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. N

    Closed under addition and closed under scalar

    Could someone explain to me about what closed under addition and closed under scalar multiplication means? I have a patchy idea of what it is but how does it relates to A = {(x,y) | x^2 + y^2 <= 1}? What does A stands for? What does the language implies? Edit: My interpretation: Let's...
  2. Z

    Why is the parallelogram rule for the addition of forces as it is?

    Why is the parallelogram rule for the addition of forces as it is? I feel it must have some deep origin and pointing to something fundamental. Though I know this problem may have no answer: God design it as such. But I wonder how the first person came up with this rule, where does his/her...
  3. W

    Velocity Addition: Solving Minkowski Space with Metric Techniques

    Hi all. I'm taking a course in GR and trying to get my intuition and mathematical techniques up to speed. I've been trying to derive the velocity addition formula in Minkowski space, but for some reason I can't do it. Here's what I have: I'll use the Minkowski metric of signature...
  4. V

    Addition of carbon to cast iron beyond 6.67%

    can anyone please give me the answer for addition of carbon to cast iron beyond 6.67%..
  5. T

    Is Scalar Addition Equivalent to Adding Parallel Vectors?

    Homework Statement T/F: The addition of two scalars is equivalent to the addition of parallel vectors. Select one: a. False b. True Homework Equations The Attempt at a Solution i said true reason being the magnitude of vectors can be added if they are in the same direction by...
  6. D

    Special Relativity: Velocity Addition

    Homework Statement Two objects fly toward you, one from the east with speed u, and the other from the west with speed v. Is it correct that their relative speed, as measured by you, is v+u? Or should you use the velocity addition formula, V=(u+v)/(1+uv/c2)? Is it possible for their relative...
  7. C

    Proof about addition commuting

    Homework Statement Prove by induction that 1+a=a+1 for all natural numbers The Attempt at a Solution basis step 1+1=1+1 Now we assume that it is true for a=k and then look at k+1 I first wanted to do this (k+1)+(k+1)=(k+1)+1 but that doesn't give me the right answer. so should it be...
  8. D

    Proof for Closure of Vector Addition - Can You Help?

    If the tangent space at p is a true vector space, then it must be that the sum of two vectors is itself a directional derivative operator along some path passing through p. I've been trying to prove that this is true without any luck. My textbook "proves" that vector addition is closed by...
  9. G

    Addition of orbital angular momentum and spin

    Homework Statement Consider an electron with spin \frac{1}{2} and orbital angular momentum l=1. Write down all possible total angular momentum states as a combination of the product states | l=1 , m_l > | s = \frac{1}{2} , m_s > Homework Equations Lowering operator : J_- |j, m> =...
  10. F

    Opposite Directions: Addition vs. Multiplication

    Instead of i2 = -1, is it useful to think that (-i)i = 1, i.e. opposites cancel each others, kind of doing an addition instead of the multiplication ? If I replace i with "direction" it kind of makes sense. (I like to think that this is not just math but also physics.)
  11. M

    Velocity addition formula question

    In the Galilean formula u=u'+v, the velocities are bold so we know they're vectors, so for instance if u'=(a,b), and v=(c,d), we say that u=(a+c,b+d). But, in SR u=(u'+v)/(1+vu'/c2), none of the velocities are shown in bold, at least not not the way I saw it written. So, if we used the same two...
  12. H

    Finding Net Force [w/ Vector Addition]

    Homework Statement A 20 kg object is being acted on by a 90 N [E60N] force, a 40 N [S] force, and a 40 N [E27S] force. Find the net force acting on the object and its acceleration. Homework Equations ƩF = ma The Attempt at a Solution I understand that I have to add vector...
  13. stripes

    Searching for Closure: Mult. Inverses & Addition

    Homework Statement Homework Equations The Attempt at a Solution Well thankfully I just have to present closure under mult. inverses and closure under addition. But I seem to be going in circles...if a is in G, then we need to show that a-1 is also in G. So a*a-1 = 1F, but is...
  14. S

    MHB Addition with eight bit 2's complement numbers. [Check my Work]

    Perform the following operations involving eight-bit 2's complement numbers and indicate whether arithmetic overflow occurs. 1) 01110101 + 11011110. So I have a +Positive + a -Negative number. So I just added normally, and got the result of 01010011. I realized before I started the problem I...
  15. S

    MHB Addition involving eight-bit 2's complement numbers and indicating overflow.

