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Date: 05/25/97 at 21:04:01
From: Leif
Subject: The square root of i
What is the square root of i?
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Date: 05/26/97 at 07:58:07
From: Doctor Anthony
Subject: Re: The square root of i
We have i = cos(pi/2) + i.sin(pi/2)
sqrt(i) = (cos(pi/2) + i.sin(pi/2))^(1/2)
By DeMoivre's theorem:
= cos(pi/4) + i.sin(pi/4)
1 + i
= ----------
sqrt(2)
To learn more about DeMoivre and his theorem, look at:
http://mathforum.org/library/drmath/view/53975.html
-Doctor Anthony, The Math Forum
Check out our web site! http://mathforum.org/dr.math/
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Date: 12/08/2003 at 07:58:07
From: Doctor Schwa
Subject: Re: The square root of i
Good question! When I was in high school and confronted with this same
problem, it seemed obvious to me that the answer must be "j". Just like
like we needed to invent a new number "i" to be the square root of -1, it
seemed like we'd need yet another new kind of number to be the square root
of "i", and so on forever.
The amazing thing is that you don't: once you have "i", any equations
with addition, multiplication, exponents and so on (in short, any polynomial)'
can be solved without inventing any new types of numbers! An equation
like x+3 = 2 makes you invent negatives, x*3 = 2 makes you invent fractions,
x*x = 2 makes you invent irrationals, x*x = -2 makes you invent imaginaries...
but then you're done!
So, once I knew it was possible, and that the answer had to be some complex
number (a+bi), the question is, how do you find out values of a and b that
will make (a+bi)^2 = i?
Well, squaring out the left side gives
a^2 + 2ab * i - b^2 = i,
and the only way for that to work is if the real number part is 0,
a^2 - b^2 = 0,
and the imaginary part is 1*i,
2ab = 1.
Since a^2 = b^2, a = b or -b ... but since 2ab = 1, a and b must be
both positive or both negative, so a = b.
Then since 2ab = 1, and a = b, 2aa = 1, so
a^2 = 1/2,
and
a = b = sqrt(1/2) or a = b = -sqrt(1/2)!
Does that make sense?
-Doctor Schwa, The Math Forum
Check out our web site!
TrinityTutors.com
Instead of simply looking at algebra as the study of the properties of operations on numbers, this textbook will endeavor to focus more on how algebraic concepts and skills can be used in a wide variety of problem-solving situations, especially thoses involving relationships between things that vary over time. (For example: the relationship between the supply of an item and its price.) Since the quantities involved are not fixed amounts, when using mathematical statements to describe such relationships, algebra uses letter symbols, known as variables, in place of numbers.
Algebra has been around for multiple hundreds of years. “Old algebra” includes such topics as polynomials, complex numbers, prime numbers,and the quadratic formula. Roughly speaking, it works like this:
Basic arithmetic deals with three primary concepts: (1) numbers; (2) the four operations (addition, subtraction, multiplication, and division); and (3) the rules of operation, such as the associative property, the commutative property, etc.
What “old” algebra does is take away some of the numbers and replace them with variables so that it’s easier to see what is going on with the structures that remain (the four operations and the corresponding rules).
A PRACTICAL APPROACH1
Advances in abstract algebra are still being made today -- its concepts and methodology currently being used by physicists, chemists, computer scientists and of course, mathematicians. We will be studying both new and old algebra (which is also still being used today).
With regard to groups, rings, and fields, groups have the fewest requirements, and include lots of crazy structures like remainders-by-seven, the Rubik's cube configuration space, the set of square matrices of nonzero determinant, the real number system, the set of spatial rotations, and elliptic curves. Groups are like vector spaces, which you may have seen in “linear algebra.”
Rings are like polynomials, but more abstract.
Fields are like fractions. Fields are the most like the normal number system, but can still be strange: For example there is a field with sixteen elements in which + and – are two names for the same operation!
Since it is impossible to define the concept of a set mathematically, we will characterize it informally as a well-defined collection of objects. Lets begin by logging what we need to know about sets: First of all, we call the members of a set elements. Capital letters are normally used as names for sets. Typically, we will use a capital S to name our generic set. The normal conven is to use a pair of set braces { } to enclose the elements of a set (or the description thereof), using commas to separate the individual elements. If a set is described by a characterizing property P(x) of its elements x, the brace notation {x l P(x)} often called "set builder notation," is also often used, and is read "the set of all x such that the statement P(x) about x is true."
