The Definition of “Undefined”

In learning mathematics, we often encounter terms that are not always clear. Example of such term is the word undefined. What do we mean by undefined?

The word undefined may slightly differ in meaning depending on the context. In plain language, it means something which has no sensible meaning. For instance, during the time when the negative numbers were not yet invented, the numerical expression 5 – 8 has no meaning. In our time, we can say that 5 – 8 is undefined in the set of positive integers.

Below are some examples of the different contexts where the different meanings of “undefined” can be drawn.

Numbers

Square Root of Negative Numbers. The \sqrt{-1} is undefined in the set of real numbers. This means that no real number exists that when multiply it by itself, the product is equal to -1. Note, however, that some operations may be undefined under some sets, but defined in other sets. We know from high school mathematics that square root of – 1 equals i in the set of complex numbers.

Algebra

Division by 0. Since we do not know the answer if a number divided by 0, let us examine the quotient of numbers when divided by numbers close to 0. To make it simple, let us try 1/0.

Figure 1 – The value of the 1/x as x approaches 0.

As we can see, as x approaches 0 from the right, the quotient of 1/x is getting larger and larger. On the other hand, as x approaches 0 from the left, 1/x is getting smaller and smaller. As a consequence, there is no single number that 1/x approaches as x approaches 0.   Therefore we can say that 1/0 is undefined.

Geometry

Intersection of two lines. In Euclidean Geometry, if we talk about the intersection of two lines, we can have three cases: intersecting lines have one intersection, coinciding lines have infinitely many intersections, and parallel lines have no intersection.

Figure 2 – The three cases, in terms of intersection, two lines can be places on a plane.

We can say that if two lines are parallel, no intersection exists. Algebraically, the solution to the system of equations of the two lines is their intersection.  Hence, the solution of the two systems of linear equations of parallel lines as graphs is undefined.

 

Matrices

Matrices with Different Sizes. If A is a 2 by 2 matrix and B is a 3 x 3 matrix, then A + B has no meaning since five of the entries of matrix B have no corresponding entries in matrix A. We can say that the sum of matrix A and matrix B is not well-defined.

Sets

Intersections of Sets. If E is the set of even integers and O be the set of odd inteers, then there is no common value to both sets.

Figure 3 – The Venn Diagram of the intersection of even and odd numbers.

In set theory, we call the common values the intersection, and in this example, the intersection is the empty set.  As a consequence, we can say that the intersection of set E and O is undefined.

***

Although the word “undefined” has different meanings depending on the context, by now, you would have realized that the phrases “does not exist”, “without sensible meaning” and “cannot be determined” are somewhat synonymous to it. If the result of an operation yields no value  at all (note that 0 is a value and is not the same to no value), then it is more likely that it is undefined.

Derivative and the Maximum Area Problem

Note: This is the third part of the Derivative Concept Series. The first part is The Algebraic and Geometric Meaning of Derivative and the second part is Derivative in Real Life Context.

Introduction

The computation of derivative is often seen in maximum and minimum problems.  In this article, we will discuss why do we get the derivative of a function and equate it to 0 when we want to get its maximum or minimum. To give you a concrete example, let us consider the problem below.

Find the maximum area a rectangle with perimeter 10 units.

Without using calculus, we can substitute values for the rectangle’s length, compute for its width and its corresponding area. If we set the interval to 0.5, then we can come up with the table shown in Figure 1.

Figure 1 - Table showing the length, width, and area of a rectangle with perimeter 10.

Looking at the table above, we can observe that a rectangle of length of 2.5, a square, has the maximum area. If we have prior calculus  knowledge, however, we know that whatever the value of our perimeter, a square having the given perimeter will always have the maximum area.

Using elementary algebra, if we let x be the width of our rectangle, it follows that the length is 5-x. Let f(x) be the area of the rectangle. In effect, the area of the rectangle is described by the equation f(x) = 5x - x^2. We want to maximize the area, which implies that we want to find the maximum value of f(x).

Figure 2 – A rectangle with Perimeter 10 and width x units.

In elementary calculus, to compute for the maximum value of f(x), we get its derivative, which is equal to 5 - 2x, which we will denote f'(x). We then equate the f'(x) to 0 resulting to the equation 5-2x=0 \Rightarrow x = 2.5 which is exactly the maximum value in the table above.

Derivative and Equation to 0

In the article the Algebraic and Geometric Meaning of Derivative, we have learned that the derivative of a function is the slope of the line tangent to that function at a particular point. From elementary algebra, we also have learned the properties of slopes. If a line is rising to the right, the slope is greater than 0; if the line is rising to the to the left, then the slope is less than 0. We have also learned that a horizontal line has slope 0 and the vertical line has an undefined slope.

Figure 3 – Properties of slope of a straight line.

In the problem above, we calculated by getting the derivative (the slope of the line tangent to a function at a particular point) and equate it to 0. But a line with slope 0 is a horizontal line. In effect, we are looking for a horizontal tangent of f(x) = 5x-x^2. To give a clearer picture let us look at the graph of f(x) = 5x - x^2.

Figure 4 – Tangent lines of 5x – x2.

From the graph it is clear that the maximum point of the function is where the tangent line (red line) horizontal. In fact, there are only three possible cases that tangent line could be horizontal as shown in Figure 5: first, the minimum of a function (blue graph); second, the inflection point (red graph); and the third is the maximum of the function (green graph).

It should also be noteworthy to say that all the ordered pairs (length, area) or(width, area) in Figure 1 will be on the blue curve in Figure 4.

Figure 5 – Cases of a graph where the tangent is horizontal.

