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
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⋯
+
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
+
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=
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
=
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+
3
+
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+
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).
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...
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...
Homework Statement
Homework Equations
Is my solution correct? If not then please point out the mistakes and help me solve this question in the right way. Thanks in advance.
The Attempt at a Solution
I am exploring Gaussian integers in terms of roots, powers, primes, and composites. I understand that multiplying two integers with norm 5 result in an integer with norm 25. I get the impression that there are twelve unique integers with norm 25, and they come in two flavors:
(1) Four of them...
My question is perhaps as much about the philosophy of math as it is about the specific tools of math: is perpendicularity and rotation integral and fundamental to the concept of multiplication - in all number spaces?
As I understand it, the product of complex numbers x = (a, ib) and y = (c...
Hello everyone. I wanted to prove the following theorem, using the axioms of Peano.
Let ##a,b,c \in \mathbb{N}##. If ##ac = bc##, then ##a = b##.
I thought, this was a pretty straightforward proof, but I think I might be doing something wrong.
Proof:
Let ##G := \{c \in \mathbb{N}|## if ##a,b...
Suppose that I have an overdetermined equation system in matrix form:
Ax = b
Where x and b are column vectors, and A has the same number of rows as b, and x has less rows than both.
The least-squares method could be used here to obtain the best possible approximative solution. Let's call this...
Great question. In algebra, we do indeed avoid using the multiplication sign. We'll explain it for you here. Watch the next lesson: https://www.khanacademy.o...
I would love someone to verify the answer for equation 8 in this paper (bottom of page 263) http://onlinelibrary.wiley.com/doi/10.1046/j.1365-2028.2002.00368.x/epdf
For the sake of clarity here is the equation is LaTeX which you can render at the following link
\frac{QC + Q\lambda \sigma -...
I was tutoring a student and I came across the following question. I feel like I'm missing something obvious, but it seems like there are too many variables for an answer to be determined. The attached picture contains all of the question details.
Homework Statement
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This is a seemingly simple problem. All I have to do is multiply two matrices:
[ 1 0 ]
[ 0 1 ] (A)
and
[ 2 ]
[ 3 ] (B)
The Attempt at a Solution
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Because the matrix A has the same number of columns as matrix B has rows, and because matrix A is an identity...
Homework Statement
The number of bacteria in a culture is given by the formula n(t) = 700e0.65t, where t is measured in hours.
What is the initial population n0 of the culture (at t = 0)?
n0 =
Approximately how many bacteria, N, will the culture contain after 5 hours?
N =
The Attempt at...