In the study of algebra, real numbers are often mentioned as a group, e.g. “the set of real numbers greater than x,” or “the collection of real numbers satisfying the equation . . .” Therefore, it is often convenient to indicate a particular group of real numbers using set notation. In this manner, the real numbers can be individually listed as part of a collection, or a continuum of real numbers can be represented concisely. For example . . .
Algebra was invented as a tool to solve real-life problems in real-world situations. It has applications to virtually every human endeavor that exists -- from art to medicine to zoology. This textbook will therefore attempt to avoid treating the subject as if it were simply a collection of rules and procedures to, for example, “find x” or “simplify.” Since such abstract symbol manipulation outside the context of real-world situations will seem to have little inherent value and feel like nothing more than busy work, this textbook will do everything possible to maintain a practical approach to algebra instruction.
Set-builder notation is generally used to represent a group of real numbers. It stipulates that sets be written in the format { x : x has property Y }, which is read as “the set of all elements x such that x has the property Y. (The colon means “such that.”) Using this notation, a set is often defined as the collection of real numbers that belong to either an open, closed, half-open, or infinite interval (of real numbers).
An open interval is a set of real numbers represented by a line segment on the real number line, whose endpoints are not included in the interval. This concept is made clear by the following definition: ( a, b ) = { x : a < x < b } where a < b
Since the endpoints of an open interval are not part of the interval,
a Ï ( a, b ) , b Ï ( a, b )
In contrast, a closed interval is a set of real numbers represented by a line segment of the real number line, whose endpoints are included in the interval. Its definition is as follows:
[ a, b ] = { x : a £ x £ b } where a < b
Since the endpoints of a closed interval are part of the interval,
a Î [ a, b], b Î [ a, b]
A half-open interval is also a set of real numbers represented by a line segment on the real number line, but with one endpoint included in the interval, and the other endpoint not included in the interval. They are defined as follows:
[ a, b ) = { x : a £ x < b } where a < b
( a, b ] = { x : a < x £ b } where a < b
An infinite interval is a set of real numbers, but it is represented by a ray or line on the real number line. As befits the name, an infinite interval does not have an endpoint in one or both directions. They are defined as follows:
(-¥, a ] = { x : x £ a }
[ a, -¥) = { x : x ³ a }
(-¥, ¥) = { x : x is a real number }
Similar definitions apply when the infinite interval is open at one end.
A PRACTICAL APPROACHWhat’s Modern About Algebra?
Our first group of numbers are integers. They include zero, all of the positive numbers representing one or more complete units, and all of their corresponding negatives.
. . . –6, –5, –4, –3, –2, –1, 0, 1, 2, 3, 4, 5, 6 . . .
We refer to inteters as either negative:
. . . –6, –5, –4, –3, –2, –1
As nonnegative:
0, 1, 2, 3, 4, 5, 6 . . .
As zero:
0
Or as positive:
1, 2, 3, 4, 5, 6 . . .
Algebra
TYPES OF NUMBERS4
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Types of Numbers
The elements of the sets whose notation you just memorized are, of course, numbers. The most basic group are those that people traditionally used when they wanted to count: counting numbers. Some people include zero. Others don't. They say that counting numbers plus zero are natural numbers, but there's not solid agreement as to whether or not zero should be included among the natural numbers either. And while whole numbers are defined as positive, one can find the term “negative whole numbers” being used to describe integers. With all this confusion and lack of standardization in terminology, lets just forget all of the above (with the single exception of integers) and use the following more precise terminology instead.
The next type is rational numbers. Rational numbers are technically regarded as ratios (divisions) of integers. In other words, a rational number is formed by dividing one integer by another integer. Consequently, they are sometimes referred to as fractional numbers. (Remember, you can turn any integer into a fraction by placing it over a fraction line and the number 1.)
After integers and rational numbers (fractions), there is another major classification of numbers: those that can't be written as fractions. Remember that fractions (rational numbers) can be written as terminating (ending) or repeating decimals (such as 0.5, 0.76, or 0.333333 . . .).