    Perform the following operations involving eight bit 2's complement numbers and indicate whether arithmetic overflow occurs. 1. 00110110 + 01000101 = 01111011 Overflow: there is none because we are adding to positive numbers. Did I do this correctly?
  16. S

    MHB Adding Binary #s, (Sign and Magnitute Addition)

    Can someone explain to me step by step what's going on here? The book acts like a 3rd grader should be able to just look at the model and be able to decipher what's going on, but this is very confusing material. Perform the following operations involving eight-bit 2's complement numbers and...
  17. S

    MHB Two's Complement Binary Addition

    (Envy) I understand 1 0 1 0 +1 1 0 1 _________ =0 0 0 1 0 1 0 1 1 0 1 ________ 0 1 1 1 which becomes 1000 BUT, how does 1 0 1 0 +0 0 1 0 _________ not become 1 0 1 0 1 0 1 0 + 0 0 1 0 ___________ ?
  18. I

    Linear Algebra Matrix Addition Algorithm

    Homework Statement let L and M be two symmetric nxn matrices. develop an algorithm to compute C=L+M, taking advantage of symmetry for each matrix. Your algorithm should overite B and C. What is the flop-count? Homework Equations How to minimize the number of flop count? I want to make...
  19. S

    Non-parallel Relativistic Velocity Addition

    Homework Statement (d) A spaceship moves with velocity (4/5)c relative to Earth. The astronaut throws his empty beer bottle out the window with velocity (3/5)c relative to the ship in the sideways direction. What is the velocity of the bottle (magnitude and direction) relative to the Earth (i)...
  20. B

    Proving trig equations using addition formulae

    SOLVED! :) Homework Statement The question is as follows; Prove that (tan(A+B)-tanA)/1+tan(A+B)tanA = tanB Homework Equations I'm certain the addition fomulae need to be applied, although I'm not entirely sure how. I genuinely have tried many times! The Attempt at a Solution...
  21. MI5

    MHB Counting proof of the addition rule

    Let $ \left\{A_1, A_2, \cdots , A_n\right\}$ be a system of subsets of a finite set $A$ such that these subsets are pairwise disjoint and their union $A = \cup_{i=1}^{n}A_{i}$. Then $ |A| = \sum_{i=1}^{n}|A_i|$. (1) Proof: According to the hypothesis, each $a \in A$ belongs to exactly one of...
  22. M

    Extremely confusing vector addition problem

    Homework Statement Suppose a plane flies at a constant groundspeed of 500 miles per hour due east and encounters a 50 mile-per-hour wind from the northwest. Both the airspeed and the compass direction must change to for the plane to maintain its groundspeed and eastward direction. Find the...
  23. M

    Calc 3: Vector Addition Problem Solution Example

    I'm trying to do this homework problem but I keep getting an answer that is not given. Can someone tell me what I am doing wrong please. This is for University level calc 3. Homework Statement Homework Equations A vector U=<x,y> ||U|| means the length of U = √(x^2 +y^2). vectors can also...
  24. PsychonautQQ

    Understanding Vector Addition in Acceleration Problems

    Homework Statement http://grephysics.net/ans/0177/23 The answer to this problem is confusing me. So you can calculate the normal acceleration and tangential acceleration to each by equal and 90° apart from each other, so their final vector has an angel of 45°. However, if the speed travels...
  25. C

    Addition reactions: Alkenes vs Alkynes

    Homework Statement Alkenes are more reactive than alkynes toward addition of electrophilic reagents like HCl. Yet when alkynes are treated with one molar equivalent of these reagents, it is easy to stop the reaction at the alkene stage. This appears to be paradox. Explain. The Attempt at a...
  26. J

    Why multiplication produces new uniits but not addition or subtraction

    In physics new units are usually multiplication or division of some physical quantities. Why is there no unit which is a result of addiing or subtracting other units? I'm sensing this is a rather stupid question but still I'm going to ask anyway. Thank you!
  27. D

    Why is addition of angular momenta important in Quantum Mechanics?

