Varieties of Multiplication
2012-05-14 | 2023-11-22
Estimated Reading Time: 61 minutes
This was my first blog—written in 2012—under the Mathematical Musings tag. The intention was to re-visit topics in mathematics that trigger concern or disquiet in the earnest student of the subject. My hope was that ideas that appeared puzzling or forbidding at first sight could be coaxed to become friendly and helpful. Unhurried explanations and a different perspective would underpin the approach. I have retained, substantially unchanged, what I first wrote, to maintain the freshness, flavour, and vintage of the original blog, even if it is a little rough around the edges.
Prologue
This blog is experimental in three ways.
First, this is my maiden attempt to display mathematics on a web page. It might look simple, but believe me, it is no mean task. Thanks to the concerted efforts of many generous people, I am using MathJax to render the mathematics via Pandoc, and its flavour of Markdown.
The second experimental feature is what I have called “slicing the orange of knowledge with a different cut” in my book Secrets of Academic Success. The idea of multiplication runs like a thread through much of mathematics, from the most elementary stages of counting to what constitutes cutting edge research. Unfortunately, in the way mathematics is taught at present, multiplication is bundled with each stage of mathematics and viewed separately as an operation in that context.
By concentrating on the single unifying idea of multiplication, and viewing it across the whole of mathematics, we are indeed “slicing the orange of knowledge with a different cut”. Even if you have not encountered some of the varieties of multiplication mentioned here, this exposure will help you grasp those varieties better when you do encounter them. Please send me feedback on whether this approach works for you.
Third, this is an extremely long blog. In fact, I call it a slog 😉 . It took me some weeks to write it. So, take your time reading it. It is unlikely that you will finish it in one sitting. Read it in parts at your own pace. After having read it once, cast your eyes and mind over the whole to get an overall view of the main ideas.
I thought of splitting the blog into three or four manageable parts, but decided against it because I wanted the evolution of multiplication as an idea to be left whole in a single post. Tell me if it puts you to sleep 😄 .
With that out of the way, let us begin. I want to look at some of the varieties of multiplication that mathematicians have developed over time. It is a survey that will serve as a pinhole through which we can view how a single, simple mathematical idea has been expanded and elaborated into uses far beyond its historical moorings.
Multiplication as a binary operation
Consistency is valued more in mathematics than in other disciplines. The idea is not to upset the apple cart but to expand it. Definitions, conventions, rules, facts, and fallacies—once established—are usually above dispute, and do not vary with time or place. So, let us start by defining some terms.
Multiplication is a binary operation: it is something that we do with two mathematical objects, whatever they might be. Usually, the two are similar objects or at least compatible objects. The whole numbers are an example. We can and do multiply two whole numbers.
Multiplication as repeated addition
Practically and historically, multiplication arose as an arithmetic
convenience for repeated addition. If we add the number
When we see the arithmetic expression
We say that the “something” which is repeatedly added, is the multiplicand. The number of times that “something” is added is the multiplier. And the result of this operation is the product. Thus far we are in perfect harmony with accepted usage.
Commutativity and multiplication
Multiplication of numbers is commutative,
i.e., the multiplier and multiplicand can change roles without affecting
the result.
To accommodate our zeitgeist, the distinction between multiplier and multiplicand is fading away, in favour of the symmetrical and neutral term factor. The result of multiplying two factors is still the product, as before.
Rectangular numbers
Historically, stones were used to count. And the stones representing any number may be arranged in geometric shapes, like lines, triangles, rectangles, and so on. This gives us a geometrical or pictorial representation of numbers.
All numbers which are the products of two whole numbers, neither of
which is one, may be expressed as rectangular numbers. The
symbolic operation
Factorization is not unique
There is a subtle but important point to grasp here. The product
Square numbers
A square is a special case of a rectangle whose length and width are
equal. When we write
One could carry this analogy further and move into three dimensions
to represent a number like
Prime numbers
A number which cannot be expressed as the product of two numbers other than one and itself is called a prime number. Prime numbers can only be arranged in a line, never in a rectangle. Seven is a prime number as illustrated below.
Try to rearrange the seven icons into a rectangle to convince yourself that it is not possible and that seven is prime. Experimenting like this will help you better understand what testing for primality entails.
Prime numbers are like building blocks that may be used to build larger numbers by multiplication.
Prime factorization is unique
Let us look at the number
Symbols for multiplication
If you are sharp-eyed, you might have come across the multiplication of two negative numbers by enclosing each of them in parentheses: (). The same symbols are also used to define associativity and distributivity later in this blog. We now look at the chequered history of how the notation for multiplication has changed with time, need, and context.
Times sign
The symbol for multiplication that we first learn is the rotated plus
sign “
Parentheses
Parentheses, written in pairs as (), have traditionally denoted
precedence during evaluation of an expression. Division and
multiplication are evaluated before subtraction and addition.
