What is Addition: Definition and 651 Discussions

Addition (usually signified by the plus symbol +) is one of the four basic operations of arithmetic, the other three being subtraction, multiplication and division. The addition of two whole numbers results in the total amount or sum of those values combined. The example in the adjacent image shows a combination of three apples and two apples, making a total of five apples. This observation is equivalent to the mathematical expression "3 + 2 = 5" (that is, "3 plus 2 is equal to 5").
Besides counting items, addition can also be defined and executed without referring to concrete objects, using abstractions called numbers instead, such as integers, real numbers and complex numbers. Addition belongs to arithmetic, a branch of mathematics. In algebra, another area of mathematics, addition can also be performed on abstract objects such as vectors, matrices, subspaces and subgroups.Addition has several important properties. It is commutative, meaning that order does not matter, and it is associative, meaning that when one adds more than two numbers, the order in which addition is performed does not matter (see Summation). Repeated addition of 1 is the same as counting; addition of 0 does not change a number. Addition also obeys predictable rules concerning related operations such as subtraction and multiplication.
Performing addition is one of the simplest numerical tasks. Addition of very small numbers is accessible to toddlers; the most basic task, 1 + 1, can be performed by infants as young as five months, and even some members of other animal species. In primary education, students are taught to add numbers in the decimal system, starting with single digits and progressively tackling more difficult problems. Mechanical aids range from the ancient abacus to the modern computer, where research on the most efficient implementations of addition continues to this day.

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  1. M

    I Velocity Addition: Angled Motion Explained

    Any ideas?
  2. V

    B Justification of addition in Spivak, Ch.1

    I am 100% reading too much into this, but I am curious which of the properties provided by Spivak allow one to justify a specific argument. For reference/context, the properties are: P1: If a, b, and c are any numbers, then $$a +(b + c) = (a + b) +c$$ P2: If a is any number, then $$a + 0 = 0 +...
  3. Benjamin_harsh

    Calculating Net Force & Acceleration: Addition of Forces Explained

    ##\sum F_{X} = F - F_k## (The net force, ##\sum F_{X}## is always equal to m.a) ##ma = F - \mu_{k}.F_n## ##ma = F - \mu_{k}.(mg)## [Here ##F_n = mg## when body is on flat surface] ##20(a) = 150 - 0.25(20)(10)## ##\large a = \frac {150 - 0.25(20)(10)}{20}## ##\large a = 5\frac{m}{sec^2}##...
  4. F

    I Proof of Addition Reversibility

    How can you prove that ##f(x)=g(x) \Leftrightarrow f(x)+C=g(x)+C##
  5. jim mcnamara

    How Dihalogens Catalyze Michael Addition Reactions. - NCBI - piectron?

    How Dihalogens Catalyze Michael Addition Reactions. - NCBI What Michael Addition is about: https://en.wikipedia.org/wiki/Michael_reaction I do not know what the definition of 'piectron' is. In the abstract of the paper, I assume it is a misspell. Which makes me wonder? I learned about...
  6. Charles Link

    B A problem involving the addition of two cubes

    This post is meant to be a fun one. It is at the beginner level, because I think most people at the intermediate level would know what the solution is, but they might also find it entertaining as well. ## \\ ## A hypothetical situation: An algebra teacher observed that ## 3^2+4^2=5^2 ## and...
  7. R

    Addition of Subscript F_4k+4: Explained with Examples

    Can somebody please explain below ##F_{4k+4} = F_{4k+3} + F_{4k+2}## I thought it would be ##F_{4k+4} = F_{4k+3} +F_{4k+1}## Thanks!
  8. E

    MHB Proving Addition/Subtraction Formulas - Tan(x-y)+Tan(y-z)+Tan(z-x)

    tan(x-y)+tan(y-z)+tan(z-x)=tan(x-y)tan(y-z)tan(z-x) I have redone this problem two or three times and all the steps just make my head spin. I've tried looking up tutorials online but they introduce things into the problem that we haven't been taught yet and that just confuses me more. Help!
  9. hnnhcmmngs

    But, as I said, you don't actually need the coordinates at all.