There are number however that can be written as non-repeating, non-terminating decimals. These numbers are non-rational and are therefore called irrationals numbers. Examples would be √ 2 or π.
The rationals and irrationals are totally separate number types; there is no overlap.
Putting these two major classifications together gives you the real numbers, which make up the real number system. They are numbers that are represented by points on the number line. A real number is also defined as any number that can be written as a fraction. (But I thought that was rational numbers, and that real numbers and rational numbers don't overlap! You need to check this out!!!)
“But why,” you ask, “are they called ‘real’ numbers? Are there ‘pretend’ numbers?” Well yes, there are. They’re actually called “imaginary” numbers. They are used to make complex numbers. and always include the lower-case letter "i". (If a number does not have an “i” in it, it's a real number.)
TYPES OF NUMBERS (continued . . .)
THE COMMUTATIVE PROPERTIES5
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Properties
of Numbers
One of the "jobs" assigned to this textbook is to insure that you know how to use properties of numbers to demonstrate whether assertions are true or false -- An impossible task unless you know what the properties of numbers are. So, let's take a look at five of the most basic properties: commutative, associative, distributive, identity, and inverse. These properties lay the foundation for working with equations, functions, and formulas.
Generally speaking, the associative property states that if you change the grouping of numbers upon which some operation is being performed, the result remains the same. (Neither the commutative nor the associative properties work with subtraction or division.)
Let's look at the associative property of addition next: a + (b + c) = (a + b) + c
Again, the two sides of the equation are equivalent to one another.
PROBLEM:
Use the associative property to write an equivalent expression to (a + 7b) + 3c.
SOLUTION:
Using the associative property of addition (where changing the grouping of the addends does not change the value of the sum) we get (a + 7b) + 3c = a + (7b + 3c).
a(bc) = (ab)c
Now let's turn to the associative property of multiplication: a(bc) = (ab)c
Of course, both sides are equivalent.
PROBLEM:
Use the associative property to write an equivalent expression to (3 ∙ n)x.
SOLUTION:
Using the associative property of multiplication (where changing the grouping of the factors does not change the value of the product) we get (3 ∙ n)x = 3(nx)
THE ASSOCIATIVE PROPERTIESTHE DISTRIBUTIVE PROPERTYThe distributive property applies when you are multiplying some term that is outside a set of parentheses times two or more other terms that are being added or subtracted inside of the parentheses ( ). The distributive property states that you multiply the outside term times every term on the inside. (The terms on the inside must be separated by + or –.)
EXAMPLE: a(b + c – d) = ab + ac – ad
PROBLEM:
Use the distributive property to write 2(x - y) without parenthesis..
SOLUTION:
Distribute the 2 to each term on the inside the parentheses, or in other words, multiply every term inside of ( ) by 2, which give you . . .
2(x – y) =
2(x) – 2(y) =
2x – 2y
PROBLEM:
Use the distributive property to write –(2x + 5) without parenthesis.
SOLUTION:
Basically, when you have a negative sign in front of a ( ), like this example, you can think of it as taking a -1 times the ( ). What you end up doing in the end is taking the opposite of every term in the ( ).
–(2x + 5) =
(–1)(2x + 5) =
(–1)(2x) + (–1)(5) =
(–2x – 5)
PROBLEM:
Use the distributive property to find the product of 4(2a + b + 3c).
SOLUTION:
As mentioned above, you can extend the distributive property to as many terms as are inside the parentheses. The basic idea is that you multiply the outside term times each term on the inside.
4(2a + b + 3c) =
4(2a) + 4(b) + 4(3c) =
8a + 4b + 12c =
a(b + c – d) = ab + ac – ad
or
(a + b)c = ac + bc
the negative sign outside the parentheses is equivalent to (–1) so . . .
distributing the (-1) to each term inside the parentheses gives us . . .
and multiplying we get . . .
for our solution
distributing the 4 to each term inside the parentheses gives us . . .
and multiplying we get . . .
for our solution
The additive identity is 0, meaning that when you add 0 to any number, you end up with that number as a result.
