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
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+
b
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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
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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).
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...
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 ?
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##...
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...
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...
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...
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...
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!
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...
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...
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...
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...
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...
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?
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...
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...
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...
Let $$\mathcal{A}$$ be a $C*$-algebra which may or may not have a unit with norm $$||.||$$, and put $$\mathcal{\overline{A}} = \mathcal{A} \oplus \mathbb{C}$$ as a vector space with mupltiplication:
$$(a, \lambda) (b, \mu) = (ab + \lambda b + \mu a , \lambda \mu)$$,
$$(a, \lambda)^{*} =...
Hey! 😊
I want to write a function in Python that returns the multiplication table $20\times 20$.
We do that using lists in lists, right? I have written the following :
def mul_table() :
Prod = []
Table = []
for i in range(21) :
for j in range(21) ...
The definitions of them seem like arbitrary choices or an abuse of notation. I wonder what the reasons behind the definitions are. Thanks.
PS. My instructor said such defs simplify the process of solving modular equations.
Prove by induction that for any natural numbers n and m , n x (m++)= (n x m) + n
The base case, n=0 gives 0 x m++=(0 x m) +0 gives 0=0
Now assume n x (m++) = (n x m) +n
For n++ we get
n++(m++)=((n++)m) + n++
from this point I am stuck, how can I prove both sides are the same?
This may be very simple but I'm having trouble working it out and the calculator isn't giveing me the result I need.
Below is the example calculation:
1020000*0.5*[(1.10)1.5-1]= 78382
Here is the one I am having trouble working on
207559*0.5*[(1.10)1.5-1]=
If someone could also show me how...
Matrix multiplication is defined by
\sum_{k}a_{ik}b_{kj} where ##a_{ik}## and ##b_{kj}## are entries of the matrices ##A## and ##B##. In definition of orthogonal matrix I saw
\sum_{k=1}^n a_{ki}a_{kj}=\delta_{ij}
This is because ##A^TA=I##. How to know how many independent parameters we have in...
I have referred to this page: https://taoanalysis.wordpress.com/2020/03/26/exercise-5-3-2/ to check my answer.
The way I thought of the problem:
I know ##xy = \mathrm{LIM}_{n\to\infty} a_n b_n## and I know ##x'y = \mathrm{LIM}_{n\to\infty} a'_n b_n##. Thus if ##xy=x'y##, maybe I can try showing...
Divergence & curl are written as the dot/cross product of a gradient.
If we take the dot product or cross product of a gradient, we have to multiply a function by a partial derivative operator.
is multiplication by a partial derivative operator allowed? Or is this just an abuse of notation
is this right
Q) Determine this voltage in its simplest complex number form.
v = (2xj6)(3-j8)
2x3=6
2x-j8=-16
j6x3=j18
j6x-j8=-j48
v=6 +(j18-j16) - J(^2)48 (j^2 = -1)
v=6 +j2 +48
V=54 + j2
Hi everyone,
I'm working through some group theory questions online. But unfortunately they don't have answers to go with them. So, I'm hoping you can say if I'm on the right track.
If this is a binary operation on ℝ, am I right in thinking it satisfies the closure and associativity axioms...
Hi PF!
I am trying to multiply each component of B by the matrix A and then solve A\C. See the code below.
A = rand(4);
B = rand(5,1);
C = rand(4,1);
for i = 1:5
sol(:,i) = (B(i)*A)\C
end
But there has to be a way to do this without a for-loop, right? I'd really appreciate any help you have!
Dear Everyone,
I am having trouble with finding a formula of the multiplication 3 formula power series.
\[ \left(\sum_{n=0}^{\infty} a_nx^n \right)\left(\sum_{k=0}^{\infty} b_kx^k \right)\left(\sum_{m=0}^{\infty} c_mx^m \right) \]
Work:
For the constant term:
$a_0b_0c_0$
For The linear...
The below matrix represents a rotation.
0 0 -1
0 1 0
1 0 0
Im trying to obtain the general point (x y z) when rotated by the above rotation matrix? So visited https://www.andre-gaschler.com/rotationconverter/ entered the above figures and not sure which entry would be x y z but assume it...
The Karatsuba multiplication algorithm is a faster-than-O(n2) (approximately O(n1.58)) multiplication method of two large numbers. I have been working on a small project where I have implemented it (among other things), and I noticed something curious about it that I'm uncertain how to prove or...
Please refer to the screenshot below. Every step is justified with an axiom. Please see the link to the origal document at the bottom.
I am trying to understand why the proof was not stopped at the encircled step.
1. Is there no axiom that says ## x \cdot 0 = 0 ## ?
2. Isn't the sixth...
I have tried to do this using arrays and do loops:
program matrixmul
implicit none
real A(2, 2), B (2, 2), C (2, 2)
integer i, j, k
write (*, *) 'Input: First matrix'
do i = 1, 2
do j = 1, 2
read (*, *) A (i, j)
enddo
enddo
write (*, *) 'Input: Second...
I'm aware of the axioms of real numbers, the constructions of real number using the rational numbers (Cauchy sequence and Dedekind cut). But I can't relate the arithmetic of irrational numbers to real world usage.
I can think the negative and positive irrational numbers to represent...
If I had to guess what the complex matrix would look like, it would be:
##T(x+iy)=(xa-by)+i(ya+bx)=\begin{pmatrix}
a+bi & 0 \\
0 & -b+ai\end{pmatrix}\begin{pmatrix}
x \\
y \end{pmatrix}##
I'm not too sure on where everything goes; it's my first time fiddling with complex numbers, and moreover...
According to me matrix multiplication is not commutative. Therefore A^2.A^3=A^3.A^2 should be false. But at the same time matrix multiplication is associative so we can take whatever no. of A's we want to multiply i.e A^5=A.A^4 OR A^5=A^2.A^3
I was drawing out the multiplication table in "matrix" form (a 12 by 12 matrix) for a friend trying to pass the GED (yes, sad, I know) and noticed for the first time that the entries on the diagonal are real, i.e. the squares (1, 4, 9, 16, ...), and the off diagonal elements are real and complex...
A new way to multiply one that only a computer could love if only it had enough bits to do it:
https://www.sciencenews.org/article/mathematicians-may-have-found-fastest-way-multiply-huge-numbers
Dear Everyone,
$\newcommand{\Z}{\mathbb{Z}}$Suppose the set is defined as:
$\begin{equation*}
{(\Z/n\Z)}^{\times}=\left\{\bar{a}\in \Z/n\Z|\ \text{there exists a}\ \bar{c}\in \Z/n\Z\ \text{with}\ \bar{a}\cdot\bar{c}=1\right\}
\end{equation*}$
for $n>1$
I am having some trouble
Proving that...
Hi!
When calculating ##(\hat{a} \hat{a}^{\dagger})^2## i get ##\hat{a} \hat{a} \hat{a}^{\dagger} \hat{a}^{\dagger}## which is perfectly fine.
But how do I end up with the ultimate simplified expression $$\hat{ a}^{\dagger} \hat{a} \hat{a}^{\dagger} \hat{a} + \hat{a}^{\dagger} \hat_{a} +...
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...
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
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 ||...
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...
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