    Homework Statement Calculate |u+v+w|, knowing that u, v, and w are vectors in space such that |u|=√2, |v|=√3, u is perpendicular to v, w=u×v. Homework Equations |w|=|u×v|=|u|*|v|*sinΘ The Attempt at a Solution [/B] Θ=90° |w|=(√2)*(√3)*sin(90°)=√(6) Then I tried to use u={√2,0,0}...
  10. karush

    MHB Matrix Addition: OK - No Examples Found

    OK from the text bk I did not see any example of this the circle red is mine ... why is this here so not sure how these questions are to be answered. Much Mahalo
  11. nothing909

    Binary Addition: How to Add Binary Numbers with Examples

    Homework Statement Shown is two binary numbers adding with the answer The Attempt at a Solution Looking at the end side 1.1+1.1, shouldn't that be 11.00... instead of 10.00...
  12. opus

    B Vector addition- Positive x axis.

    Please see the attached page to see what I'm talking about. In the top right paragraph, it states to use "the positive direction of the x-axis". It is given that ##θ_2=30°## and it shown visually at the bottom of the page. In the problem it's using -60° and I'm not sure how they're getting that.
  13. A

    B Addition, multiplication, divison and subtraction of error

    If given two variable of the form p=x±δx and q=y±δy where δx and δy are the error obtained while measuring p and q and x and y are it's absolute value obtained. We define R=p+q,p-q,p/q,pq In each of this case I want to know what will be the error in R.(e.g δR) Thank you
  14. N

    B Velocity Addition Q: Ball Thrown from Moving Ship?

    Suppose a spaceship is flying past me, an observer on earth, at .86c. A traveler on the ship has a ball which is attached to a string which is 100 feet long in the ship frame. The traveler grabs the string and throws the ball out of the front window of the ship at a speed of 100 feet per second...
  15. CDL

    Einstein Velocity Addition for a Moving Charge in a Wire

    Homework Statement I am reading through Griffiths' Electrodynamics, and I have come to the scenario in the Relativity chapter where in an inertial reference frame ##S##, we have a wire, with positive charges (linear density ##\lambda##) moving to the right at speed ##v##, and negative charges...
  16. C

    A Is it possible to locate prime numbers through addition only

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  17. M

    Fractional polynomial addition

    Homework Statement Determine whether there exist ##A## and ##B## such that: $$\frac{1}{3x^2-5x-2} = \frac{A}{3x+1} + \frac{B}{x-2}$$Homework Equations None The Attempt at a Solution [/B] First I divided the polynomial ##3x^2-5x-2## by ##3x+1## and got ##x-2## as a result without a...
  18. Specter

    Proving the associative property of vector addition

    Homework Statement Give an example of the associative property of vector addition using vectors in Cartesion form. Homework Equations (u+v)+w=u+(v+w) The Attempt at a Solution I can't figure out how to get the arrow on top of my work so I wrote it without it. I'm somewhat confused on why I...
  19. C

    MHB Proving matrix group under addition for associative axiom

    Dear Everyone, I have some feeling some uncertainty proving one of the axioms for a group. Here is the proof to show this is a group: Let the set T be defined as a set of 2x2 square matrices with coefficients of integral values and all the entries are the same. We want to show that T is an...
  20. C

    Simple Vector Addition Problem

    Homework Statement "A bird flew 40 m to the west, then 100 m in a direction 36.9 degrees to the north of east. Use the algebraic addition of vectors to fins the magnitude of the bird's net displacement" Use sin (36.9) =0.6 and cos (36.9) = 0.8 Homework Equations Vector addition? A =...
  21. alexi_b

    Vector Addition Question: find angle (A+B & A-B)

    Homework Statement Two vectors A and B have precisely equal magnitudes. For the magnitude of A + B to be 65 times greater than the magnitude of A - B, what must be the angle between them? Homework EquationsThe Attempt at a Solution I tried using the dot product and solving for the angle but i...
  22. ExplosivePete

    Capacitance Addition Formula Origin

    Hello, Could someone help me find a general derivation of the formula for addition of capacitance of conductors? I'm hoping to find something that could be applied to a general assortment of conductors, not just parallel plate capacitors. I look forward to your response, Pete
  23. karush

    MHB *aa3.2 Let Q be the group of rational numbers under addition

    aa3.2 Let Q be the group of rational numbers under addition and let $Q^∗$ be the group of nonzero rational numbers under multiplication. In $Q$, list the elements in $\langle\frac{1}{2} \rangle$, In ${Q^∗}$ list elements in $\langle\frac{1}{2}\rangle $ ok just had time to post and clueless
  24. R

