## Friday, November 15, 2013

## Monday, July 1, 2013

### Unizor - Trigonometry - ARC... Introduction

We have introduced six trigonometric functions in the previous lectures:

In this lecture we would like to address the issue of inverse functions for these six trigonometric functions. Inverse function, in general, is a function with a domain being a range of an original function, with a range being a domain of an original function and a nice property that, applied in composition with an original function, it results in an identity function either over a domain or a range of an original function, depending on the order of composition.

From this perspective an inverse of any trigonometric function is a function defined on some subset of real numbers (that is, on a range of an original trigonometric function) and values among the angles, which are a domain of an original trigonometric function (usually, expressed in radians).

The main tool to define trigonometric function was a unit circle. In this circle an angle in radians corresponds to the length of an arc that corresponds to this angle. That is why all inverse trigonometric functions are called by the name that combines a prefix

There is a problem, however, which makes this definition plainly wrong. All trigonometric functions are periodic and, therefore, there are multiple (actually, infinite) number of angles with the same value of any trigonometric function. For example, sin(π/6)=1/2, sin(5π/6)=1/2, sin(π/6+2π)=1/2 etc. So, what is arcsin(1/2)? It can be π/6, it can be 5π/6, π/6+2π etc. - an infinite number of values. This cannot be a definition of a function that for each argument has to be fully and uniquely defined with one value. To overcome this difficulty in defining inverse trigonometric functions we have to re-examine a concept of an inverse function, which is a subject of this lecture.

Recall that a

It's important to emphasize that the rule of transformation must be applicable to each element of a domain with no exceptions and the rule must map each element of a domain into one and only one element of a codomain. Also important to understand that a range does not necessarily coincide with a codomain, it can be its subset.

Notice that it is possible for two different elements of a domain to be mapped into the same element of a codomain, thus a value of a function for two different arguments can be the same.

Now let's address inverse functions.

In short, we would like to know an argument of a function (an element of its domain) if we know the value from a codomain this argument is mapped into.

Unfortunately, it's not always easy. It is easy in case a function establishes one-to-one correspondence between its domain and its range that completely fills the codomain. For example, an exponential function

Let's consider another familiar example,

There is however a way to define a function that is almost what we need. Recall that

The above technique of restricting the domain of an original function in order to be able to define an inverse function is used to define inverse trigonometric functions. If we narrow the domain of a trigonometric function to an interval where there is a one-to-one correspondence between thus restricted domain and the full range of a function, the inverse function can be defined. If this interval exists, we can consider an original trigonometric function only on this interval and define an inverse trigonometric function as having codomain coinciding with this interval.

**,***y=sin(x)***,***y=cos(x)***,***y=tan(x)***,***y=cot(x)***and***y=sec(x)***. Each is a function of an angle (usually expressed in radians) chosen from a domain of real numbers with a range (also a real number) depending on a specific function.***y=csc(x)*In this lecture we would like to address the issue of inverse functions for these six trigonometric functions. Inverse function, in general, is a function with a domain being a range of an original function, with a range being a domain of an original function and a nice property that, applied in composition with an original function, it results in an identity function either over a domain or a range of an original function, depending on the order of composition.

From this perspective an inverse of any trigonometric function is a function defined on some subset of real numbers (that is, on a range of an original trigonometric function) and values among the angles, which are a domain of an original trigonometric function (usually, expressed in radians).

The main tool to define trigonometric function was a unit circle. In this circle an angle in radians corresponds to the length of an arc that corresponds to this angle. That is why all inverse trigonometric functions are called by the name that combines a prefix

*arc*(implying that we are seeking an arc or a corresponding angle in a unit circle) and the original name of a trigonometric function. Thus, we can talk about**as an angle, a sine of which equals to***y=arcsin(x)***, or about***x***as an angle, a cosine of which equals to***y=arccos(x)***, etc.***x*There is a problem, however, which makes this definition plainly wrong. All trigonometric functions are periodic and, therefore, there are multiple (actually, infinite) number of angles with the same value of any trigonometric function. For example, sin(π/6)=1/2, sin(5π/6)=1/2, sin(π/6+2π)=1/2 etc. So, what is arcsin(1/2)? It can be π/6, it can be 5π/6, π/6+2π etc. - an infinite number of values. This cannot be a definition of a function that for each argument has to be fully and uniquely defined with one value. To overcome this difficulty in defining inverse trigonometric functions we have to re-examine a concept of an inverse function, which is a subject of this lecture.