This order may be altered by including terms in parentheses, which are
accorded highest priority during evaluation. So,
When our discourse embraces negative quantities, in order to avoid
ambiguity, we need something to enclose both the negative sign,
Juxtaposition without any symbol
The archetypal symbol for the unknown in algebra is
To avoid confusion, a convention was adopted that when two algebraic variables, representing unknown quantities, were written next to each other or juxtaposed, it indicated the multiplication of the two quantities. There is no intervening symbol between the two variables.
Thus,
Centred dot
As more exotic objects entered the mathematical collection, yet
another symbol was devised to show (at least one form of)
multiplication. The vertically centred dot
So, there is both a rationale and a mathematical context for the introduction of each symbol for multiplication, according to time, need, and circumstance.
Asterisk
The latter half of the twentieth century saw the introduction of yet
another symbol for multiplication, this time for use in programming
languages. Because the ASCII character
set, devised during the era of teleprinters, did
not include the symbol
Repeated multiplication—or exponentiation—is usually represented by a double asterisk ** in most computing languages, although a caret ^ sometimes assumes this function in some languages.
Laws of arithmetic
We now return to the assertion, made at the start of this blog, that
multiplication is a binary operation: something that happens
between two mathematical objects. You might object that we can
and do multiply three numbers. For example,
Early mathematics was eminently practical, concerned with computing areas and volumes, profit and loss, and so on. In the course of time, mathematicians started to systematize their body of knowledge by introducing logic and rigour into their subject. They wanted to move beyond ad hoc methodology and construct an intellectual edifice that was stable, durable, and generalizable.
The real numbers as a field
Some of the most unpleasant experiences of school mathematics are the sudden and unexpected changes that intrude into the familiar arithmetic of primary school. Division, fractions, negative numbers, multiplication by zero, product of two negatives being positive, etc. are a few examples. When rule upon unanticipated rule is foisted on the young student, with no rhyme or reason, the student gets overwhelmed and develops a distaste for mathematics or even a reflex fear of it. This need not be so.
One way out is a quick but easy introduction to some ideas of abstract algebra which lay the foundation for all subsequent mathematics. This way, all the rules are bunched together as unquestioned assumptions or axioms. Then, based on those assumptions, we build a mathematical edifice that is logical, consistent, and extensible. Mathematics will then be changed from a mysterious game with ever-changing rules into a trustworthy friend who can be relied upon.
The numbers we use every day are drawn from a set called the real numbers
denoted by
The set
Property | Addition | Multiplication |
---|---|---|
Associativity | ||
Commutativity | ||
Identity | ||
Inverse |
For the record, formal definitions for the above terms are available on the Web from reputable sites whose links are listed below:
Distributivity. Multiplication is distributive over addition. For completeness, we define
In our case, both conditions hold, and we may simply say that multiplication is distributive over addition for the reals.L e f t D i s t r i b u t i v i t y 𝑎 ( 𝑏 + 𝑐 ) = 𝑎 𝑏 + 𝑎 𝑐 R i g h t D i s t r i b u t i v i t y ( 𝑎 + 𝑏 ) 𝑐 = 𝑎 𝑐 + 𝑏 𝑐
A mathematical object consisting of a set with two binary operations having the above properties is called a field. The real numbers constitute a field.
Associativity of multiplication
Because multiplication is binary, we can only multiply two numbers at any one time. If we need to multiply together three or more numbers, we have to multiply two of them first to get a single product before we can multiply it with the next number, and so on.
The associative law simply states that when we multiply three
numbers, it does not matter which two of the three we multiply first;
the result will always be the same. It is this property that allows us
to write something like
In addition to the three properties of associativity, commutativity, and distributivity, the real numbers we use every day have an additive identity and inverse in Table 1. These are considered next.
The additive
identity and inverse in ℝ
The additive identity in
When we add together a number and its inverse, we get the additive identity.
Another way of understanding the additive inverse is to look at it
geometrically as a reflection in a double-sided mirror placed
perpendicular to the real line at the position of
With reference to Figure 5, the
mirror is the silver-colored line placed at zero. The irrational number
At the risk of expounding the obvious, let us take another look at a
pictorial representation of how to obtain the additive inverse of a
number
An even more concrete algorithm to obtain the additive inverse is now
given. Suppose we want the additive inverse of
We show later in Equation 7 that
multiplying
The
multiplicative identity and inverse in ℝ
We next consider the multiplicative identity and inverse in
The multiplicative
inverse for arbitrary
If we plot
A construction to get the multiplicative inverse of some point
Note the following insights from Figure 7:
The function becomes unbounded as
approaches𝑥 . This happens both from the positive and negative sides. Symbolically,0 and equally,l i m 𝑥 → 0 + 1 𝑥 = ∞ .3 This is whyl i m 𝑥 → 0 − 1 𝑥 = − ∞ has no multiplicative inverse.0 The multiplicative inverse has the same sign as the original number, since the hyperbola has two “arms”.