  28. V

    Color addition involving wavelengths

    recently i have been thinking about how colors add when you have 2 different light sources of color and put the streams together, such as taking red and green beams and crossing them to make yellow. i was looking at wavelengths to determine if there was a mathematical way to say what color would...
  29. Y

    Matrix multiplication to addition

    I am looking for a transformation that relates a matrix product with a matrix addition, e.g. AB = PA + QB Is there any such transformation? Thnx
  30. W

    How can I solve for beta and gamma in a cartesian vector addition problem?

    I am having a really tough time trying to solve this particular problem I can find alpha easily 139 degrees which is correct. However when I go to try and solve for beta or gamma I get this huge mess and I am unable to solve for either one, Please help me I know I am overlooking something but I...
  31. S

    Understanding Vector Addition in Physics

    I am stuck on this one: According to me, the only correct answers are \vec{r} = \vec{t} - \vec{s} \\ \vec{r} + \vec{s} = \vec{t} But according to the key, these are also correct \vec{r} + \vec{t} = \vec{s} \\ \vec{s} + \vec{t} = \vec{r} I honestly do not see how, can somebody...
  32. E

    Addition to a random matrix element

    Hi all! I have no application in mind for the following question but it find it curious to think about: Say that we have a square matrix where the sum of the elements in each row and each column is zero. Clearly such a matrix is singular. Suppose that no row or column of the matrix is the...
  33. J

    Why addition? Integral problem

    Hello, I can't seem to paste this in. So here is the link. http://www.calcchat.com/book/Calculus-ETF-5e/ It is chapter 7, section 1, question 17. Why are you adding ∫ [(x+2) +√4 -x] dx + ∫ 2(√4 -x) dx ? I'm confused for 2 reasons. Why can you not just evaluate from -5 to 4? ( you will see the...
  34. pratikaman

    Real mechanism behind addition, subtraction, multiplication, division

    we all know the basic rules for operations of addition, subtraction, multiplication and division. but what i don't know is why these rules (of addition, subtraction, multiplication and division) works. as if we have been given algorithm to do these operations but not explained how they...
  35. C

    Is (nearly) all mathematics addition?

    So far after observing the math I've been learning for the past 10 years of my school career it seems to me like everything can be simplified into addition. Ill show by example; Addition: obviously Subtraction: Addition of negative numbers Multiplication: Repeated Addition Division...
  36. M

    How Do You Calculate Vector Magnitudes and Directions from Given Angles?

    Homework Statement Three vectors, A, B and C each have a magnitude of 50 units. Their directions relative to the positive direction of the x-axis are 20°, 160° and 270°, respectively. Calculate the magnitude and direction of each of the following vectors. 1.) A + B + C 2.) A-B+C 3.)...
  37. C

    Analytical method of vector addition

    Homework Statement Joe pushes on the handle of a 10 kg lawn spreader. The handle makes a 45° angle with the horizontal. Joe wishes to accelerate the spreader from rest to 1.39 m/s in 1.5s. What force must Joe apply to the handle? Neglect friction. Homework Equations A=Vf-Vi/t. F=ma...
  38. M

    Organic chemistry addition reaction

    Homework Statement Number 8 is the one I was unsure of in my exam. I know a hydride shift would give it a better carbocation, however it was not tetra hedral, so I panicked, and quickly did a alkyl shift, it worked out on the exam, but I was on the verge of running out of time so I...
  39. D

    Is all Math nothing more than addition?