Recall that a

*function*is a rule that puts into a correspondence (or "maps", as sometimes say) each element of one set (called*domain*) to a single element of another set (called*codomain*) with all the elements of a codomain that actually have a prototype in a domain (that is, actually are mapped into) called a*range*of a function. So, a function is a triplet of a domain, a codomain and a rule that transforms each element of a domain into some (single) element of a codomain. It is important to understand that we can change the definition of a function by just changing a domain or a codomain without changing the rule of transformation of elements of one into elements of another. Range specification is not part of a function definition, it's the consequence of it.It's important to emphasize that the rule of transformation must be applicable to each element of a domain with no exceptions and the rule must map each element of a domain into one and only one element of a codomain. Also important to understand that a range does not necessarily coincide with a codomain, it can be its subset.

Notice that it is possible for two different elements of a domain to be mapped into the same element of a codomain, thus a value of a function for two different arguments can be the same.

Now let's address inverse functions.

In short, we would like to know an argument of a function (an element of its domain) if we know the value from a codomain this argument is mapped into.

Unfortunately, it's not always easy. It is easy in case a function establishes one-to-one correspondence between its domain and its range that completely fills the codomain. For example, an exponential function

**is such a function. It maps a domain of all real numbers (an argument***y=10*^{x}**can be any real number) to a codomain of all positive real numbers (a value***x***is always a positive real number) with its range completely filling a codomain and, importantly, any two different arguments***y=10*^{x}**and***x*_{1}**(***x*_{2}**≠***x*_{1}**) produce different results***x*_{2}**and***y*_{1}=10^{x1}**(***y*_{2}=10^{x2}**≠***y*_{1}**), thus establishing one-to-one correspondence between a domain of all real numbers and a codomain of all positive numbers. Therefore, we can easily find an argument if we know the value of a function, thus mapping all positive real numbers into all real numbers. This mapping is a definition of a logarithmic function***y*_{2}**which is***y=log*_{10}(x)*inverse*to an exponential function**with its domain being a codomain of an exponential function and its codomain being a domain of an exponential functions.***y=10*^{x}Let's consider another familiar example,

**defined for all real arguments. It perfectly corresponds to a definition of a function mapping all real numbers to all non-negative real numbers. However, it's not a one-to-one correspondence. Numbers 2 and −2 are both mapped to a number 4. Generally, two numbers A and −A, where A is any real number, are mapped to the same value A***y=x*^{2}^{2}. How can we find an argument in this case if we know the value of a function? Strictly speaking, without any additional conditions, we cannot. Defined as above, the function**has no inverse one.***y=x*^{2}There is however a way to define a function that is almost what we need. Recall that

*monotonic*(or*monotonous*) function of real argument with real value always defines a one-to-one correspondence between its domain and its range. Let's narrow a definition of a function**to an interval where it is monotonic. We have two such intervals: (−∞,0] and [0,+∞). On the former the function is monotonously decreasing, on the latter it is monotonously increasing. Let's concentrate on the latter, that is***y=x*^{2}**and use only this interval as a domain. Since on this interval function***x≥0***is monotonously increasing and still fills completely the range of an original function***y=x*^{2}**defined for all real arguments, the narrowly defined function establishes one-to-one correspondence between all non-negative arguments and all non-negative values of a function. Therefore, an inverse function exists and can be defined. Its domain will be a set of all non-negative real values and the same is true for its codomain and a range. Thus, the value for such inverse function for an argument 4 will be 2 (and not −2, since −2 does not belong to its codomain, regardless of the fact that (−2)***y=x*^{2}^{2}=4). This restriction for the codomain of our newly defined function, which, incidentally, is called*principal square root*, allows to define a function inverse to a narrowly define function**. More precisely, if we narrow the domain of a function***y=x*^{2}**to non-negative arguments***y=x*^{2}**, the principal square root function***x**y=**√*represents an inverse function to it.**x**The above technique of restricting the domain of an original function in order to be able to define an inverse function is used to define inverse trigonometric functions. If we narrow the domain of a trigonometric function to an interval where there is a one-to-one correspondence between thus restricted domain and the full range of a function, the inverse function can be defined. If this interval exists, we can consider an original trigonometric function only on this interval and define an inverse trigonometric function as having codomain coinciding with this interval.

## Sunday, June 16, 2013

### Unizor Trigonometry

Just published an Introduction lecture to Trigonometry part of UNIZOR.COM. Presented trigonometric functions on the unit circle and explained the connection with right triangles. Also explained why we can use trigonometric functions as functions of angles referring to a property of similar triangles to have congruent angles and proportional sides.

Video on this topic is coming soon.

Video on this topic is coming soon.

