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.

View More On Wikipedia.org
Recent contents Top users
  1. M

    Is the Inverse Calculation Correct for Matrix Polynomial Equation?

    I suppose the title should be "Matrix polynomial T or F" but whatever. Homework Statement True or false: if A2-2A+I=0, then A-1=2I2-I The Attempt at a Solution My thought: A2-2A+I=0 becomes (A-I)(A-I)=0 so A = I The inverse of I is I. So the second equation: A-1=2I2-I becomes I = 2I2-I...
  2. P

    Basic arithmetic through to university lvl textbook recommendations

    Hi all, newbie here. So I've tried asking this question on Reddit with barely a reply and now another forum I frequent with also not much of a reply. And a while back an acquaintance recommended this forum to me. So my request is can you recommend me some mathematics textbooks that go from...
  3. F

    Question regarding modular arithmetic from discrete math

    Homework Statement Homework Equations a. I know that x*a mod y should be the same as y*b mod x but I don't understand why b. I know that an inverse can be constructed because x and y are mutually prime and gcd(x,y) = 1 , but I have no clue at what pair x and m is possible c. I have no idea...
  4. D

    Sum of alternating series using four-digit chopping arithmetic

    Homework Statement Let a_{n} be an alternating series whose terms are decreasing in magnitude. How to compute the sum as precisely as possible using four-digit chopping arithmetic? In particular, apply the method to compute \sum\limits_{n = 0}^\infty {\frac{{{{( - 1)}^n}}}{{(2n)!}}} and...
  5. C

    Exploring the Dot Product: Arithmetic and Magnitude of Vectors

    If you square the magnitude of a vector you get the dot product, correct? ||v||^2 = v . v Can you also say that ||v|| = sqrt(v . v)?
  6. N

    Inequality of arithmetic and geometric means

    Hey! I have this: 2(√(1-a^2 ))+ 2a How to determine the maximum value of this? I think good for this is Inequality of arithmetic and geometric means, but I don't know how use this, because I don't calculate with this yet. So, have you got any ideas? Poor Czech Numeriprimi... If you...
  7. N

    MHB Geometric Progression sequence with an Arithmetic Progression grouping problem

    Good Day, My friends and I are stuck on solving the last part of the attached problem. The solution is 2^[(n^2 + n)/2] - 1. Can anyone help us with solving this? Thanks & Regards, Nicodemus
  8. F

    Medical Neuroscientist says PSAT score related to simple arithmetic skills

    https://www.youtube.com/watch?v=uIjMIo8Lsrw Do I have this right? It sounds like the people who didn't do so well on the PSAT, who used their right brain, "overthought" the simple arithmetic problems, by using the (subjective?) quantity-related part of their brain. the students who did...
  9. Government$

    Another arithmetic progression problem

    Homework Statement Sum of first three members of increasing arithmetic progression is 30 and sum of their squares is 692. What is the sum of the first 15 members?The Attempt at a Solution So i have system of equations: a1 + a2 + a3 = 30 (a1)^2 + (a2^2) + (a3^2) = 692...
  10. Government$

    Arithmetic progression problem

    Homework Statement Let a_{m+n}=A and a_{m-n}=B be members of arithmetic progression then a_{m} and a_{n} are? (m>n).The Attempt at a Solution I fugured that a_{m}=\frac{A+B}{2} but i have no idea what a_{n} is. In my textbook solution is a_{n}=\frac{(2n-m)A + mB}{2} How did they arrived to...
  11. T

    Modulo Arithmetic: Is a^{\varphi(n)}\equiv 1 (mod \;n) for gcd(a,n)=1?