    B Addition of Perpendicular Vectors in two ways

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  25. xaratustra

    Vector addition and Newton's law

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  26. L

    Maximizing Localization in Adding Multiple Plane Waves

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  27. V

    Vector addition and the force applied to the shaft of the pulley

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  28. R

    I Relativistic addition question

    Observer A measures B moving east at .8c and measures C moving west at .8c . Then wouldn't the impact of the two will be a result of relative motion greater than c in frame A. I understand B and C frame will not see any object speed greater than c. So the impact in frame A and frame B will not...
  29. Poetria

    Addition of power series and radius of convergence

    Homework Statement ##f(x)=\sum_{n=0}^\infty x^n## ##g(x)=\sum_{n=253}^\infty x^n## The radius of convergence of both is 1. ## \lim_{N \rightarrow +\infty} \sum_{n=0}^N x^n - \sum_{n=253}^N x^n## 2. The attempt at a solution I got: ## \frac {x^{253}} {x-1}+\frac 1 {1-x}## for ##|x| \lt 1##...
  30. Hamza Abbasi

    Relativistic Velocity Addition

    Homework Statement Two spaceships pass each other. Spaceship A moves relative to nearby planet at velocity v1 , while spaceship B moves at velocity v2 relative to the planet . How fast does spaceship A moves relative to spaceship B. Homework Equations Velocity Transformation : v* =...
  31. Physics345

    Sketching resultants using vector addition.

    Homework Statement Given the following diagram, use vector addition to sketch the resultants: a) u→−2v→ b) 3v→−u→ Homework EquationsThe Attempt at a Solution For this question I am fairly certain that it is correct but for some reason I'm doubting myself. I was wondering if I could get some...
  32. srfriggen

    Partition the integers under "anti-closure" of addition

    Homework Statement Can you partition the positive integers in such a way that if x, y are member of A, then x+y is not a member of A. x and y have to be distinct. That is, {1, 2, 3} cannot be in the same set, since 1+2 = 3, but 1 and 2 can be, since 1+1=2, but 1 and 1 are not distinct...
  33. isukatphysics69

    General question about vector addition -- using graphical versus mathematical methods

    Homework Statement I have an exam on vectors and 2 dimensional motion today. I NEED to know if it is possible to solve any GRAPHICAL VECTOR PROBLEM using only INTEGRATION AND MATH WITHOUT ADDING VECTORS TOGETHER GRAPHICALLY. So given any word problem that is supposed to be solved using that...
  34. R

    Does x = -1 Make the Function FoG Undefined Even If Part of It Equals Zero?

    Homework Statement Say you have a fn FoG = x+2/x+1 + x+1/x+2 if x = -1, the first one is undefined. But the second one would end up as 0, a real number I'm trying to understand, would x = -1 make the whole term FoG undefined? because only x+2/x+1 would be undefined, x+1/x+2 would be =...
  35. A

    B Addition of velocities at speeds near C - any differences?

    Good evening, If a ship is moving at 0.5C east in a gravity free environment, and fires a bullet perpendicular at 0.5c, is it just the hypotenuse (a squared + b squared)/c squared that we use for the speed? Or is there more to it since we are at great speed?
  36. H

    Strategies for Solving Complex Addition Problems

    Homework Statement Homework EquationsThe Attempt at a Solution I tried to rationalize the fractions by multiplied it by $$\sqrt{n + 1} - \sqrt{n} $$ it will be sum of $$ \frac {\sqrt{n + 1} - \sqrt{n}} {2n +1} $$ also tried to see the arithmetic structure $$ 1 + \frac{1}{\sqrt{2} +...
  37. S

    I Velocity addition and conservation of the energy

    Hello everyone, For some time I'm a little bit confused about (at the first view) a very simple question, which is about the conversation of the energy of moving objects (in terms of special relativity). As an example let's talk about firearms. If the mass of the gun M1 is infinitely higher...
  38. M

    How Do You Calculate Eigenvalues for Combined Spins in Quantum Mechanics?

    Homework Statement Given 3 spins, #1 and #3 are spin-1/2 and #2 is spin-1. The particles have spin operators ## \vec{S}_i, i=1,2,3 ##. The particles are fixed in space. Let ## \vec{S} = \vec{S}_1 + \vec{S}_2 + \vec{S}_3 ## be the total spin operator for the particles. (ii) Find the eigenvalues...
  39. S

    Addition reaction to produce 3-methyl-3-hexanol

    Homework Statement Write a chemical equation for addition reactions to produce 3-methyl-3-hexanol. Condensed structural formulas should be used for each organic compound.Homework Equations none The Attempt at a Solution https://i.imgur.com/REIpKHK.jpg Is this correct? My friend tried...
  40. D

    I Velocity Addition Formula: u_y > c?

