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. Robin04

    Triangle determined by arithmetic and geometric sequences

    Homework Statement Determine the triangles where the sides are consecutive elements of a geometric sequence and the angles are consecutive elements of an arithmetic sequence. Homework Equations The Attempt at a Solution I don't really know how to approach this problem, what the solution would...
  2. Physiona

    I'm struggling with an Arithmetic Geometrical question.

    Homework Statement The angles in triangle ABC form an increasing arithmetic sequence. The ratio of angles A:B:C can be written in the form 185:370:555 respectively. You are told that the total area of the triangle is 9 Length BC is Given the area of...
  3. F

    Finding harmonic components with basic arithmetic

    Homework Statement Homework Equations I'm guessing trigonometric identities such as sin(a)cos(b) = 1/2(sin(a+b)+sin(a-b)) might be relevant. The Attempt at a Solution I've been thinking of some way to get an approximation of each harmonic by working with the Fourier series representation...
  4. Physiona

    Find the Missing Angle in Triangle ABC: Use Arithmetic Sequence and Sine Rule

    QUESTION: The angles in triangle ABC form an increasing arithmetic sequence. The ratio of angles A:B:C can be written in the form 185:370:555 respectively. You are told that the total area of the triangle is 9 Length BC is Given the area of triangle ABC, work out the...
  5. M

    MHB Divisibility of Terms in an Arithmetic Series

    Arithmetic Series? Given the arithmetic series 5+14+23+...(to 241 terms), how many terms in the series are divisible by 5? I need a good explanation and a good start.
  6. P

    MHB Arithmetic problem: What is the capacity of each Breaker"?

    Need help with this I have 6 beaker of oil and 12 breaker of water, the total amount combined are 66 litres . What is the capacity of each Breaker"?
  7. J

    MHB Troubleshooting Euclid's Lemma Proof in Modular Arithmetic

    Encountered difficulties in proving the attached image. Greatly appreciate for the help!
  8. A

    Intro Math Best book for understanding arithmetic?

    Is there any very basic arithmetic book (for dummies) written by respected mathematicians like for example Serge Lang, Gelfand and Allendoerfer? Looking for a book that really explains the key concepts and rules of arithmetic, in a pedagogical way, from scratch, without skipping any steps when...
  9. M

    MHB Arithmetic Sequence Confusion // a{n-1} and a{n+1}

    So I have this problem I'm stuck on wrapping my head around a particular problem "In the sequence a{n}, let a{0}=2. If a{n+1} = 3 a{n} −1, then what is the value of a3?" I understand it's following the pattern of each term, and that with Arithmetic sequence a{n-1} means you would use the a{n}...
  10. L

    B Feynman explains the foundations of arithmetic

    A long time ago I read an explanation Richard Feynman did on how the concepts of arithmetic can be derived from basic principles, along the lines of Peano's axioms, but I don't remember where it was. Thanks.
  11. M

    MHB Which mean is larger when using algebraic expressions, A. M. or R. M. S.?

    Given two positive numbers a and b, we define the root mean square as follows: R. M. S. = sqrt{(a^2 + b^2)/2} The arithmetic mean is given by (a + b)/2. Given a = 1 and b = 2, which is larger, A. M. or R. M. S. ? A. M. = sqrt{1•2} A. M. = sqrt{2} R. M. S. = sqrt{(1^2 + 2^2)/2} R. M. S. =...
  12. M

    MHB Can the geometric and arithmetic means be applied to algebraic expressions?

    Given two positive numbers a and b, we define the geometric mean and the arithmetic mean as follows G. M. = sqrt{ab} A. M. = (a + b)/2 If a = 1 and b = 2, which is larger, G. M. or A. M. ? G. M. = sqrt{1•2} G. M. = sqrt{2} A. M. = (1 + 2)/2 A. M = 3/2 Conclusion: G. M. > A. M. Correct...
  13. G

    I Solving Ordinal Arithmetic: X Countably Compact but Not Compact

    Problem Statement: Show that the set X of all ordinals less than the first uncountable ordinal is countably compact but not compact. Let μ be the first uncountable ordinal. The latter question is easy to show, but I stumbled upon a curiosity while attempting the former. In showing the former...
  14. G