There are only two values of
for which the multiplicative inverse is also the original number. They are at𝑥 and𝑥 = 1 . This is because the line𝑥 = − 1 intersects the hyperbola at two points:𝑦 = 𝑥 and( 1 , 1 ) .( − 1 , − 1 )
Where have subtraction and division disappeared?
If you are wondering where subtraction and division have disappeared,
they are hiding in plain sight. Subtracting
Multiplying any number by zero always gives zero
Recall from Table 1 that
Consider an arbitrary number
We have carefully tiptoed our way to justify each step with one of the field axioms. This is the power of the axiomatic approach. There is no “wasted” axiom; there is no “missing” axiom. The set of axioms are the minimum necessary for the numbers to rest on a stable foundation.
This minimal sufficiency is at the heart of why mathematics
is so strong. It has no extra fat. There is also no deficiency. It is
frugal but sufficient. We will encounter this same idea in the statement
that a basis is a minimal spanning set in a vector space.4 Any other algebraic structures, like
the complex numbers
If you feel that Equation 4 is too
much sleight of hand, and you would like something more concrete, you
can always console yourself with the convenient but specific example of
Product of a positive and a negative number
Negative numbers arose when loans had to be given and taken. They also find use in describing the depth of an ocean trench as being so much below sea level. Other applications arise naturally with temperatures below the freezing point or with electric charges of a negative type, etc.
The signs of products featuring negative numbers are not intuitively
comprehensible. So, we have to rely on the axioms to guide us to
consistent results. What is the sign of the product of a positive and a
negative number? To find out, we first prove that multiplying a
number by the additive inverse of another number gives the additive
inverse of their product, i.e.,
If we now assume that
One interesting corollary from Equation 5 and Equation 6 is:
These slow but careful explanations might seem contrived, but they provide guardrails against falling off when future mathematical objects are encountered. And it is a whole lot more satisfying than hand-waving or saying “Just take it on faith.”
Why is the product of two negative numbers positive?
Let us use the field axioms to navigate our way to this result as well.
Let
The scheme we have followed so far is to add something to the object
of interest and use the axioms to prove that the sum is zero. The result
we are after will then pop out. Let us apply that method again, using
the final result fron Equation 6:
I have dealt with the arithmetic of fractions and negative numbers in my freely downloadable chapter “Arithmetic Revisited” from Secrets of Academic Success. I urge you to read it if you feel the need.
Exponentiation
Just as multiplication with whole numbers is repeated addition,
exponentiation is repeated multiplication. A new notation is used to
indicate repeated multiplication. We denote it by a superscript
indicating how many times the number is multiplied:
What is the exponent in the number written plainly as
Logarithms: multiplying by adding
We may reduce multiplication to addition if we focused on the exponents of a common base. This is exactly what was done by an eccentric Scottish laird called John Napier whose labours have made all our lives much less tedious. The books of logarithmic tables, affectionately called “log books” when I was at school, along with the slide rule were the mainstay of physicists and engineers before the advent of the electronic calculator in the mid-1970s. And they all relied on Napier’s scheme of reducing multiplication to addition.
Multiplying by adding is simple. Suppose we want to multiply
Express the number
as a power of2 :1 0 .2 ≈ 1 0 0 . 3 0 1 0 3 Express the number
as a power of3 :1 0 .3 ≈ 1 0 0 . 4 7 7 1 2 Add the two powers or exponents:
.0 . 3 0 1 0 3 + 0 . 4 7 7 1 2 = 0 . 7 7 8 1 5 Find out what number equals
. In our case,1 0 0 . 7 7 8 1 5 .1 0 0 . 7 7 8 1 5 ≈ 5 . 9 9 9 9 8
“Aha!” you might say. “But the answer is not exactly
If you had to compute
Square roots
Taking a square root is a form of exponentiation. If I gave
you a number like
If instead, I asked you to find that number, which when
multiplied by itself gives
But is that the whole story? Recall that the product of two negative
numbers is positive. So,
Complex numbers
Talking about square roots and negative numbers, can we ever take the square root of a negative number? Do such numbers exist? If so, where? And are they useful?
The squares of real numbers give rise to only two possibilities. The square of zero is zero. The square of any non-zero real number, whether positive or negative, is always positive, as we have just seen. So, the possibility of a negative number being the square of a real number does not ever arise.
But we do encounter a different situation when solving an algebraic
equation like
In the course of time, these pesky numbers—whose square is a negative number—kept popping up insistently in unlikely places. They were then reluctantly assigned mathematical existence with the somewhat pejorative term imaginary numbers. They were after all not real numbers!
In the course of time, a sandwich number was invented which was composed of the sum of a real number and an imaginary number. This number was called a complex number. It is the first of several mathematical objects we will encounter in this blog that are different from the real numbers of everyday life.