    If a frame S' moves with velocity v in x direction wrt S, and you have v = 0.5c, u_x' = -0.9c and u_y' = 0.9c then it seems you find that u_y = 1.417c, greater than c. The relevant formula is: u_y = u_y'/(gamma * [1+v u_x' / c^2]) Apologies for formatting. Details: gamma = 1/sqrt(1-v^2/c^2)...
  41. M

    What Are the Forces at Play in Antarctic Snowcat Towing?

    Homework Statement Two snowcats in Antarctica are towing a housing unit north, (an example is shown in Figure 4-50 on page 102). Snowcat A pulls at an angle 49 o west of north, and snow cat B pulls at an angle 31 o east of north. The sum of the forces FA and FB exerted on the unit by the...
  42. A

    Expressing the addition of two sinusoidal waves this form.

    Homework Statement Express the following in the form x = Re{Aeiαeiωt} (a) x= cos(ωt) + sin(wt) (b) x= sin(ωt +π/4) + cos(ωt) (c) x= 2cos(ωt+π/3) + (√3)sin(ωt)-cos(ωt) Homework Equations cos x = 1/2 e^ix + 1/2 e^-ix sin x = − i/ 2e^ix + i/2 e^−ix The Attempt at a Solution To be honest, I have...
  43. parshyaa

    I Why could we represent the addition of two vectors like this?

    Do we have any proof to show that we can represent the addition of two vectors like this, i mean do we have proof for triangle law of vector addition(or its a law that is why we can't have its proof, then please give me a satisfying reason for this)
  44. Mr Davis 97

    I Addition of exponents proof in group theory

    Just out of curiosity, what would a proof of ##a^m a^n = a^{m+n}## amount to? Of course obviously if you have n of one thing and m of another you get m+n, but I am wondering if this is rigorous enough, or if you need induction.
  45. I

    Center of Mass (Subtraction over addition)

    Homework Statement Find the coordinates for the center of mass in the picture above. Plate is uniform.Homework Equations Since it's uniform, the lengths can be converted to the mass. So: coord x =(M1X1 +M2X2 + M3X3 + ... +MiXi)/(M1 +M2 + M3 + ... +Mi) coord y =(M1Y1 +M2Y2 + M3Y3 + ...
  46. Math Amateur

    MHB Addition and Multiplication of Natural Numbers - Bloch Th. 1.2.7 .... ....

    I am reading Ethan D. Bloch's book: The Real Numbers and Real Analysis ... I am currently focused on Chapter 1: Construction of the Real Numbers ... I need help/clarification with an aspect of Theorem 1.2.7 (1) ... Theorem 1.2.7 reads as follows: https://www.physicsforums.com/attachments/6976...
  47. G

    Addition of Angular Momentum for identical particles

    This is the problem I'm trying to understand: Consider two particles with spin 1 without orbital angular momentum. If they are distinguishable, from the rule of addition of angular momentum applied to spin, we'll have states of total spin j=0,1,2. If we have, however, identical particles which...
  48. A

    A Multiplication and Addition to get an integer

    Hi, to understand finally the Laue equation for diffraction I am missing something : h*p+k*q+l*r = integer. Given that p,q,r are integers how come h,k,l MUST BE INTEGERS as well? Say p=q=r=2, than h=k=l=1/2 works just fine. I understand that there is something about a common...
  49. R

    How will the addition of phenol to a solution affect the O2

    I asked a question regarding oxygen meters recently in another section of the forum but unfortunately no one seems to have used them before. So I thought I would break the problem down and ask a direct chemistry question... If I introduce phenol to a water solution will there be any reaction...
  50. N

    Addition of spin and degeneracy

    Homework Statement S=1/2 I1=I2=1/2 (nuclear spin) I=I1+I2 J=S+I Homework Equations H= a S . I1 + a S.I2 The Attempt at a Solution A) Find the values of I, the eigenvalues of I2, their degree of degeneracy and show that [H,I2]=0Using momentum addition I get I = 0 or 1 I2 has for...
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