    I Ordinal Arithmetic: Proving X is Countably Compact

    Problem Statement: Show that the set X of all ordinals less than the first uncountable ordinal is countably compact but not compact. Let μ be the first uncountable ordinal. The latter question is easy to show, but I stumbled upon a curiosity while attempting the former. In showing the former...
  15. R

    I Arithmetic Block in Shor Algorithm

    Hello everyone! So I was looking at Shor Algorithm for prime factorization and I have some doubts in the arithmetic part. Let's define a function f that : f(x) = ax mod N. The middle step in shor algorithm is to calculate, simultaneously, all values of f. In some papers and books, I saw some...
  16. F

    MHB Determining if a sequence is arithmetic

    Question: Find the first 5 terms of this series and determine if it is an arithmetic sequence. An= 2 + 6n Help please!
  17. MartinTheStudent

    I Do I use instrument error or arithmetic mean error?

    Hi. Let's say I have data which I have measured. For example I measured a length of an object and the measurment was repeated 5 times. An instrument which I used to measure has an error, value of which I know. My options are to either to just go with the instrument error (probably not, right?)...
  18. C

    MHB Arithmetic Concepts: Get Help Understanding Problems

    Hi everyone! I'm having trouble solving this problem, its set up in a way that I don't understand and I was hoping someone could help clarify it with me..Thanks a bunch!
  19. S

    Is my code suffering from an arithmetic error?

    What it does is described well in my comments. I'm using decimal, which is more precise than double but has a smaller range of values. https://msdn.microsoft.com/en-us/library/ms228360(v=vs.90).aspx const decimal pi =...
  20. Vitani11

    I Understanding Scalar Quantity in Basic Bra-Ket Arithmetic

    Say I have a vector product |x+a⟩⟨x| and I multiplied it by a ket vector |x'⟩. Can I pull the |x'⟩ into the ket vector |x+a⟩? also could you split up the ket vector |x+a⟩ into two ket vectors added together?
  21. S

    MHB Integer Arithmetic for Precise Calculation of Irrational Numbers

    I have authored documents of 40 years of computer software development with a mind to collect them into a publication at some point. They have been built around several software topics but mathemetics is a favorite of mine. I find a point of inspiration and write a piece of software around it...
  22. T

    MHB Using limits to solve an arithmetic problem?

    Supposing $x = 0$, do I need limits to solve $\frac{lnx }{x^2} + \frac{1}{2x^2}$? Since $lnx$ does not exist at $x = 0$, then the best we can do is $\lim_{{x}\to{0}} lnx$ which is $0$. But then, $x^2$ at zero equals 0, so we have $0/0$ which is indeterminate, so I need to find...
  23. M

    MHB Find Lowest Value for A: a1, a2, a3 & 4 | Arithmetic Progression

    a1, a2, a3 and 4 make an arithmetic progression with difference d. For which values of d, A = a1a2 + a2a3 + a3a1 has the lowest value?I don't know if I went with the right approach, but I managed to get this : A=3x2 +6xd + 2d2 for a1= x, a2 = x + d, etc... But I don't know what else to do.
  24. Thiru07

    Need help in understanding a percentage problem.

    Homework Statement A grocer sells items by mixing small stones into grains. He has 50 kg each of two varieties of rice. He mixes 7.5 kg of stones in all. For one kg of rice, he mixes 100 grams of stones in the first variety and the remaining in the other. Peter purchases 5 kg of rice of each...
  25. M

    MHB Show that the number of questions he answered is 14, Arithmetic progressions

    A competitor participating in a programme organised by a certain telelvision channel, wins by answering 15 questions correctly. The price money of 50 for the first question, 75 for second question , 100 for third question etc ... given for correct answers , are in an arithmetic progressionA...
  26. donaldparida

    B Can modular arithmetic help us find remainders and unit digits?