The additive identity is 0
a + 0 = a (or 0 + a = a)
The multiplication identity is 1, meaning that when you multiply any number by 1, you wind up with that number as your answer.
a(1) = a (or 1(a) = a)
The additive inverse (or negative) states that for each real number a, there is a unique real number, denoted –a, such that a + (–a) = 0.
In other words, when you add a number to its additive inverse, the result is zero. Other terms that are synonymous with additive inverse are negative and opposite.
PROBLEM:
Write the opposite (or additive inverse) of -3.
SOLUTION:
The opposite of -3 is 3, since -3 + 3 = 0.
The multiplicative inverse (or reciprocal) states that for each real number a, except 0, there is a unique real number such that a ∙ ½ = 1 (or ½ ∙ a = 1).
In other words, when you multiply a number by its multiplicative inverse the result is one. A more common term used to indicate a multiplicative inverse is the reciprocal. A multiplicative inverse or reciprocal of a real number a (except 0) is found by "flip-flopping" it. The numerator of a becomes the denominator of the reciprocal of a and the denominator of a becomes the numerator of the reciprocal of a.
PROBLEM:
Write the reciprocal (or multiplicative inverse) of -3.
SOLUTION:
The reciprocal of -3 is -1/3, since -3(-1/3) = 1.
When you take the reciprocal, the sign of the original number stays intact. Remember that you need a number that when you multiply times the given number you get 1. If you change the sign when you take the reciprocal, you would get a -1, instead of 1, and that is a no no.
These additive and multiplicative inverses will come in handy big time when you go to solve equations later on, so keep them in your memory bank until that time.
THE IDENTITY AND INVERSE PROPERTIESa
a
PROBLEM:
Example 9: Write the opposite (or additive inverse) of ⅛.
SOLUTION:
The opposite of ⅛ is -⅛, since ⅛ + (-⅛) = 0.
PROBLEM:
Example 9: Write the reciprocal (or multiplicative inverse) of ⅛.
SOLUTION:
The reciprocal of ⅛ is 8, since 8(⅛) = 1.
These problems are from West Texas A&M University's beginning algebra website. They will allow you to check your understanding of these properties. To get the most out of them, you should work the problem out on your own and then check your answer by clicking on the link to West Texas A&M University (www.wtamu.edu) for the answer/discussion for that problem. At the link you will find the answer as well as any steps that went into finding that answer.
1. Use a commutative property to write an expression equivalent to: xy
2. Use a commutative property to write an expression equivalent to: 1 + 3x
3. Use an associative property to write an expression equivalent to: (a + b) + 1.5
4. Use an associative property to write an expression equivalent to: 5(xy)
5. Use the distributive property to find the product: -2(x - 5)
6. Use the distributive property to find the product: 7(5a + 4b + 3c).
7. Write the opposite (additive inverse) and the reciprocal (multiplicative inverse) of -7.
8. Write the opposite (additive inverse) and the reciprocal (multiplicative inverse) of 3/5
PRACTICE PROBLEMS6
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Properties
of Integers
Much of abstract algebra involves properties of integers and sets. We will now begin collecting some of the ones we will need for future reference.
An important property of the integers, which we will often use, is the so-called Well Ordering Principle. Since this property cannot be proved from the usual properties of arithmetic, we will take it as an axiom. The Well Ordering Principle states that: Every nonempty set of positive integers contains a smallest member.
THE DIVISION ALGORITHMThe concept of the visibility plays the fundamental role in the theory of numbers. As you well know, we cannot divide without having certain entities. First of all, we need something to divide (dividend), and something to divide by (divisor). After performing the operation, we end up with an answer (quotient), and oftentimes, a remainder as well.
At this point, we will substitute the names of these entities with variables. Let a stand for our dividend, b for our divisor, q for our quotient, and r for our remainder.
Undoubtedly, you are aware that after having obtained the solution to a division problem, one may check his or her answer by multiplying the quotient times the divisor and adding the remainder. In other words, a = bq + r
The above expression is known as the Division Algorithm.
But, wait just a second here! If we're talking about division, it seems like it would make a lot more sense to simply write:
a ÷ b = q + r
I mean, that’s what we’re talking about, isn’t it?
Well . . . yes, of course it is. But the only problem is, we're describing a property of integers, and integers are an example of something called an "integral domain," and in an integral domain, the only operations that are defined are addition and multiplication -- not division.
