# What is Multiplication: Definition and 496 Discussions

Multiplication (often denoted by the cross symbol ×, by the mid-line dot operator ⋅, by juxtaposition, or, on computers, by an asterisk *) is one of the four elementary mathematical operations of arithmetic, with the other ones being addition, subtraction and division. The result of a multiplication operation is called a product.
The multiplication of whole numbers may be thought of as a repeated addition; that is, the multiplication of two numbers is equivalent to adding as many copies of one of them, the multiplicand, as the quantity of the other one, the multiplier. Both numbers can be referred to as factors.

a
×
b
=

b
+

+
b

a

times

{\displaystyle a\times b=\underbrace {b+\cdots +b} _{a{\text{ times}}}}
For example, 4 multiplied by 3, often written as

3
×
4

{\displaystyle 3\times 4}
and spoken as "3 times 4", can be calculated by adding 3 copies of 4 together:

3
×
4
=
4
+
4
+
4
=
12

{\displaystyle 3\times 4=4+4+4=12}
Here, 3 (the multiplier) and 4 (the multiplicand) are the factors, and 12 is the product.
One of the main properties of multiplication is the commutative property, which states in this case that adding 3 copies of 4 gives the same result as adding 4 copies of 3:

4
×
3
=
3
+
3
+
3
+
3
=
12

{\displaystyle 4\times 3=3+3+3+3=12}
Thus the designation of multiplier and multiplicand does not affect the result of the multiplication.The multiplication of integers (including negative numbers), rational numbers (fractions) and real numbers is defined by a systematic generalization of this basic definition.
Multiplication can also be visualized as counting objects arranged in a rectangle (for whole numbers), or as finding the area of a rectangle whose sides have some given lengths. The area of a rectangle does not depend on which side is measured first—a consequence of the commutative property.
The product of two measurements is a new type of measurement. For example, multiplying the lengths of the two sides of a rectangle gives its area. Such products is the subject of dimensional analysis.
The inverse operation of multiplication is division. For example, since 4 multiplied by 3 equals 12, 12 divided by 3 equals 4. Indeed, multiplication by 3, followed by division by 3, yields the original number. The division of a number other than 0 by itself equals 1.
Multiplication is also defined for other types of numbers, such as complex numbers, and more abstract constructs like matrices. For some of these more abstract constructs, the order in which the operands are multiplied together matters. A listing of the many different kinds of products used in mathematics is given in Product (mathematics).

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1. ### I Formal definition of multiplication for real and complex numbers

I know that the definition of multiplication for integers is just repeated addition. For example, 5 times 3 means 5 + 5 + 5, but what about if we want to extend this definition to real or complex numbers ? Like for example, what does pi times e mean ? How are we supposed to add pi to itself e...
2. ### B A question about rules of multiplication

This might sound like a stupid question but I am just wondering why is it that x times yz equals xyz and not xyxz ? Why don't we distribute multiplication in this case ?
3. ### Number of Multiplications in the FFT Algorithm

Hello everyone, maybe some of you know the formula for the number of multiplications in the FFT algorithm. This is again given as ##N/2 \cdot log(N)##. Why is that so? Can you really "prove" this? I can only deduce this from what I know, because we have ##log(N)## levels and ##N/2##...
4. ### Prove ##(a\cdot b)\cdot c =a\cdot (b \cdot c)## using Peano postulates

with this background, we proceed to the proof. Let us define a set $$G = \{ z \in \mathbb{N} | \; x, y \in \mathbb{N}\; (x \cdot y) \cdot z = x \cdot (y \cdot z) \}$$ We want to prove that ##G = \mathbb{N} ##. For this purpose, we will use part 3) of Peano postulates given above...
5. ### Prove ##a\cdot b = b \cdot a ##using Peano postulates

with this background, we proceed to the proof. Let us define a set $$G = \{ z \in \mathbb{N} | \mbox{ if } y \in \mathbb{N}, y\cdot z = z \cdot y \}$$ We want to prove that ##G = \mathbb{N} ##. For this purpose, we will use part 3) of Peano postulates given above. Obviously, ## G...
6. ### B How Can I Calculate Negative Multiplication Without Following Traditional Rules?

Say I have 6 pencils. I want to times this by negative two. Now ignoring the rules that your teacher taught you work this out. 6 pencils negative 2 times. Negative one time would be 0 and another negative times would be -6 right? So 6 x - 2 = -6 according to simple logic. The calculator will say...
7. ### B Are there two kinds of inverse with respect to closure?

For every instance of addition or multiplication there is an inverse, closed on the naturals. Not every instance of subtraction and division is defined, so not closed on the naturals. This looks like two kinds of inverse. Instance inverse - the inverse of instances of addition and...
8. ### Proof of ##M^n## (matrix multiplication problem)

For, Does anybody please know why they did not change the order in the second line of the proof? For example, why did they not rearrange the order to be ##M^n = (DP^{-1}P)(DP^{-1}P)(DP^{-1}P)(DP^{-1}P)---(DP^{-1}P)## for to get ##M^n = (DI)(DI)(DI)(DI)---(DI) = D^n## Many thanks!
9. ### B Relation between Division and multiplication

For example what is ##\frac {169}{13} = ?## This says “When ##169## is divided into ##13## groups how many there are in each group?” This can be converted into a multiplication problem like this “##13## groups of how many in each group makes ##169##?” This is ##13 * ? = 169##. It can be solved...
10. ### B What is the link between proportion and multiplication?