    I am new to number theory and I heard from my friend that we can use modular arithmetic to conveniently find the unit digit of a number or the remainder obtained on dividing a number by another number such as the remainder obtained on dividing (x^y) by a. Is it possible?How can we do this?
  27. L

    B Forgotten correct method of basic arithmetic (subtraction)

    What happens when you have to subtract: 30 -9 Basically 30 - 9 but the 3 must be made 13, so what happens to the zero? I know if it was any other number we would reduce its value by one. Also, if possible, can anyone please show me how 30 in base 5 is 110? I understand that its 1*5*5 + 1*5...
  28. Math Amateur

    MHB Fundamental Theorem of Arithmetic - Bhattacharya et al - Ch. 2, Section 1

    I am reading the book, Basic Abstract Algebra by P.B. Bhattacharya, S.K. Jain, and S.R. Nagpaul ... ... and am currently focused on Chapter 2: Integers, Real Numbers and Complex Numbers ... I need help with an aspect of the proof of the Fundamental Theorem of Arithmetic in Section 1.3 ... ...
  29. Math Amateur

    I Fundamental Theorem of Arithmetic - Bhattacharya et al

    I am reading the book, Basic Abstract Algebra by P.B. Bhattacharya, S.K. Jain, and S.R. Nagpaul ... ... and am currently focused on Chapter 2: Integers, Real Numbers and Complex Numbers ...I need help with an aspect of the proof of the Fundamental Theorem of Arithmetic in Section 1.3 ... ...The...
  30. M

    MHB A problem on arithmetic progressions.

    Okay, So far I have seen and know to find the common difference of arithmetic terms using the consecutive terms. But this progression looks different (Shake) Find the common difference , Value of Z and the first term of it. Many Thanks (Happy)
  31. alexandria

    Arithmetic and Geometric Series

    Homework Statement Homework Equations no equations required 3. The Attempt at a Solution a) so for part c) i came up with two formula's for the tortoise series: the first formula (for the toroise series) is Sn = 20n This formula makes sense and agrees with part a). for example, if the...
  32. L

    Courses Course choice - bad at arithmetic

    Greetings. I've come with yet another issue which is causing me some distress. So I took everyone's advice and started self-teaching calculus (or at least I started trying... Limits are still confusing). However, my problem doesn't stem there. I decided to review my algebra before year 10...
  33. J

    I Congruence vs equality in mod arithmetic

    I've encountered what seems to be two different notations for modular arithmetic and I'm confused as to whether they mean the same thing. My abstract algebra textbook (Pinter) and professor would write, for example, 5 = 15mod(10), as though mod(10) is an operation that returns the amount by...
  34. nomadreid

    I Gödel's 1st Incompleteness Thm: Min Arithmetic Req'd?

    I often read (for example, in Wikipedia on "Rosser's Trick") that in order for a proof of Gödel's First Incompleteness Theorem, one assumes an efficient consistent theory of numbers which includes a "sufficient fragment of elementary arithmetic". What minimum would qualify? Is Robinson's Q a...
  35. adjacent

    Arithmetic sequence involving years

    Homework Statement A 25 year old programme for building new houses began in Core Town in the year 1986 and finished in 2010. The number of houses built each year form an arithmetic sequence. Given that 238 houses were built in 2000 and 108 in 2010, find the number of houses built in 1986...
  36. Biker

    B Arithmetic Series and Geometric Series

    Here is a question that I have a problem with, It doesn't seem to have a solution: An increasing sequence that is made of 4 positive numbers, The first three of it are arithmetic series. and the last three are geometric series. The last number minus the first number is equal to 30. Find the sum...
  37. K

    Arithmetic progression

    Homework Statement A finite arithmetic progression is given such that ##S_n>0## and ##d>0##. If the first member of the progression remains the same but ##d## increases by 2, then ##S_n## increases 3 times. If the first member of the progression remains the same but ##d## increases 4 times...
  38. karush

    MHB Find the sum of the first 17 terms of the arithmetic series

    Find the sum of the first 17 terms of the arithmetic series $$8+\sqrt{7}, \ 6,\ 4-\sqrt{7}$$ $$u=8+\sqrt{7}$$ $$S_{17} =\frac{u\left(1-\frac{{6}^{17}} {u} \right)}{u}$$ My first shot at this
  39. P