    Is it true that if A \equiv B \mod{\varphi(N)} where \varphi (N) is Euler's totient function then a^A \equiv a^B \mod{N}? I'm not after a proof or anything but I didn't do a number theory course and it seems that this fact is used in many questions I'm currently doing.
  12. BobG

    Just a simple arithmetic problem

    Easy arithmetic problem: x=222,222,222,222,222,222,222^2-222,222,222,222,222,222,221^2 Find x.
  13. J

    Understanding arithmetic and the Newton

    A Newton is: ( (one kilogram times one meter) per second) per second) I am trying to get at the basic logic of how we can apply numbers to reality. I have a good understanding of how we use a ratio to express things. A ratio separates one quantity into the amount of another quantity. In fact...
  14. S

    Is ℝ a Subset of ℂ? Understanding Complex Number Arithmetic and Isomorphism

    Homework Statement A textbook of mine asserts that ℝ is a subset of ℂ. The motivation for this is drawn by defining complex addition and multiplication and then showing that these operations on complex numbers of the form (x,0), with x an element of ℝ, are isomorphic to the field ℝ witih...
  15. trollcast

    Arithmetic Progression - show that question

    Homework Statement Given that a2, b2 and c 2 are in arithmetic progression show that: $$\frac{1}{b+c} , \frac{1}{c+a} , \frac{1}{a+b} $$ ,are also in arthimetic progression. Homework Equations The Attempt at a Solution So I assume by "in arithmetic progression" they mean those...
  16. D

    Executing Arithmetic Operations w/ 6-Bit Binary Numbers

    Homework Statement Execute the arithmetic operations using 6-bit binary numbers in 2's complement representation. 18+11 (just going to list this first task) Homework Equations To convert from regular binary number, from LSB, do not invert initial o's, or first 1, but invert all other...
  17. H

    Primes, pigeon holes, modular arithmetic

    Homework Statement The Attempt at a Solution Don't have a clue how to even start this one, sorry.
  18. PeteyCoco

    Can number theory help improve mental arithmetic?

    I decided to try some of the problems in my Mechanics text without a calculator to see how well I could approximate the answers using differential equations and mental arithmetic. I was a bit slow, but I remembered some tricks I used as a kid in grade school that I hadn't used in ages. Does...
  19. F

    Modular arithmetic with a variable modulus and fractions

    (This is my first post.) I can't seem to find a good way of solving this sort of congruence for x: x^2 / 3 + 11 \equiv 5 (mod x) Through trial and error it appears at least 3 and 6 are answers, but how can you reach them regularly? (I'm heard conflicting things about fractions being...
  20. D

    MHB Problem involving arithmetic and geometric mean.

    $a,b,c$ are any three positive numbers such that $a+b+c=1$. Prove that $$ab^2c^3 \leq \frac{1}{432}$$
  21. V

    Arithmetic Sequence: Finding the First 3 Terms Using tn= t1+(n-1)d

    Homework Statement In an arithmetic sequence, the 11th term is 53 and the sum ofof the 5th and 7th terms is 56. Find the first 3 terms of the sequence. Homework Equations The Attempt at a Solution I'm trying to use the formula: tn= t1+(n-1)d but don't have right numbers. please...
  22. N

    Find Sum of Arithmetic Series Sn: Σ 200 r=5 5r-2

    Find the sum of \Sigma 200 r=5 5r-2 Sn = n/2 [2a + (n-1)d ]I used S 200 and I got about 101400 but then when I verified on my calculator it was 100058, my calculator has the sigma notation for working out the sum of , how do you get 100058?
  23. J

    Can you use modular arithmetic to solve this?

    I came across a problem like this (not homework) x^2+y^2-k For example, x^2+y^2-24 \text{ ,n=4} x^2+y^2-45 \text{ ,n=8} If x and y are any positive integers (not given) and k is a positive integer (given), is this expression divisible by n (a positive integer that is given). A...
  24. U

    Find the least value of a

    Homework Statement If four distinct points on the curve y=2x^4+7x^3+3x-5 are collinear, then find the arithmetic mean of x-coordinates of the aforesaid points. Homework Equations The Attempt at a Solution I think that the four points mentioned must be the roots of the equation.
  25. D

    Modular Arithmetic and Diophantine Equations

    If one is solving a modular equation: 4k \equiv 1 \: (\text{mod } n) with n even, known, for k, then one needs to find the inverse of 4 modulo n: 4x - 1 = nc 4x - nc = 1 But this only has solutions iif (4,n) = 2 (n is even, but not a multiple of 4), which doesn't divide 1, so...
  26. M

    Number Theory (Modular Arithmetic and Perfect Squares)