Okay, but why are we only allowed to use operations that can be defined in an integral domain? Ah, good question -- but one to which I do not have the answer! Nonetheless, the powers that be say students studying abstract algebra must learn about the Division Algorithm, so let us push on, for it is a fundamental property of integers that we will use often!
Our first task is to establish (prove) this fundamental property of. In order to do so, we will need the Well Ordering Principle, which is why we started there. Let us proceed!
Division Algorithm
Division Algorithm:
Let a and b be integers with b > 0. Then there exist unique integers q and r with the property that a = bq + r where 0 £ r < b.
Consider the set S = {a – bq | q is an integer and a – bq ≥ 0}
If 0 Î S, then b divides a and we may obtain the desired result with q = a/b and r = 0
Now assume 0 Ï S.
Since S is not empty
Then a = bq + r and r ≥ 0, so all that remains to be proved is that r < b.
If r > b, then a – b(q + 1)
= a – bq – b
= r – b > 0 so that a – b(q + 1) Î S.
To establish the uniqueness of q and r, let's suppose there are integers q, q´, r, and r´ such that
a = bq + r, 0 ≤ r < b and a = bq´ + r´, 0 ≤ r´ < b.
For convenience, we may also suppose that r´ ≥ r.
Then bq + r = bq´ + r´ and b(q - q´) = r´ – r.
So, b divides r´ – r and 0 ≤ r´ – r ≤ r´ < b.
It follows that r´ – r = 0 and therefore r´ = r and q = q´.
As previously stated, the Division Algorithm is a property of integers. It states that integers are “critters” such that you can take any negative or nonnegative integer a, along with some other positive integer b, and get the original integer by multiplying b times some third integer q, (then adding the resulting product to a final nonnegative integer r, which is less then b).
To prove that integers really do behave in this way, we begin with the existence portion of the theorem, which means we must demostrate that q and r really do exist under the given conditions.
If we are going to use the Well Ordering Principle to prove that our theorem is correct -- and indeed we are -- we’ll need to limit ourselves to positive integers, since that is the kind of set to which the Well Ordering Principle refers. We’ll call our set of positive integers S.
Our stating that a – bq ≥ 0 is just another way of saying that none of the members of this set are negative (which is not permissible if we’re going to apply the Well Ordering Principle). And since the algorithm says that q has to be an integer, we're making that a condition here as well.
Moreover, “nonempty” is another characteristic of any set to which the Well Ordering Principle applies, so let’s proceed by proving that S isn't empty. To do so, it is sufficient to show that S contains at least one element.
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All About
Groups
Group theory might be described as the study of symmetry, since collecting symmetries is the way in which groups are formed, such as finding all of the possible ways that a regular polygon can be repositioned (rotated, flipped, etc.) and yet, still fill the space it originally occupied. Group theory is a useful tool in areas of study such as physics and engineering. And they turn out to have a lot to do with puzzles like Rubik's Cube. As a matter of fact, the Rubik's cube configuration space is a group. To see group theory in action, let’s consider the symmetires of a square . . .
SYMMETRIES OF A SQUARE“” ‘’A group consists of three “entities” if you will.
The first entity is a nonempty set.
The second is a binary operation, which is a function (a rule or formula) that assigns to a given pair of elements some third element, with all three elements coming from the same set.
And the third and final entity that is necessary in order to have a group has already been mentioned: a pair of elements.
The object from which we will be collecting symmetries is a transparent square.with the corners on one side color-coded as follows:
Note that, were we to rotate the square, the maximum number of different positions in which it would still occupy the space it occupies now is four (4), and the total number of possible orientations is two (2), namely, face up and face down. That means there are a total of eight different positions ( 4 x 2 = 8) which, when assumed by the square, will fill the space it originally occupied.
However, to form a group we need a nonempty set -- and it must be a set that contains elements which can form pairs.
Two form such a set, let’s assign the following codes to the following manipulations:
CODE
- R0 = rotate the square 0° (no change in position)
- R90 = rotate the square 90° (counter clockwise)
- R180 = rotate the square 180°
- R270 = rotate the square 270°
- H = flip the square 180° about the horizontal axis
- V = flip the square 180° about the vertical axis
- D = flip the square 180° about the main diagonal
- D′ = flip the square 180° about the other diagonal
Any of the eight positions previously mentioned can be obtained by performing one of the above manipulations. Moreover, we can perform a sequence of two manipulations and get the same result as a single manipulation all by itself. Consequently, we are now able to form a group.