## Friday, May 31, 2013

__Myth and Truth about Mathematics__

As I see it, math is viewed by majority of students in America as an unneeded subject. Indeed, they realize that in their practical life they will unlikely have to solve a system of linear equations or calculate an area of a trapezoid. In 99% of the cases they are right, so they are negatively predisposed to math and consider it to be just a bunch of facts they have to memorize to pass an exam and then they can happily forget it. Nothing is left from this knowledge after a year or two. Looks like the system of education focused on giving the students prepackaged recipes to as many math topics as possible fails because students do not go any deeper in understanding these topics, just learn a rule "if you have to find ... do ...", which, as I mentioned before, is easily forgotten as unneeded. As one person characterized it, this system is "a mile wide and an inch deep". Not withstanding some memory training, this system bears no useful purpose.

So, the need to drop all the mathematical facts onto student's head counting that something will stick and might be useful in their future life is a myth.

What I think the focus of math education should be is the development of the mind. Does not matter what math topic is studied. What matters is, how does it help to develop abilities of a student's mind. What differentiates math from other subjects is its completely artificial character. It's a product of bright minds. No other subject has this degree of brain participation because all other subjects are dealing with real objects. Studying chemistry or literature, foreign language or biology - it's all about what exists in the world, already created. Studying math allows to get a student involved in the creative process. Prove this theorem, solve this problem, construct this geometrical figure - these are creative assignments not available anywhere else. Artificial character of math allows to train the student's mind to be logical, analytical, creative, intelligent. Math is an ultimate form of art because it is completely a product of artful minds.

If I were a teacher, I would start my first lesson from asking students about why should they study math. I am sure there will be very few, if any, who would answer that it's needed to develop their mind or simply to get smarter. Most likely, I would hear no answer at all. And then I would compare studying math to develop the mind based on problem solving approach to exercises in gym to develop the muscles, stamina and other physical abilities. What gym is for physical development, math is for intellectual development. I realize that not many students will be interested in their intellectual development, after all people in general are not satisfied with their physical appearance but quite satisfied with how their brain functions, but whoever will be interested, will benefit tremendously.

So, mathematics is a tool to develop a student's mind. This is the truth. And to accomplish this, a student must be challenged with proofs of theorems, problem solving, logical approach to theory and creativity. This approach is not in the focus of most schools and UNIZOR.COM attempts to compensate for this.

Our goal with Unizor is not as much to change an entire system of education, we realize impossibility of this, but to present a choice for those real knowledge seekers, who are still there in the world, deprived from proper intellectual development by existing educational system.

## Thursday, April 4, 2013

### Unizor - Algebra - Fundamental Theorem

This theorem was called "Fundamental" at the time when the most important (if not the only) purpose of algebra was solving polynomial equations, and the higher degree - the better.

Nowadays algebra is concerned with much broader spectrum of problems, but the historical name of this theorem is retained.

The Fundamental Theorem of Algebra states that any polynomial equation has at least one solution in the field of complex numbers.

More precisely, assume we have an equation of the following type:

a[0]·z^n + a[1]·z^(n-1) + a[2]·z^(n-2) + ...

+ a[n-2]·z^2 + a[n-1]·z + a[n] = 0

where n is an integer, all coefficients ai are constant complex numbers, z is a complex variable the value of which we have to find and the coefficient a[0] at z^n is non-zero (so, the polynomial is truly of the nth degree).

Or, using Σ-notation,

Σa[i]·z^(n-i) = 0

where index i ∈ [0,n] (that is, an integer index i is changing in the interval from 0 to n).

This equation is called polynomial equation of nth degree in the field of complex numbers.

Then the Fundamental Theorem of Algebra states that such a polynomial equation must have at least one complex solution, oftentimes called the root of this polynomial.

Proof of this theorem is beyond the scope of this course (which is rather exception than a rule) because of its complexity. However, this theorem has a few interesting corollaries (simple consequences) that we will prove.

Corollary 1

Let P(n)(z) signify a general polynomial of the nth degree as a function of a complex argument z with some complex coefficients ai:

P(n)(z) = a[0]·z^n + a[1]·z^(n-1) + ... + a[n-1]·z + a[n]

Let c be one of the roots of this polynomial, that is c is a solution of the corresponding polynomial equation, thus P(n)(c) = 0.

Then this polynomial of the nth degree can be represented as a product of (z - c) and some other polynomial of the (n-1)th degree:

P(n)(z) = (z-c)·P(n-1)(z)

Corollary 2

Any polynomial

P(n)(z) = Σa[i]·z^(n-i)

of the nth degree has exactly n roots in the field of complex numbers and can be represented as a product of expressions (z-c[i]) and a constant multiplier A, where i∈[1,n], c[i] are its roots and a multiplier A equals to the coefficient a[0] at z^n.