    MHB Arithmetic and geometric sequence

    *I am struggling with arithmetic and geometric sequences. if the 4th term is m-8, 6th term 8m+3 and 8th term is 10m-5 Calculate the 1st and 5th term Which term will have a value of -70The 4th term of geometric sequence is -16 and the 6th term is -64. Calculate the 3rd and 5th terms. thank you...
  40. G

    Modular arithmetic on vector spaces

    Homework Statement Let U is the set of all polynomials u on field \mathbb F such that u(3)=u(-2)=0. Check if U is the subspace of the set of all polynomials P(x) on \mathbb F and if it is, determine the set W such that P(x)=U\oplus W. Homework Equations -Polynomial vector spaces -Subspaces...
  41. I

    MHB Sum of an Infinite Arithmetic Series

    Somewhere I saw that the sum of the infinite arithmetic series \sum_{n=1}^{\infty}n = \frac{-1}{12} Why exactly is this? I thought infinite arithmetic series had no solution? Also... WHY is it negative? Seems counter-intuitive that the sum of all the NATURAL numbers is a decimal, a negative...
  42. C

    Prove that ordinal arithmetic is associative

    Homework Statement Let a, b, c be ordinals. Prove that a+(b+c)=(a+b)+c Homework EquationsThe Attempt at a Solution I looked at a set theory book by Jech and he says Prove by induction on c. Should I look at the case where its true for c+1[/B]
  43. TheMathNoob

    I can't do arithmetic operations in my head quickly

    I am currently in upper division math courses in my career and I can't do something like 109-64 quickly in my head. I obviously try to break every number apart to make hard subtractions, but I can't do it fast!. Should I worry about this?. Overall, I have done well in my math life, but I feel...
  44. M

    Advanced Arithmetic: Find Min Digits for Fraction Decimal

    Homework Statement What is the minimum number of digits to the right of the decimal point needed to express the fraction as a decimal? a) 4 b) 22 c) 26 d) 30 e) 104 Homework EquationsThe Attempt at a Solution One possible solution is: "We can rewrite the fraction as . Since the last digit of...
  45. Mark44

    Insights Why Can't My Computer Do Simple Arithmetic? - Comments

    Mark44 submitted a new PF Insights post Why Can't My Computer Do Simple Arithmetic? Continue reading the Original PF Insights Post.
  46. G

    Solving Arithmetic on Large Positive Integers with Bignum Strings

    Homework Statement Write a program (without using GMP library - https://gmplib.org) which performs arithmetic operations on large positive integers (addition, subtraction, multiplication and division). Maximum number of digits in one number is 100. Large number is the number that can't be...
  47. REVIANNA

    Solving Arithmetic Progression: Sum and Product of Four Integers

    Homework Statement the sum of four integers in A.P is 24 and their product is 945.find themHomework Equations ##(a-d)+a+(a+d)+(a+2*d)=24## ##2a+d=12## ##(a+d)(a-d)(a)(a+2d)=945## ##(a^2-d^2)(a^2+2*a*d)=945## The Attempt at a Solution there are two equations and two unknowns a(one of the...
  48. stungheld

    Arithmetic and Geometric sequence problem

    Homework Statement The sum of first three numbers of the arithmetic sequence is 54. If you subtract 3 from the first one, leave the second one unchanged and add 12 to the third one you get the first three numbers of the geometric sequence of the form ##ar + ar^2 + ar^3 + ... ar^n ## Find r...
  49. Chrono G. Xay

    Predict Digits of Irrational Numbers with Modular Arithmetic Summation?

    Would it be possible to write an equation utilizing a summation of a modular function of a Cartesian function, whose degree is dependent upon the index of the root, in that it predicts the digits less than 1 of the root, that when summed equals the computed value sqrt( n )? I already have what...
  50. Ackbach

    MHB Trachtenberg Speed System of Arithmetic

    I was very excited to learn that you can download the entire Trachtenberg Speed System of Arithmetic book for free from this website. (https://archive.org/details/TheTrachtenbergSpeedSystemOfBasicMathematics_201803) Why spend such an inordinate amount of time on arithmetic? Don't we have...
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