    Homework Statement If k is an integer, explain why 5k +2 cannot be a perfect square. Homework Equations n/a The Attempt at a Solution I'm in way over my head and not really sure what type of proof I should be using. In my course, we just went over some number theory and modular algebra so...
  27. D

    MHB Finding the Formula for Partial Sums of an Arithmetic Sequence

    Use a geometric or algebraic argument to find a formula for the partial sums $A_n$ of an arithmetic sequence. I know that the partial sum is $S_n = n/2(2a_1+(n-1)d)$ where d is the difference. $A_n = \sum\limits_{k = 1}^n a_k$ I can come up with $n/2(a_1+a_n)$ but how do I get the difference?
  28. C

    Modular arithmetic with cardinals.

    Can I do operations like {\aleph_0}^{\aleph_0}mod {\aleph_0} and would this equal \aleph_0
  29. FeDeX_LaTeX

    Prime Number Arithmetic Progression

    "Determine the least possible value of the largest term in an arithmetic progression of seven distinct primes." I really have no clue what to do here. Is there a general tactic that you can use to do this, other than trial and error? Some experimenting gives you these of arithmetic...
  30. J

    Complex Arithmetic - Mathematica agrees with me, textbook says I'm wrong.

    Mathematica agrees with my second solution (not the first one though). The back of my textbook says: "\sqrt[4]{8}[\cos(\frac{5\pi}{8}) + i\sin(\frac{5\pi}{8})] and \sqrt[4]{8}[\cos(\frac{13\pi}{8}) + i\sin(\frac{13\pi}{8})]" Edit: The second z in my picture should be |z|, the modulus...
  31. G

    How many arithmetic and basic algebra errors do you make

    when i do higher maths i make a tone of basic algebra and arithmetic mistakes. i was going a basic AX = B using LU decomposition in linear algebra and I had to go back and check my basic math about 6 times before i got the right answer. is it just me or do a lot of people do this?
  32. F

    MATLAB Add Packages to MATLAB for Higher Order Arithmetic

    Basically I'm writing my MSc dissertation right now, and I've been doing a lot on primes I've written all my code in MATLAB, but my supervisor told me today that MATLAB is crap for higher order arithmetic and the primes only go up to something like 10 digits long. SO I'm kinda screwed...
  33. F

    Efficient Prediction of Linear Congruential Pseudorandom Numbers

    Homework Statement Hi everyone! I have a homework question given below: One scheme of pseudorandom number generator is the linear congruential generator where we pick some modulus m, constant a and b and a seed x0, then generate sequence x1, x2, x3, ... according to the equation: x(t+1) =...
  34. F

    Modular Arithmetic Homework: 1/2*(x-4)(x-5) = 4(x-4)(x-5) (mod 7)

    Homework Statement Hi everyone, I have a problem in the following modular arithmetic operation 1/2*(x-4)(x-5) = 4(x-4)(x-5) (mod 7) ("=" means congruent in this expression) Homework Equations The Attempt at a Solution I am completely lost on how the operation is valid. If...
  35. B

    Interesting problem involving arithmetic progression

    I just came up with a problem I hope you will find interesting, but I can't seem it solve it myself. I thought of induction as some guide, but am not sure how to proceed. There are N terms in some finite arithmetic progression. Two of those terms are equal to 3. Prove that all terms in this...
  36. H

    Discrete Math - Modular Arithmetic

    Homework Statement For which values of n≥2 does the implication: axb=0 ⇔ a=0 or b=0 For some Zn (n should be a subscript) NOTE: For the a x b, the x should be the x that has a circle around it. I didn't see that symbol in the "quick symbols" :) Homework Equations I know that this...
  37. L

    Sum of Finite Arithmetic Series: Formula & Examples

    1. Write out in full and determine the sum of the finite arithmetic series r = 9 Ʃ 5r 5,10,15,20,25,30,35,40 <-- until 9 r=1 But how do I determine the sum of the finite arithmetic series? I forget the formula :/
  38. T

    Terms of a geometric series and arithmetic series, find common ratio

    Homework Statement Different numbers x, y and z are the first three terms of a geometric progression with common ratio r, and also the first, second and fourth terms of an arithmetic progression. a. Find the value of r. b. Find which term of the arithmetic progression will next be equal to...
  39. I

    Is (n+1)(2n+1) Always a Multiple of 6 for All Natural Numbers n?