The set of complex numbers is denoted by
But there is a little asymmetry in the expression
Multiplication of complex numbers
When we multiply complex numbers, we are really performing
multiplication on two pairs of real numbers with the imaginary
unit sandwiched in between. Because of the property
Although the rules of multiplication of complex numbers differ from those of real numbers, the field axioms still hold. This is the purpose behind the development of abstract algebra. The principle is DRY (Don’t Repeat Yourself) or, equivalently, do not re-invent the wheel.
Polar form of complex numbers
We have resorted to the unit circle to unravel the meaning of the tangent ratio and function in a previous blog.
We now take recourse to the same unit circle to better understand the
multiplication of complex numbers. Let us press the
The formula
Referring to Figure 8, we may
say:
What exactly is the advantage of all this jiggery-pokery? It makes multiplication easier because exponents are added when the numbers they represent are multiplied.
If we have two complex numbers
Consistency between real and complex multiplication
What happens if we used the rule for complex multiplication above but
set the imaginary parts to zero so that complex multiplication reverts
to the multiplication of two real numbers? Do we get consistent results?
Let us try it by substituting
It is a primary requirement in mathematics that when we extend the definition of an operation on a simpler object to encompass a more complicated mathematical object, the new definition should revert to the accepted definition for the simpler object when the complicated object reverts into the simpler object. This is the point about consistency that I made at the start of this blog.
Complex numbers as ordered pairs
Arithmetic operations on complex numbers result only in complex numbers and do not give rise to new types of numbers. And all complex numbers consist of two parts: a real part and an imaginary part.
Hence, if we develop a new notation for this two-part number for
purposes of arithmetic, we may dispense altogether with the symbol
The simplest notation is to represent complex numbers as ordered pairs of real numbers with the convention that the first number is the real part and the second number is the imaginary part. Once we have done this, we need to re-define all the arithmetic operations for these ordered pairs.
We may thus use the ordered pair
Drawing upon our previous results, we may then define
multiplication for this ordered pair as being
Vectors
Ordered pairs lead rather neatly to the idea of vectors.6 Indeed, there is more than a passing resemblance between complex numbers and two-dimensional vectors.
Both may be represented by ordered pairs of real numbers and the rules for addition and subtraction of these ordered pairs are identical. Moreover, they may both be represented as points on a Cartesian plane. Vectors are the second new mathematical object, after complex numbers, that we are meeting in this blog.
A vector is traditionally defined as a quantity having two attributes: magnitude and direction. A simple everyday example is the wind which has both a speed and direction, and may therefore be represented by a vector. Indeed, if you have already encountered vectors, you might think of them exclusively as directed line segments or lines of specific lengths with arrow tips as shown in Figure 10.
Addition and subtraction of vectors: geometric viewpoint
How do we add and subtract these geometric entities, let alone multiply and divide them? If you have done physics at high school, you will know that we use something called the parallelogram law.
We draw a pair of two-dimensional vectors so that both originate from the same point. We then complete the parallelogram formed by these two vectors by drawing in the other two sides. The sum of the two vectors, or their resultant, is the diagonal of the parallelogram that starts at the same origin as the two vectors. This is something best grasped from a picture: see Figure 11.
The origin of the Cartesian plane is labelled
: the vector from𝐮 to the point𝑂 ( 8 , 4 ) ; : the vector from𝐯 to the point𝑂 and( 2 , 6 ) ; : the vector from𝐰 to the point𝑂 .( 1 0 , 1 0 )
The dotted grey lines indicate the two sides of the parallelogram
that we draw to close the figure. The vector
So, we may represent the addition of
The parallelogram law is a geometric statement of what happens when
we add two ordered pairs the way we would two complex numbers:
Addition and subtraction of vectors: algebraic viewpoint
We may identify two-dimensional vectors uniquely by an ordered pair representing their co-ordinates on the Cartesian plane. This is the algebraic viewpoint. It is less cumbersome and more powerful as we have already seen from the addition of two vectors.
Subtraction is equally simple. We may add the additive
inverse of each component of the vector being subtracted to get:
Vectors as algebraic entities
We have just seen that two-dimensional vectors may be represented by ordered pairs on the Cartesian plane. This representation might be extrapolated to include vectors of dimensions greater than two. Obviously, we would then be moving from ordered pairs to ordered triples, etc.8
To generalize, we may think of vectors as a list of “numbers in a
slim teabag” where their order matters. Formally, an
The components of a vector may be written within parentheses or
brackets. They may be arranged vertically as a column vector of
a single column and
It is conventional to assume that an arbitrary vector is a column vector. Row vectors are then the transposes of the column vectors. This is the convention we will follow.
Row-column nomenclature
The size of a vector is denoted by writing down the number of rows
followed by a
By definition, a vector must have at least one dimension equal to
Addition and subtraction of vectors
Addition and subtraction for the ordered n-tuples representing two vectors may be defined as the addition or subtraction of their respective components.