Nowadays algebra is concerned with much broader spectrum of problems, but the historical name of this theorem is retained.

The Fundamental Theorem of Algebra states that any polynomial equation has at least one solution in the field of complex numbers.

More precisely, assume we have an equation of the following type:

a[0]·z^n + a[1]·z^(n-1) + a[2]·z^(n-2) + ...

+ a[n-2]·z^2 + a[n-1]·z + a[n] = 0

where n is an integer, all coefficients ai are constant complex numbers, z is a complex variable the value of which we have to find and the coefficient a[0] at z^n is non-zero (so, the polynomial is truly of the nth degree).

Or, using Σ-notation,

Σa[i]·z^(n-i) = 0

where index i ∈ [0,n] (that is, an integer index i is changing in the interval from 0 to n).

This equation is called polynomial equation of nth degree in the field of complex numbers.

Then the Fundamental Theorem of Algebra states that such a polynomial equation must have at least one complex solution, oftentimes called the root of this polynomial.

Proof of this theorem is beyond the scope of this course (which is rather exception than a rule) because of its complexity. However, this theorem has a few interesting corollaries (simple consequences) that we will prove.

Corollary 1

Let P(n)(z) signify a general polynomial of the nth degree as a function of a complex argument z with some complex coefficients ai:

P(n)(z) = a[0]·z^n + a[1]·z^(n-1) + ... + a[n-1]·z + a[n]

Let c be one of the roots of this polynomial, that is c is a solution of the corresponding polynomial equation, thus P(n)(c) = 0.

Then this polynomial of the nth degree can be represented as a product of (z - c) and some other polynomial of the (n-1)th degree:

P(n)(z) = (z-c)·P(n-1)(z)

Corollary 2

Any polynomial

P(n)(z) = Σa[i]·z^(n-i)

of the nth degree has exactly n roots in the field of complex numbers and can be represented as a product of expressions (z-c[i]) and a constant multiplier A, where i∈[1,n], c[i] are its roots and a multiplier A equals to the coefficient a[0] at z^n.

## Wednesday, March 27, 2013

### The Purpose of Math Education

__Myth and Truth__

As I see it, math is viewed by majority of students in America as an unneeded subject. Indeed, they realize that in their practical life they will unlikely have to solve a system of linear equations or calculate an area of a trapezoid. In 99% of the cases they are right, so they are negatively predisposed to math and consider it to be just a bunch of facts they have to memorize to pass an exam and then they can happily forget it. Nothing is left from this knowledge after a year or two. Looks like the system of education focused on giving the students prepackaged recipes to as many math topics as possible fails because students do not go any deeper in understanding these topics, just learn a rule "if you have to find ... do ...", which, as I mentioned before, is easily forgotten as unneeded. As one person characterised it, this system is "a mile wide and an inch deep". Not withstanding some memory training, this system bears no useful purpose.

So, the need to drop all the mathematical facts onto student's head counting that something will stick and might be useful in their future life is a myth.

What I think the focus of math education should be is the development of the mind. Does not matter what math topic is studied. What matters is, how does it help to develop abilities of a student's mind. What differentiates math from other subjects is its completely artificial character. It's a product of bright minds. No other subject has this degree of brain participation because all other subjects are dealing with real objects. Studying chemistry or literature, foreign language or biology - it's all about what exists in the world, already created. Studying math allows to get a student involved in the creative process. Prove this theorem, solve this problem, construct this geometrical figure - these are creative assignments not available anywhere else. Artificial character of math allows to train the student's mind to be logical, analytical, creative, intelligent. Math is an ultimate form of art because it is completely a product of artful minds.

If I were a teacher, I would start my first lesson from asking students about why should they study math. I am sure there will be very few, if any, who would answer that it's needed to develop their mind or simply to get smarter. Most likely, I would hear no answer at all. And then I would compare studying math to develop the mind based on problem solving approach to exercises in gym to develop the muscles, stamina and other physical abilities. What gym is for physical development, math is for intellectual development. I realize that not many students will be interested in their intellectual development, after all people in general are not satisfied with their physical appearance but quite satisfied with how their brain functions, but whoever will be interested, will benefit tremendously.

So, mathematics is a tool to develop a student's mind. This is the truth. And to accomplish this, a student must be challanged with proofs of theorems, problem solving, logical approach to theory and creativity. This approach is not in the focus of most schools and UNIZOR.COM attempts to compensate this.

Our goal with Unizor is not as much to change an entire system of education, we realize impossibility of this, but to present a choice for those real knowledge seekers, who are still there in the world, deprived from proper intellectual development by existing educational system.

Subscribe to:
Posts (Atom)