    Hello, The formulation of the question says: Show, using congruences and disjunction of cases, which,for all natural n є Ninteger, the integern (n+1) (2n+1) is a multiple of 6. Simultaneously, themultiple of 3 is a multiple of 2 (number), then it is divisible by 6 3 cases are...
  40. M

    Proving Impossibility of Integer n in Modular Arithmetic

    Homework Statement Prove that it is impossible to find an integer n, such that n^2 = 2 mod(4) or n^2 = 3 mod(4) Hence or otherwise, prove that there do not exist integers m and n such that 3m^2 - 1 = n^2 Homework Equations The Attempt at a Solution Since n must be...
  41. M

    Solving Modulo Arithmetic Equations in Z/5Z^2

    Homework Statement Solve in K=\mathbb{Z}/_{5}\mathbb{Z}\cdot\mathbb{Z}/_{5}\mathbb{Z} the following system of equations: \begin{cases} 2^{-}x-3^{-}y=1^{-} & 1^{-}x+2^{-}y=2^{-}\end{cases} Not that the ^- means a number with a bar over it. ( I don't know how to input it in the latex software...
  42. B

    MHB What Are the Solutions to These Basic Arithmetic Problems?

    8(6-5) +10 equal?4(-2)to the 2nd degree plus 8 (-2) + 3(-2) + 6 equal? 3 (1/3) (9) equals? 598%/ 26% what is answer?
  43. E

    How Accurate Are the Bounds for Eigenvalues in Circulant Matrices?

    Hi, I have the following equation: \gamma=\frac{1}{\frac{1}{N}\sum_{n=1}^N|\lambda_n|^{-2}} where lambdas are the eigenvalues of an N-by-N circulant matrix A. I used two properties to bound the above equation...
  44. M

    Chinese Remainder Theorem: Divisibility in Arithmetic

    Chinese remainder theorem help Homework Statement Solve in Z^{2}:6x-5y=1 Conclude the solution to the system: X≡2(mod5) , X≡1(mod6) The Attempt at a Solution 1- solved the equation and found one unique solution which was S={(1,1)} Given: X≡2(mod5) , X≡1(mod6) X≡2(mod5) means X=5t+2...
  45. K

    Arithmetic Progression formula proof

    The proof says that - Let, Sn= a+(a+d)+(a+2d)+...+(a+(n-2)d)+(a+(n-1)d)----->1 Sn= (a+(n-2)d)+(a+(n-1)d)+...+a+(a+d)+(a+2d)------>2 Now if we have to add such things(1 and 2) how would we do that?
  46. 1

    Lunatic Rantings about Modular Arithmetic.

    So we're doing modular arithmetic in my proof class. I have a weird cycle when learning something new in pure math, I think "wow, this is just exceedingly indepth version of something learned by gradeschool children." Then I find something (on my own, not in the textbook, just thinking about it...
  47. R

    Number Theory- arithmetic functions

    Problem: Show that for each k, the function σk(n)=Ʃd|n dk is multiplicative. The attempt at a solution: What I know is that I am supposed to use the Lemma which states that if g is a multiplicative function and f(n)=Ʃd|n g(d) for all n, then f is multiplicative. I am just very confused...
  48. C

    Particle Motion: Retardation & Arithmetic Progression

    Homework Statement A particle moves in a straight line away from a fixed point O in the line, such that when its distance from O is x its speed v is given by v=k/x , for some constant k. (a) show that the particle has a retardation which is inversely proportional to x3 The answer is...
  49. Loren Booda

    Arithmetic vs. geometric uncertainties

    Rather than arithmetic ("plus or minus") uncertainties, are there classical (not of Heisenberg uncertainty principle) measurements whose uncertainties otherwise appear as geometric ("times or divided by")?
  50. Dembadon

    Comp Sci Programming: Fraction Arithmetic in C++

    This program has been assigned in my intro to programming course, which assumes no previous knowledge of programming. Up to this point we've only been required to write functions that accomplish specific tasks within a program. This is the first time we've been asked to design and write a full...
Back
Top