Just to free ourselves from geometrical thinking about vectors, let
us add two four-dimensional vectors whose components are given
by algebraic variables representing real numbers.
Subtraction is equally “commonsensical”:
These vector sums and differences would be difficult to visualize geometrically, but they are trivially routine algebraically.
Multiplication of vectors
It is easy to think of the addition or subtraction of vectors, say in the context of “wind speed” and “air speed” of an aircraft. But what does the multiplication of vectors consist of and what meaning could we extract from this operation?
Vector multiplication is a strange, many-headed beast. It is important to know what it is and what it is not. Here is a quick run down:
Vector multiplication is different from real number multiplication.
Vector multiplication is different from complex number multiplication.
There are several varieties of vector multiplication, some of which give us scalars, others vectors, and still others matrices:9
- multiplication of a vector by a scalar to yield a vector
- dot product or scalar product or inner product of two vectors to yield a scalar
- cross product of two vectors to yield a third vector orthogonal to the other two
- tensor product or outer product of two vectors to yield a matrix
Each variety of vector product was devised as an operation because it is useful and has a ready meaning in a particular context.
When it comes to multiplication, vectors reveal their nature as a class of mathematical object quite different from real or complex numbers.
Let us consider each type of multiplication in turn.
Multiplication by a scalar
Multiplication of a vector by a scalar is the easiest to understand. In this operation, we see the original arithmetic definition of real multiplication at play. We are magnifying or diminishing the magnitude of the vector by multiplying it with a scalar, while the direction of the vector is either reversed or remains unchanged.
If we have a vector
Multiplication of
When we multiply a vector by a scalar
: the vector is unchanged in magnitude and direction.𝑘 = 1 : the vector is unchanged in magnitude but reversed in direction.𝑘 = − 1 : the vector is enlarged in magnitude and reversed in direction.𝑘 < − 1 : the vector is enlarged in magnitude and unchanged in direction.𝑘 > 1 : the vector is diminished in magnitude and reversed in direction.− 1 < 𝑘 < 0 : the vector is diminished in magnitude and unchanged in direction.0 < 𝑘 < 1 : the vector has zero magnitude and its direction is undefined.𝑘 = 0
While this might seem quite a mouthful, it is really quite simple:
- a negative
reverses the direction;𝑘 - a positive
keeps the direction unchanged;𝑘 - a value of
that lies between𝑘 and− 1 or between0 and0 diminishes the magnitude; and1 - a value of
less than𝑘 or greater than− 1 increases the magnitude of the vector.+ 1
The special cases pertaining to
Did you pick up the fact that after uncoupling geometry and vectors, we finally resorted to geometry when talking about the meaning of scalar multiplication? This dual viewpoint runs through much of mathematical thinking.
Scalar division of vectors by
If we view scalar multiplication as a black box, it takes in one n-tuple and gives out another n-tuple. Like the merchant in Aladdin and the Wonderful Lamp, we are simply exchanging old vectors for new. There is no difference in kind between the input and output mathematical objects.
The zero vector is not the number zero
Distinguish carefully between the real number
As and when new mathematical objects are invented (or discovered?)
new definitions for the equivalents of zero and one for these new
objects may also be necessary. The
The four-dimensional column vector with all entries equal to
We now turn to other varieties of multiplication that may be applied to vectors.
Dot or scalar product
The centred dot
Existence of the dot product
The dot product is defined only between vectors of the same dimension. This is important to grasp. When we deal with real numbers, the multiplicand, multiplier, and product are all real numbers. They are mathematical objects of the same kind. So, we may afford to be a little careless in multiplying them together without performing any check.
We cannot afford to be equally lackadaisical with vectors. We have to respect the fact that they are not numbers per se, but a different type of mathematical object. A product of some sort might not exist between any two arbitrary vectors.
Example of dot product
It is helpful to begin with a concrete example. Let
If you look at the dot product carefully, you will see the following:
the first component of
is multiplied by the first component of𝐮 and likewise for the other components;𝐯 the individual products are then added together; and
the sum is the dot or scalar product.
It is now clear why the two vectors must have the same dimensions. If not, we will run out of either multiplier or multiplicand for pairwise multiplication.
The result, being a sum of products, is a single number, or scalar, explaining the name scalar product for this operation. We prefer the term dot product to avoid confusion with multiplication by a scalar.
It is easy to verify by direct evaluation that the dot product is commutative
and therefore symmetrical. Verify if you please that
Why did we need to write the dot product as being between a row vector and a column vector? One reason is that the product of a column vector with a row vector is actually a different type of multiplication which we will meet later. Another reason is that the row-column product mirrors matrix multiplication as explained later.