    "Nothing more" makes it sound bad. Do you think that all math is based on addition and subtraction? For example multiplication is a method for repetitive addition, is all math in a form or another just a more complicated way to use addition/subtraction?
  40. A

    The addition of the wave functions for a system

    the wave functions of individual particles can be added together to create a wave function for for system, that means quantum theory allows physicists to examine many particles at once??...how is it possible if the wave function of each particles is different??...is it based on rules of...
  41. U

    Linear equations using addition

    I'm probably making a silly mistake or Wolfram Alpha is lying to me. Question: Find the value of c and d. 3d=13-2c \frac{3c+d}{2}=8 Rearranged, simplified and multiply each equation by 2: 6d+4c=26 d+3c=16 Now find the common multiple which in my case I will use 12: 18d+12c=78 -4d-12c=-64...
  42. coolul007

    Floating Point Addition - hardware/software

    Can someone explain to me how floating point addition is implemented on a x86 in hardware or software. I would like to find out what method is used to add varying number size. if I have a 1 X 10^-100 + 2 X 10^50. are the exponents average for a common ground or does the large one rule etc. or is...
  43. M

    Matrices: lines addition master trick.

    In Jordan, Gauss-Jordan and Laplace it's necessary to miltiply a line by a constant, add the result to other line in order to obtain 'zeros' (to facilitate the process while using Laplace) or to obtain a identity matrix (Jordan and Gauss-Jordan). I take TOO long while doing this and sometimes...
  44. dexterdev

    Why XOR (mod 2 addition) and multiplication are anologous?

    Hi PF, I have a serious doubt. In spread spectrum modulation etc we multiply the message signal with PN sequence , so as to spread it in frequency domain, good and I am OK with that. But when explanation comes multiplication becomes XOR ing in most books and articles. How XOR...
  45. K

    Numerical floating point addition

    If I have a bunch of sorted numbers spanning a large range of magnitudes, is it better to add them up from smallest to largest or from largest to smallest, or something else? Let's say I'm summing an array A, which is sorted from large to small. Which gives a more accurate result: sum1 = 0...
  46. N

    Phasor Addition Simplify 5cos(wot+90°)+5cos(wot-30°)+5cos(wot-120°): Solve

    simplfy 5cos(wot+90°)+5cos(wot-30°)+5cos(wot-120°) to form Acos(cos+Ø) i did it this way 5cos(wt) = 5e^j0 = 5+J0 5cos(wot-30°) = 5e^-j30° = 5*(√3/2)+J5/2 5cos(wot-120°) = 5e^-j120° = -5/2*(√3/2)+J5*(√3/2) i have only done partially the problem just missing to add them up however when i look up...
  47. F

    How Do You Combine 3sin(ωt) and 2cos(ωt) into a Single Trigonometric Function?

    Homework Statement Express 3sin(ωt) + 2cos(ωt) in the form Rsin(ωt + α) AND verify the resultant function is the same frequency as 3sin(ωt) and 2cos(ωt) Homework Equations R = √a2+b2 α = arctan(b/a) The Attempt at a Solution My attempt using the equations above produces...
  48. C

    What would E field addition be if Gauss's law was different

    What if Gauss' Law was div(E)=q2/eps0 Then if we had two point charges, how could we calculate the resultant E field at an arbitrarily point? Obviously superposition would not work anymore, so how could it be done mathematically? This is essentially a math question
  49. J

    Non-parallel velocity addition

    Homework Statement A spaceship moves with a velocity 0.75c xhat with respect to the earth. The ship releases a probe with a velocity 0.75c yhat relative to the ship. What is the velocity of the probe as viewed on earth, magnitude and direction? Homework Equations The velocity-addition...
  50. S

    Vector Addition? Which Vectors Do I Draw and Where?

    Homework Statement Vectors a and b have magnitudes 2 and 3. The angle between them is 50 degrees. Determine 5a - 2b and determine its magnitude and direction Homework Equations Cosine Law and Sine Law The Attempt at a Solution I can solve for the magnitude using the cos law and...
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