I found this quote online: “Multiplication is the mathematical manifestation of the fundamental physical phenomenon of proportionality (as addition is to combination).” Question 1: How are multiplication and proportion linked? How can and WHY DOES multiplication model proportion? (My...
11. ### B How and why can multiplication combine physical quantities?

I am on a journey to not just understand how to manipulate physics equations but to understand why they work , and how they describe physical phenomena. I understand how division combines physical quantities. I have this much physical quantity 'per' this much physical quantity. It puts 2...
12. ### A Determining if a list of numbers is a result of multiplication

Suppose I have 2 collections of lists. In the first collection the lists consists of random integers, with most (but not all) in the range 0-1000. In the second collection the lists consist of integers calculated in the following way: a. start with a random integer of similar range to the...
13. ### Linearity and non-linearity in addition and multiplication

Hello friends. Excuse my ignorance. Why is addition linear and not multiplication?
14. ### I Reasoning behind Infinitesimal multiplication

Hello everyone! I have quite a bit of experience with standard calculus methods of differentiation and integration, but after seeing some of Walter Lewin's lectures I noticed in his derivation of change in momentum for a rocket ejecting a mass dm, with a change in velocity of the rockey dv, he...
15. ### Is there a mistake in this tensor multiplication problem?

ep_{ijkl} M^{ij} N^{kl} + ep_{ijkl}N^{ij} M^{kl} The second term can be rewritten with indices swapped ep_{klij} N^{kl}M^{ij} Shuffle indices around in epsilon ep{klij} = ep{ijkl} Therefore the expression becomes 2ep_{ijkl}M^{ij}N^{kl} Not zero. What is wrong here?
16. ### Is result of vector inner product retained after matrix multiplication?

Hi, I was thinking about the following problem, but I couldn't think of any conclusive reasons to support my idea. Question: Let us imagine that we have two vectors ## \vec{a} ## and ## \vec{b} ## and they point in similar directions, such that the inner-product is evaluated to be a +ve...
17. ### MHB SQL commands with subtraction and multiplication

Hey! :giggle: The following relations of a project administration of a company are given, where the primary key of the respective relations are underlined. An employee can be assigned to several projects. Furthermore, an employee can have different competencies, which are billed at different...
18. ### Multiplication of Taylor and Laurent series

First series \frac{1}{2}\sum^{\infty}_{n=0}\frac{(-1)^n}{n+1}(\frac{1}{p^2})^{n+1}= \frac{1}{2}(\frac{1}{p^2}-\frac{1}{2p^4}+\frac{1}{3p^6}-\frac{1}{4p^8}+...) whereas second one is...

44. ### I Amplitude and Multiplication

When we multiply psi sub x, psi sub y, psi sub z and psi sub t together to get a function of all four variables, does each separate wavefunction have a radius of one such that the radius is unchanged after the multiplication or is their radius far smaller than one? Secondarily, can this...
45. ### 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
46. ### Hexadecimal Multiplication using 2's complement

Homework Statement Perform the following operations using 2's complement method. ##FFFD_{16} * FFF1_{16}## Homework Equations - The Attempt at a Solution ##FFFD_{16}## ----> (1's comp.) = ##0002_{16}## -----> (2's comp.) = ##0003_{16}## ##FFF1_{16}## ----> (1's comp.) = ##000E_{16}## ----->...
47. ### B Vector multiplication and division

what is the use of multiplying and dividing a vector by a scalar?
48. ### MHB Tweaking a Multiplication Table

I recently posted a couple of multiplication tables and I feel it needs a tweak: \begin{array}{c||c|c|c|c|} V & e & a & b & c \\ \hline \hline e & e & a & b & c \\ a & a & e & c & b \\ b & b & c & e & a \\ c & c & b & a & e \end{array} The LaTeX on the forum doesn't like the double lines ||...
49. ### MHB Real Analysis, liminf/limsup Equality and Multiplication

Here are a couple more problems I am working on! Problem 1: Prove that, $limsupa_n+liminfb_n \leq limsup(a_n+b_n) \leq limsupa_n+limsupb_n$ Provided that the right and the left sides are not of the form $\infty - \infty$. Proof: Consider $(a_n)$ and $(b_n)$, sequences of real numbers...
50. ### Mathematica Non-commutative multiplication in Mathematica

Does anyone know if it's possible to tell Mathematica to do calculations with non-Abelian groups, such as the quaternions? For example, how do you tell it to do (3 + j)(4 - i)? I would like to extend this beyond quaternions so is there is a way to define arbitrary group definitions? Thanks! -Dan