General case and formula
These results for the dot product may be generalized by taking
It is conventional to write the vector itself in boldface as
Observe that the vector
Consistency with real multiplication
What happens if our two vectors degenerate into scalars having single
components
Geometric viewpoint
What does the dot product mean? What does it signify given that vectors originated as physical abstractions? We need to put on our “geometric glasses” and view the dot product geometrically. We will need a little bit of trigonometry on the way.11
Let us consider a two-dimensional vector
By the Theorem of Pythagoras, the magnitude of the vector
, denoted by𝐮 , is given by‖ 𝐮 ‖ . The symbol√ 𝑢 2 𝑥 + 𝑢 2 𝑦 denoting a pair of double vertical lines represents the norm or magnitude of the vector written within it.‖ ‖ The magnitudes of the projections of
in the directions of the𝐮 and𝑥 axes are respectively𝑦 𝑢 𝑥 = ‖ 𝐮 ‖ c o s 𝛼 𝑢 𝑦 = ‖ 𝐮 ‖ s i n 𝛼 = ‖ 𝐮 ‖ c o s ( 9 0 ° − 𝛼 )
The magnitude of the projection of a vector in a particular direction is equal the magnitude of the vector multiplied by the cosine of the angle made by the vector with that direction.
We could make similar claims for a vector
Let the angle between the two vectors be denoted by
The dot product of two vectors is equal to the product of their magnitudes and the cosine of the angle between them. It is a scalar.
Applications of the dot product
Unlike straightforward multiplication of real or complex numbers, the dot product seems a little contrived. Why is it so defined? And is it useful?
The answers to both questions lie in the practical utility of the dot product. Vectors are used to represent forces, displacements, momenta, and a whole host of other abstractions that are the bread and butter of physics. And the dot product neatly dovetails with a recurring pattern of relationships in physics where two vectors give rise to a scalar in a multiplicative fashion.
For example, mechanical
work
The dot product is succinct, precise, notationally crisp, and practically useful. That is why it has been defined and that is why it still exists.
The cosine and sine functions
The cosine and sine of an angle are trigonometric
ratios from right-angled triangles that were later expanded in scope
to become mathematical
functions. Graphs of
For now, we only need to focus on these facts, bearing in mind that
The values of
andc o s 𝑥 lie only betweens i n 𝑥 and− 1 .1 andc o s 0 = 1 .c o s ( 𝜋 2 ) = 0 ands i n 0 = 0 .s i n ( 𝜋 2 ) = 1 .s i n 𝑥 = c o s ( 𝜋 2 − 𝑥 ) .c o s 𝑥 = s i n ( 𝜋 2 − 𝑥 )
Oddness and evenness
Observe from these graphs that
Although not apparent from the graphs, both the cosine and sine
functions are periodic
and repeat themselves every
Orthogonality
To recapitulate, the cosine of an angle is
For example, a force vector
The dot product therefore measures the degree of alignment
or similarity between two vectors. When the angle between them
is zero degrees, this alignment is at its greatest. When the vectors are
orthogonal, each vector has no component in the direction of the other;
so they are independent. When the two vectors make an angle
greater than
Orthogonality—and the independence of vectors it implies—is a very powerful property that finds application daily whenever we talk over the telephone or download a compressed image from the Web.
The idea of projecting some mathematical object onto another and the idea of one mathematical object being orthogonal to another are both fundamental to many areas of mathematics and are well worth keeping in mind.
We now move on to the next type of vector product.
Cross product
The third type of vector product is the cross product. Because the dot product gave a scalar result that involved the cosine function, you might ask tongue in cheek, whether the cross product produces a vector result that involves the sine function in its definition. And facetious or not, you are actually right. 😄
The cross product is a vector and it does involve the sine of the angle between the two vectors. In addition, just as in the dot product, orthogonality peeps at us again through the cross product.
We will consider three-dimensional vectors. Any pair of
three-dimensional vectors
The result of a cross product is orthogonal to the two vectors giving rise to it. There are two directions orthogonal to the plane. Think of the flat table again. An arrow at right angles to the table coming out of it and pointing upwards is in one direction. Now reverse the direction of the arrow so that it goes into the table pointing downwards. This is the other orthogonal direction. They both lie along the same straight line but are oriented in opposite directions.
We are now ready to define the cross product as
Both the vectors
The expression
Imagine that you are rotating a corkscrew starting at
Since the corkscrew would then move upwards, that is the
direction of both
In the cross product, we have just met the
Anti-commutativity
If we were to compute
Applications of the cross product
Like the dot product, the cross product owes its ubiquity to its
usefulness in physics. For example, the torque vector
Outer product of two vectors
The outer product is the last of the four varieties of multiplication for vectors that we will consider here.
Recall that the dot product is defined as the scalar resulting from
the multiplication of a
What happens if we swap the order and start multiplying an
Indeed we do. And the resulting product is a different mathematical object called a matrix. This is an example of a mathematical operation involving two known mathematical objects whose result gives rise to a new kind of mathematical object which then acquires a life and personality of its own.