Suppose that the square flipped 180° about its horizontal axis and then rotated 90° counter clockwise as follows . . .
R90→
H→
--------------------
The result is that the square ends up in the same position it would have been in had we flipped it 180° about the other diagonal (running from the upper right-hand corner to the lower left-hand corner), for which the code it is D. Consequently, what we have is R90H = D′
So, we now have a set with the following elements:
S = {R0, R90, R180, R270, H, V, D, D′}
However, before we can label this set as a group there are four conditions that it must meet.
- It must be closed. That means that every result obtained by applying the binary operation to each pair of elements from the set is a member of the set as well. So far, so good!
- It must have identity. In other words, there must be an element e (call the identity) such that ae = ea = a. Sense HR0 = R0H = H we’re still okay.
- It must have inverses. For each element a there is an element b (called the inverse of a) such that ab = ba = e. Given that DD′ = D′D = R180 of we’re still in the clear.
- And finally, the set must have a associativity . . . that is, (ab)c = a(bc) for all a, b, c in the set.
Since grouping the manipulations will not affect the outcome, as long as the order remains the same, all four conditions have been met. Hence, the symmetries of a square form a group.
DEFINITION FOR A GROUP
A group is a nonempty set, together with a binary operation (usually called multiplication) that assigns to each ordered pair of elements (a,b) some element from the same set, denoted by ab. (The set is a group under the given binary operation if and only if the properties of closure, associativity, identity, and inverses are satisfied.)
The idea here is to show that S is nonempty for a particular case -- that case being a – bq = 0.
Again, why should I care if the set includes zero or not? Whether or not the set is empty only matters if I can demonstrate that it has at least one positive member. Am I right?
So, lets assume S is a nonempty set with zero as an element. This would mean that . . .
a − bq = 0
a = bq
a/b = q
There is no contradiction here, so we have proven that S is a nonempty set with 0 belonging to it. One last time . . . so what?
Again, let's assume that S is not empty. But this time, we will consider the case of 0 not being in S. Consequently, we need for a – bq to be greater than zero. Remember that the following conditions apply to the Division Algorithm:
- a is allowed to be any integer whatsoever
- b has to be greater than zero
So a can be positive or negative. (It can also be zero too, couldn't it? I never saw that mentioned anywhere. Is that because it's impossible to prove anything if you let a = 0?)
Suppose that a > 0 and we let q = 0.
Then a − bq = a − (b ∙ q) = a − (b ∙ 0) = a − 0 = a.
Since a > 0, and since a − bq = a, it follows that a − bq > 0 as well. There is no contradiction here, so S is not empty when a > 0.
Now, if a < 0 and we let q = 2a.
Then a − b(2a) = a(1 − 2b) = −a(2b − 1). Since the negative of a is positive, we have
lal(2b − 1). And since the expression (2b − 1) must be positive, once again we end up with a positive outcome.
Another typical argument to prove S is not empty is as follows . . .
Once again start with a < 0, but this time let q = a. Given this case, we have a − ba = a(1 − b) = −a(b − 1). Since the negative of a is positive, we have lal(b − 1), and by the expression (by assumption) b − 1 ≥ 0.
But again, since the Well Ordering Principle call for the elements of S to be positive integers, don't we want b − 1 to be greater than 0 and not equal to it?
If the Well Ordering Principle states that every nonempty set of positive integers contains a smallest member why are we considering the following set . . .
S = {a – bq | q is an integer and a – bq ≥ 0}
Zero is neither positive nor negative, right? Or am I mistaken? Since zero isn't positive, shouldn't we be considering this set instead?
S = {a – bq | q is an integer and a – bq > 0}
I need clarification on this point!
Modern/Abstract
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What “modern” algebra (called abstract algebra) and physics do is take this process another step, replacing the variables with other things, like matrices or polynomials for example, producing pseudo-numerical knowledge about structures which are not at all numerical in nature: (1) groups, (2) rings, and (3) fields, tangible examples of the abstract rules with various levels of requirements that make them more and more like real numbers. If mathematics were, let’s say, a car, then old algebra would perhaps be a covered wagon, while abstract algebra might be something along the lines of a Pontiac Solstice.
