48. R

    I Geometric mean versus arithmetic mean

    The Beer-Lambert law gives the intensity of monochromatic light as a function of depth ##z## in the form of an exponential attenuation: $$I(z)=I_{0}e^{-\gamma z},$$ where ##\gamma## is the wavelength-dependent attenuation coefficient. However, if two different wavelengths are present...
  49. opus

    Python troubles- says I'm using a string in arithmetic

    Homework Statement I'm making a program to display what I have posted in the image and my program needs to look just like it. The idea is to have an employee enter information such as their pay and hours worked. Then taxes get calculated and it shows a net amount that the employee has made for...
  50. M

    A Atiyah's arithmetic physics

    Sir Michael Atiyah just gave a livestreamed talk claiming to prove the Riemann hypothesis. But it turns out that this is part of a larger research program in which he also claims to have an apriori calculation of the fine-structure constant and possibly other physical constants. Atiyah is 89...