This type of multiplication is called an outer product, in contrast to the dot product which is a type of inner product. It is also sometimes called a tensor product in honour of the fact that we are ascending a hierarchy in linear algebra that starts with scalars and moves on to vectors and then to matrices and on to tensors with progressive generalizations at each step.
Outer product: symbol and example
The symbol for the outer product is
As already presaged, the outer product results from the
multiplication of an
In contrast to the dot product, however, the two vectors may have
different numbers of elements. This is why the resulting matrix is not
necessarily a square matrix with equal numbers of rows and columns, but
rather has
Here is an example that will help you decipher how the outer product
is computed:
The outer product is non-commutative
The outer product is not commutative. To see why, consider
Applications of the outer product
The outer product finds application in fields like physics, electrical engineering, and statistics. Whether application precedes or follows the original mathematical development, whenever a new mathematical object persists, it is almost always due to its usefulness for some purpose or other.
The outer product of two vectors leads to matrices and the multiplication of matrices is yet another variety of multiplication. It is the last we will consider in this blog.
Matrices
We have already seen that a matrix consists of a lot more “numbers in
a teabag” in which order is respected. An
Hark back to complex numbers and remember how the real numbers are merely complex numbers whose imaginary parts are zero. We hear a similar refrain with matrices, vectors, and scalars. Vectors are matrices with one column or one row. A matrix with a single column and row is a scalar.
Each time a mathematical object is generalized, we will see a previously defined object appearing as a degenerate case of the new object. This provides a link between the new and the old and also ensures that consistency is maintained in this evolutionary spiral.
It is customary to refer to a matrix by an uppercase letter. The individual numbers, or elements, of a matrix are usually denoted by a lowercase letter and given double subscripts denoting their position in the matrix.
The element
Applications of matrices
Matrices arose naturally from the study and solution of systems of linear equations. They are also useful in succinctly embodying geometric transformations of points in the two-dimensional Cartesian plane. They are profoundly useful in electrical engineering, physics, economics, and many other fields.
Indeed, if one considers matrices as a class of mathematical object, what we do with them and the meanings we assign to these actions are largely limited only by our imagination and the mathematical consistency of the results. This is how new mathematics is built up from the old, and constantly expanded in scope, variety, and application.
Matrix multiplication
The product of matrix
Any two real numbers may be multiplied together, but the product of any two matrices need not necessarily be defined. As the mathematical objects that we deal with become increasingly complex, additional constraints often apply to operations on them.
Example of matrix multiplication
Here is an example of matrix multiplication. We group a whole row on
the left matrix and multiply it element-wise with a whole column on the
right matrix and add all the products. In this case, we compute
This is reminiscent of the dot product. Indeed, matrix multiplication may be viewed as a generalization of the dot product for matrices and the dot product as a degenerate case of matrix multiplication in which the left matrix is a row vector and the right matrix is a column vector.
Non-commutativity
For any two matrices
If both matrices are square and of the same dimensions, is their
multiplication commutative? In other words, does
In mathematics a single exception falsifies the rule. Let us consider
the following simple example:
Geometric effects of matrix multiplication: 2D case
A
How might a matrix accomplish this? If we post-multiply a
matrix by a vector, we will get another vector. We need to transform
Pay attention to the interplay between the symbolic and the pictorial, between the algebraic and the geometric aspects of the one operation. If you develop the ability to maintain this “dual vision” as you study mathematics, it will be helpful for your own unfolding understanding. A strange algebraic object correctly used might work geometric miracles right under your nose, and vice versa.
And that completes my survey of varieties of multiplication. I do not know if you are heaving a sigh of relief but I certainly am! We have only scratched the surface here. There are many more varieties of multiplication and each serves a purpose. You will discover them in the course of your studies.
Summary
This blog has been a journey through mathematics tracking multiplication as the single theme.
Multiplication happens between two mathematical objects to yield a third. In this survey, we have encountered four different mathematical objects:
- Real numbers
- Complex numbers
- Vectors
- Matrices
The way the multiplication is accomplished as well as its meaning differ with context. We have met seven different varieties of multiplication here:
- Real multiplication
- product is real
- commutative
- Complex multiplication
- product is complex
- commutative
- Multiplication of a vector by a scalar
- product is a vector
- magnitude and direction depend on value of scalar
- Dot product of two vectors
- product is a scalar
- commutative
- measures “similarity” or “alignment” between the two vectors
- involves cosine of angle between the two vectors
- Cross product of two vectors
- product is a third vector orthogonal to the two vectors
- anti-commutative
- involves the sine of the angle between the two vectors
- Outer product of two vectors
- product is a matrix
- not commutative
- Matrix product
- product is another matrix
- not commutative
We have made glancing acquaintance with logarithms and how they transform multiplication into addition. We have also skimmed over the trigonometric functions, given their place in the theory of vectors.