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Why Study Algebra?
To make algebra as interesting as possible and demonstrate the relevance of as many of the concepts that are being taught as feasible, this book, whenever practical, will include applications from the real-world. It will also define virtually every term as it is introduced, in case you are unable to remember their definitions from previous encounters, starting with the term axiom, which is a law, truth, or fundamental principle that is established, self-evident, universally recognized, taken for granted, or accepted without proof.
Followed back to its foundations, mathematics is based on the axioms of set theory. So, in our study of algebra it only makes sense to begin with sets, the concept of which is itself and axiom.
All About Sets
SET NOTATIONSay no more . . .
An empty set is denoted by Æ.
If a set B is a subset of a set A (if every elemment of B is in A), we denote that fact by B Í A or A Ê B.
(NOTE: According to the definition of subset, any set is a subset of itself. In such cases, the subset is referred to as an improper subset. Also, every set has Æ as a subset.)
And finally, if A and B are both sets, the set A ´ B = {(a, b) ½ a Î A and b Î B} is the Cartesian product of A and B.
For example, if A = {2, 4, 6} and B = {6, 8} then we have A ´ B = {(2, 6), (2, 8), (4, 6), (4, 8), (6, 6), (6, 8)}.
Appearing on the next page is the notation we will be using for the various familiar sets of numbers that will be coming up regularly and repeatedly throughout this textbook. It would be wise indeed to commit them to memory right now, once and for all.
{2, 4, 6, 8}
and
{x l x is an even whole positive number ≤ 8}
and
{2x l x = 1, 2, 3, 4}
. . . all are three different ways to represent the same set.
A lot of the work throughout his textbook is going to involve familiar sets of numbers. Therefore, let's make sure that we learn the notation for these sets right here in now.
Z
Z
Q
Q
R
R
R
R
+
Q
Q
+
Z
Z
+
Z
Z
Q
Q
R
R
* is this set of nonzero integers
* is this set of nonzero rational numbers
* is the set of nonzero real numbers.
is the set of positive integers.
is the set of positive rational numbers.
is the set of positive real numbers.
is this set of all integers.
is this set of all rational numbers.
is the set of all real numbers.
Also, do not proceed to any of the subsequent pages until you can answer all of the following questions correctly without referring to any previous pages or any notes that you may have taken.
What is the definition of a set?
What do we used to denote a set?
How do we denote the fact that a particular element is a member of a particular set?
Illustrate an example of how is that might be defined using set notation.
Illustrate two ways of denoting that set see is a subset of set D.
How do we denote an empty set?
What do we call a set that is a subset of itself?
What is a Cartesian product?
GROUP
RING
FIELD
Nonempty Set
Nonempty Set
Nonempty Set
1 Binary Operation
2 Binary Operations
2 Binary Operations
Commutative Under Addition
2 Binary Operations
2 Binary Operations
Imaginary numbers are the square roots of negative numbers. They are called imaginary because the square root of a negative number cannot be real. The square root of - 1 is called i
Imaginary numbers are numbers that can be written as a real number times i.
i was invented because people wanted to be able to take square roots of negative numbers, and you can't do that if you limit yourself to real numbers.
So we can make an imaginary number by taking a real number like 5 and multiplying it by i. That gives us 5i. Some other imaginary numbers are
Complex numbers are numbers like 7 + .4i; they're a real number plus an imaginary number.
But some mathematicians decided they wanted to take the square roots of
negative numbers anyway. They were kind of renegade mathematicians at
the time. What they did is they said "hey, let's make up a new number
and call it i. It will have the property that when you square i, you
get -1."
Now, with this new number, you can take the square root of negative
numbers. For instance, the square root of -4 is 2 times i (which most
mathematicians just write as 2i). The square root of -2 is 1.41421i.
And so on.
Footnote: actually, there are TWO numbers that are the square root of -1, and those numbers are i and -i , just as there are two numbers that are the square root of 4, 2 and -2.
