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.

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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. Let$b_1, b_2, \cdots b_n$ be measures of the exterior angles of the same polygon where all angle names with the same subscripts are adjacent angles from $a_1$ and$b_1$ 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.

An Intuitive Introduction to Limits

Limits is one of the most fundamental concepts of calculus. The foundation of calculus was not entirely solid during the time of Leibniz and Newton, but later developments on the concept, particularly the $\epsilon-\delta$ definition by Cauchy, Weierstrass and other mathematicians established its firm foundation. In the discussion below, I shall introduce the concept of limits intuitively as it appears in common problems. For a more rigorous discussion, you can read the post article titled “An extensive explanation about the $\epsilon-\delta$ definition of limits”.

Circumference and Limits

If we are going to approximate the circumference of a circle using the perimeter of an inscribed polygon, even without computation, we can observe that as the number of sides of the polygon increases, the better the approximation. In fact, we can make the perimeter of the polygon as close as we please to the circumference of the circle by choosing a sufficiently large number of sides.  Notice that no matter how large the number of sides our polygon has, its perimeter will never exceed or equal the circumference of the circle.

Figure 1 – As the number of side of the polygons increases, its perimeter gets closer to the circumference of the circle.

In a more technical term, we say that the limit of the perimeter of the inscribed polygon as the number of its sides increases without bound (or as the number of sides of the inscribed polygon approaches infinity) is equal to the circumference of the circle.  In symbol, if we let $n$ be the number of sides of the inscribed polygon, $P_n$ be the perimeter of a polygon with $n$ sides, and $C$ be the circumference of the circle, we can say that the limit of $P_n$ as $n \to \infty$ is equal to $C$. Compactly, we can write $\lim_{n \to \infty} P_n = C$.

Functions and Limits

Consider the function $f(x) = \frac{1}{x}$ where $x$ is a natural number. Calculating the values of the function using the first 20 natural numbers and plotting the points in the $xy$-plane, we arrive at the table and the graph in Figure 2.

Figure 2 – As x increases, f(x) gets closer and closer to 0.

First, we see that as the value of $x$ increases, the value of $f(x)$ decreases and approaches $0$. Furthermore, we can make the value of $f(x)$ as close to $0$ as we please by choosing a sufficiently large $x$. We also notice that no matter how large the value of $x$ is, the value of $f(x)$ will never reach $0$.

Hence, we say that the limit of $f(x) = \frac{1}{x}$ as the value of $x$ increases without bound is equal to $0$, or equivalently the limit of $f(x) = \frac{1}{x}$ as $x$ approaches infinity is equal to $0$. In symbol, we write the limit of $f(x) \to \infty$ as $x \to 0$ or more compactly the $\lim_{x \to \infty} \frac{1}{x} = 0$.

Tangent line and Limits

Recall that the slope of a line is its “rise” over its “run”. The formula of slope $m$ of a line is $m = \displaystyle\frac{y_2 - y_1}{x_2 - x_1}$, given two points with coordinates $(x_1,y_1)$ and $(x_2,y_2)$.  One of the famous ancient problems in mathematics was the tangent problem, which is getting the slope of a line tangent to a function at a point.  In the Figure 3, line $n$ is tangent to the function $f$ at point $P$.

Figure 3 – Line n is tangent to the function f at point P.

If we are going to compute for the slope of the line tangent line, we have a big problem because we only have one point, and the slope formula requires two points.  To deal with this problem, we select a point $Q$ on the graph of $f$, draw the secant line $PQ$ and move $Q$ along the graph of $f$ towards $P$. Notice that as $Q$ approaches $P$ (shown as $Q'$ and $Q''$), the secant line gets closer and closer to the tangent line. This is the same as saying that the slope the secant line is getting closer and closer to the slope of the tangent line. Similarly, we can say that as the distance between the x-coordinates of $P$ and $Q$ is getting closer and closer to $0$, the slope of the secant line is getting closer and closer to the slope of the tangent line.

Figure 4 – As point Q approaches P, the slope of the secant line is getting closer and closer to the slope of the tangent line.

If we let $h$ be the distance between the x-coordinates of $P$ and $Q$, $m_s$ be the slope of the secant line $PQ$ and $m_t$ be the slope of the tangent line, we can say that the limit of the slope of secant line as $h$ approaches $0$ is equal to the slope of the tangent line. Concisely, we can write $\lim_{h \to 0}m_s = m_t$.

Area and Limits

Another ancient problem is about finding the area under a curve as shown in the leftmost graph in Figure 5. During the ancient time, finding the area of a curved plane was impossible.

Figure 5 – As the number of rectangles increases, the sum of the area of the rectangles is getting closer and closer to the area of the bounded plane under the curve.

We can approximate the area above in the first graph in Figure 5 by constructing rectangles under the curve such that one of the corners of the rectangle touches the graph as shown in the second and third graph in Figure 5. We can see that as we increase the number of rectangles, the better is our approximation of the area under the curve. We can also see that no matter how large the number of rectangles is, the sum its areas will never exceed (or equal) the area of the plane under the curve. Hence, we say that as the number of rectangles increases without bound, the sum of the areas of the rectangles is equal to the area under the curve; or the limit of the sum of the areas of the rectangles as the number of rectangles approaches infinity is equal to the area of the plane under the curve.

If we let $A$ be the area under the curve, $S_n$ be the sum of the areas of $n$ rectangles, then we can say that the limit of $S_n$ as n approaches infinity is equal to $A$. Concisely, we can write $\lim_{n \to\infty} S_n = A$.

Numbers and Limits

We end with a more familiar example usually found in books. What if we want to find the limit of $2x + 1$ as $x$ approaches $3$?

To answer the question, we must find the value $2x + 1$ where $x$ is very close to $3$. Those values would be numbers that are very close to $3$ – some slightly greater than $3$ and some slightly less than $3$. Place the  values in a table we have

Figure 6 – As x approaches 3, 2x + 1 approaches 7.

From the table, we can clearly see that as the value of $x$ approaches $3$, the value of $2x + 1$ approaches $7$.  Concisely, we can write the $\lim_{x \to 3} 2x + 1 =7$.

Mr. Jayson Dyer, author of The Number Warrior has another excellent explanation on the concept of limits in his blog Five intuitive approaches to teaching the infinitely small.

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The area under the curve problem and the tangent problem are the ancient problems which gave birth to calculus. Calculus was independently invented by Gottfried Leibniz and Isaac Newton in the 17th century.