The derivative has many applications and it is seen in many topics in calculus.  In the next Derivative Tutorial, we are going to discuss how the derivative is used in other context.

Summary

  • The derivative of a function is the slope of the line tangent to a function at a particular point.
  • The horizontal line has slope zero.
  • In solving maxima and minima problems, we get the derivative of a function and equate to zero to get the minimum or maximum. We do this because geometrically, we want to get the line tangent to a function at a particular point that is horizontal.

The Exterior Angle Theorem

In the angle sum of a triangle post, we have discussed that the angle sum of a triangle is 180 degrees.  In the angle sum of a polygon post,  we also have discussed that  and that the angle sum of a polygon with n sides is 180(n-2). For example, a pentagon has 5 sides, so the sum of its interior angle is 180(5-2) = 180(3) = 540 degrees.

Figure 1 – The interior and exterior angles a triangle and a quadrilateral.

The angle sums that we have discussed in both blogs refer to the sum of the interior angles. What about the exterior angles?

The exterior angle is formed when we extend a side of a polygon. In the triangle above, \alpha is an exterior angle. The sum of the interior angle and the exterior angle adjacent to it is always 180  degrees (Why?).  Angles whose sum is 180 degrees are called supplementary angles.  If two angles are supplementary, we call them a linear pair.  For example, angles \alpha and a_1 are supplementary angles and at the same time a linear pair, so \alpha + a_1 = 180 degrees. Now this means, that \alpha = 180 - a_1. Therefore, if we want to compute the measure of an exterior angle adjacent to an interior angle, we can always subtract the measure of the interior angle from 180 as shown in Figure 1.

Observe the computation in the two diagrams.  If we let S_t be the angle sum of the exterior angles of a triangle, then S_t = (180 - a_1) + (180 - a_2) + (180 - a_3) = 540. Rearranging the terms, we have S_t = 540 - (a_1 + a_2 + a_3).  But a_1 + a_2 + a_3 is the sum of the interior angles of a triangle which is 180 degrees, so 540 - (a_1 + a_2 + a_3) = 540 - 180 = 360 degrees.

Now, try calculating for the sum of the exterior angles of the quadrilateral above. What is your answer?

To verify our hunch, we will try to compute for the sum of the exterior angles of a pentagon.

Let S_p be the sum of the exterior angles of the pentagon in Figure 2. Then

S_p =(180 - c_1)+ (180 - c_2) + (180 - c_3) +(180 - c_4) +(180 - c_5). Simplifying, we have S_p = 900 - (c_1 + c_2 + c_3 + c_4 + c_5). But according to the angle sum theorem for polygons, c_1 + c_2 + c_3 + c_4 + c_5 = 540. Therefore,900 - (c_1 + c_2 + c_3 + c_4 + c_5) = 900 - 540 = 360 degrees.

We have three polygons – triangle, quadrilateral, pentagon – whose angle sums of exterior angles are always 360 degrees. Now, is this true for all polygons?  Try to compute polygons up to 10 sides and see if the sum is 360 degrees.

Delving Deeper

We know that in a polygon, the number of exterior angles is equal to the number of interior angles.  Furthermore, we know that the angle sum of an interior angle and the exterior angle adjacent to each is always latex 180 degrees. If we have a polygon with 5 sides, then

interior angle sum + exterior angle sum = 180(5)

In general, this means that in a polygon with n sides

interior angle sum* + exterior angle sum = 180n

But the interior angle sum = 180(n – 2). So, substituting in the preceding equation, we have

180(n – 2) + exterior angle sum = 180n

which means that the exterior angle sum = 180n – 180(n – 2)  = 360 degrees. More formal proofs using these arguments are shown below.

Theorem: The sum of the measure of the exterior angles of a polygon with n sides is 360 degrees.

Proof 1:

Let a_1, a_2, \cdots a_n be measures of the interior angles of a polygon with n sides. Letb_1, b_2, \cdots b_n be measures of the exterior angles of the same polygon where all angle names with the same subscripts are adjacent angles from a_1 andb_1 all the way up through a_n and b_n .  We know that adjacent interior and exterior angles are supplementary angles, so this implies that their measures add up to 180 degrees. Hence,

(a1 + b1) + (a2 + b2) + … + (an + bn) = 180 + 180 + … +180 (n of them) = 180n

Regrouping the terms of the preceding equation, we have

(a1 + a2 + … + an) + (b1 + b2 + … + bn) = 180n

But the sum of the interior angles is a1 + a2 + … + an = 180(n – 2)

So,

180(n – 2) + (b1 + b2 + … + bn) = 180n

b1 + b2 + … + bn = 180n – 180(n – 2) = 360

Therefore, the sum of the exterior angles of any polygon is equal to 360 degrees.

Proof 2:

Let a1, a2, …, an be measures of the interior angles of the polygon with n sides. Since each adjacent interior and exterior angle is a linear pair, it follows that the measure of the exterior angles adjacent to them respectively are  180 – a1, 180 – a2, …, 180 – an.

If we let S, be the sum of the measure of the exterior angles, we have

S = (180 – a1) + (180 – a2) + (180 – a3) + … + (180 – an)

= (180 + 180 + 180 + … +180 (n of them)) – a – a2 – a3– … – an

S = 180n – (a1 + a2 + a3 + … + an)

But a1 + a2 + a3 + … + an is the sum of the measures of the interior angles of a polygon  with n sides which equals

180(n – 2), so, S = 180n – 180(n – 2) = 360, which is want we want to show.

Therefore, the sum of the exterior angles of any polygon is equal to 360 degrees.

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