Before I begin, let's define some notation. When used in an equation,
* is multiplication, so 4*7 = 28. x^2 means "x squared," so ^ denotes
exponentiation. Sqrt[ ] is the square root of the expression in
brackets.
Imaginary numbers were conceived in response to the question of whether
or not we could think about the square root of negative numbers, or
equivalently whether or not there existed a value that satisfied the
equation x^2 + 1 = 0. If we decide that this equation _does_ have a
solution, then we can give that solution a name: let's call it i. Then
it's not too hard to show that i^2 = -1, and that -i also satisfies the
equation x^2 + 1 = 0. The reason mathematicians chose "i" as this new
number's name is that they still questioned its validity as a number,
and questioned its right to co-mingle with the real numbers.
After all, they had a lot to worry about - what physical significance
does "i" have? (Several, in particular, in the physics of electric
circuits.) Does its introduction lead to logical inconsistencies?
(No - as a matter of fact, it opens up vast fields of study from
abstract algebra to complex analysis.) But as time went on, people
began to realize that "i" was a generally good idea - though the term
"imaginary" stuck.
How does one use "i" in calculation? Well, i follows most
mathematical conventions, though care needs to be taken occasionally
during multiplication and root extraction. For example, i+i = 2i, and
(1+i)+(3-i) = 4. i^2 = i*i = -1, from the definition; note
Sqrt[-4]*Sqrt[-1] is not equal to sqrt[(-4)(-1)] = 2, but rather
Sqrt[-4]*Sqrt[-1] = i*Sqrt[4]*i = -2. This apparent contradiction
arises simply because the rule where Sqrt[p]*Sqrt[q] = Sqrt[p*q] is
only applicable when p and q are non-negative.
Thus we have two sets of numbers: "real" and "imaginary." They are
mutually exclusive (except perhaps for 0, which could be considered as
both real and imaginary, though nearly always it is thought of as
belonging to the former). The union of these sets forms the *complex*
numbers; these are numbers of the form a+b*i, where a and b are reals.
As can be verified, addition and multiplication are well-defined (non-
ambiguous), there is an *additive identity* (zero), a *multiplicative
identity* (one), there is always an *additive inverse* (i.e., for
every complex number a+bi there exists a unique c+di such that (a+bi)+
(c+di) = 0), and a *multiplicative inverse* (i.e., for every a+bi
there is a unique c+di such that (a+bi)(c+di) = 1). Finally, addition
and multiplication are associative, commutative, and distributive.
The most common purpose of the imaginary numbers is in the
representation of roots of a polynomial equation in one variable. For
example, what are the roots of x^2 + 2*x + 5 ? Using the quadratic
formula, we find
-2 + Sqrt[4 - 4*5] -2 - Sqrt[4 - 4*5]
x = ------------------ , ------------------
2 2
or
x = {-1+2*i, -1-2*i}.
An important theorem in algebra (which I stated for a particular case
at the beginning of this letter) states that for a polynomial in one
variable of degree n, there are exactly n roots, counting multiple
roots. So a cubic equation has 3 roots, a quartic 4, and so on.
There are other applications of complex numbers, some of them quite
abstract - often they deal with the algebraic structure of these
numbers, rather than the computational aspects.
Hope this helps.... :)
-Doctor Pete, The Math Forum
Check out our web site! http://mathforum.org/dr.math/
--------------------------------------------------------------------------------
Date: 7/25/96 at 19:17:28
From: Doctor Robert
Subject: Re: imaginary numbers
I think that the moniker "imaginary" is unfortunate, since in one
sense, all numbers are imaginary in that they exist only in our minds.
But, in mathematics, there is a distinction made between real and
imaginary numbers. The real numbers are those that show up on the
number line. Imaginary numbers arise because mathematicians could not
find a solution to the equation
x^2 + 1 = 0
in the set of real numbers. So, they decided to designate the square
root of negative one by the small case letter i. If i = the square
root of -1, then the square root of -4 is 2i. Now, complex numbers
are those which have both a real part and an imaginary part. An
example would be the number 2+3i. The young man studying imaginary
numbers is probably learning to do basic operations with complex
numbers like addition, subtraction, multiplication, and division.
I hope that this helps.
-Doctor Robert, The Math Forum
Check out our web site!
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