If you carry away nothing else from this blog than a few qualitative ideas, they should include some of these:
Multiplication is a binary operation: it takes place between two compatible mathematical objects.
Mathematical objects are more varied than animals in a zoo. Each has its own nature, diet, habitat etc. Apart from the real numbers, we have encountered complex numbers, vectors, and matrices here.
Multiplication is commutative for the real and complex numbers but not for necessarily for vectors or matrices.
The meaning of a product has evolved a long way from the original “three lots of four” in the context of whole numbers. The product of a multiplication might yield an object that is quite different from the multiplicand and multiplier. We have seen scalars popping out of dot products of two vectors and matrices issuing from the outer product of two vectors.
The ideas of zero and one, of symmetry, of commutativity, of consistency of definitions, of projections, and of orthogonality, are worth remembering because they pervade much if not all of mathematics.
May the product be with you!
To explore further
We have covered a fair bit of ground in this blog, and not all of it at the same depth. For those who seek a greater acquaintance with abstract algebra, I would recommend three books that are kinder, gentler introductions to the subject, written by professional mathematicians:
W W Sawyer’s A concrete approach to abstract algebra [5] was first published in 1959, but it has retained its vigour and directness intact through more than six decades. Read it as a first introduction to the subject.
Bergen’s more recent book, A Concrete Approach to Abstract Algebra[6] is a modern successor to Sawyer, sharing the same book title. It will be a good companion to Sawyer.
The third book, by Goodman, entitled Algebra: Abstract and Concrete is available at no charge online [7]. It is very readable and concentrates a fair bit on symmetry. You could read it along with the other two or use it to complement your understanding of symmetry.
The prolific engineer-author Paul Nahin has written a
whole series of engaging popular mathematics and physics books. Two
of them are relevant to the subject of this blog. The first is aptly
entitled An Imaginary Tale: The Story of
Patrick Honner has written an engaging and easy-to-read article on imaginary numbers in Quanta Magazine that should be accessible to high school students [10].
I have made reference to Feynman’s lecture [3] in which he says, “So we have created two new functions in a purely algebraic manner, the cosine and the sine, which belong to algebra, and only to algebra.” This remarkable unification of algebra and geometry, I have not seen elsewhere. There are three volumes of the Feynman Lectures on Physics and they are a treat to read and a treasure to own [11].
Ivan Savov has written a series of irreverently named books on mathematics and physics. Of these, the one entitled No Bullshit Guide To Linear Algebra [12] is most relevant to this blog. His style is distinct from other texts, and his fascinating and painstakingly drawn diagrams that relate different branches of mathematics will help students integrate their knowledge—acquired piecemeal over the years—into the unitary whole that it really is.
Afterword
This blog started off as something that promised to be short and fizzy, tangy and piquant. But it soon became a little like hot treacle: too hot to swallow and too sticky to spit out. It transmogrified into a jumboblog or slog. If you have stuck with me this far, I applaud and thank you.
The thought crossed my mind that I could split this blog into several sub-blogs. But I soon gave up that idea because the connectedness of the thread will be lost in the segmentation. So, here you have the whole hog and the whole blog.
Mathematics is like a pastry puff: only the layers never seem to end and neither does the puff! I needed to cap the well at some point, and matrices seemed as good a place to stop as any. The pleasures of many other types of multiplication await your future explorations! 😄
As an independent scholar, I work in isolation without the benefits of a university environment or consultation with peers. So, an error of fact or fancy is all the more likely in what I write. If you are mathematically inclined, and have spotted any mistakes here, please let me know.
Feedback
Please email me your comments and corrections.
A PDF version of this article is available for download here:
References
Try this with toy blocks to convince yourself of its truth.↩︎
We use the familiar
instead of𝑥 in the graphical context.↩︎𝑎 There is no limit as
.↩︎𝑥 → 0 Which is the subject for another blog.↩︎
Because
is often associated with current, electrical engineers often use the symbol𝑖 instead.↩︎𝑗 Properly called Euclidean vectors in our context.↩︎
I leave it to you to convince yourself of this. (Hint: use graph paper, draw the co-ordinate axes, and use algebraic variables for the co-ordinates of
,𝐮 , and𝐯 .)↩︎𝐰 An ordered triple, for example, would live in our familiar three-dimensional space.↩︎
Ignore any unfamiliar terms for now.↩︎
An inner product is something more general, of which the dot product is a special case.↩︎
Which you might have to take on trust if it is unfamiliar or forgotten.↩︎
See my blog A tale of two measures: degrees and radians if you are unfamiliar with radians.↩︎
If radians bother you, keep in mind that
equals0 ° radians and that0 radians equals𝜋 2 .↩︎9 0 ° These properties make the dot product a commutative operation and the cross product an anti-commutative operation.↩︎
This is a convenient mathematical convention which is also in accord with actual physical situations.↩︎
Pronounced tau.↩︎