Why does x^0 = 1 and x^(-n) = 1/x^n? (Index)

I’m using the Twelve Days of Christmas (and perhaps a few extra days besides) to do something that I should have done a long time ago: collect past series of posts into a single, easy-to-reference post. The following posts formed my series why the rules for $x^0$ and $x^{-n}$ work the way they do.

Part 1: Multiplication and division.

Part 2: The Laws of Exponents.

Lessons from teaching gifted elementary school students (Part 3b)

Every so often, I’ll informally teach a class of gifted elementary-school students. I greatly enjoy interacting with them, and I especially enjoy the questions they pose. Often these children pose questions that no one else will think about, and answering these questions requires a surprisingly depth of mathematical knowledge.

Here’s a question I once received:

Suppose

$A \times A = B$

$B \times B \times B = C$

$C \times C \times C \times C= D$

If the pattern goes on, and if $A = 2$, what is $Z$?

In yesterday’s post, we found that the answer was

$Z =2^{26!} = 10^{26! \log_{10} 2} \approx 10^{1.214 \times 10^{26}}$,

a number with approximately $1.214 \times 10^{26}$ digits.

How can we express this number in scientific notation? We need to actually compute the integer and decimal parts of $26! \log_{10} 2$, and most calculators are not capable of making this computation.

Fortunately, Mathematica is able to do this. We find that

$Z \approx 10^{121,402,826,794,262,735,225,162,069.4418253767}$

$\approx 10^{0.4418253767} \times 10^{121,402,826,794,262,735,225,162,069}$

$\approx 2.765829324 \times 10^{121,402,826,794,262,735,225,162,069}$

Here’s the Mathematica syntax to justify this calculation. In Mathematica, $\hbox{Log}$ means natural logarithm:

Again, just how big is this number? As discussed yesterday, it would take about 12.14 quadrillion sheets of paper to print out all of the digits of this number, assuming that $Z$ was printed in a microscopic font that uses 100,000 characters per line and 100,000 lines per page. Since 250 sheets of paper is about an inch thick, the volume of the 12.14 quadrillion sheets of paper would be

$1.214 \times 10^{16} \times 8.5 \times 11 \times \displaystyle \frac{1}{250} \hbox{in}^3 \approx 1.129 \times 10^{17} \hbox{in}^3$

By comparison, assuming that the Earth is a sphere with radius 4000 miles, the surface area of the world is

$4 \pi (4000 \times 5280 \times 12) \hbox{in}^2 \approx 8.072 \times 10^{17} \hbox{in}^2$.

Dividing, all of this paper would cover the entire world with a layer of paper about $0.14$ inches thick, or about 35 sheets deep. In other words, the whole planet would look something like the top of my desk.

What if we didn’t want to print out the answer but just store the answer in a computer’s memory? When written in binary, the number $2^{26!}$ requires…

$26!$ bits of memory, or…

about $4.03 \times 10^{26}$ bits of memory, or…

about $latex 5.04 \times 10^{25} bytes of memory, or … about $5.04 \times 10^{13}$ terabytes of memory, or… about 50.4 trillion terabytes of memory. Suppose that this information is stored on 3-terabyte external hard drives, so that about $50.4/3 = 16.8$ trillion of them are required. The factory specs say that each hard drive measures $129 \hbox{mm} \times 42 \hbox{mm} \times 167 \hbox{mm}$. So the total volume of the hard drives would be $1.52 \times 10^{19} \hbox{mm}^3$, or $15.2 \hbox{km}^3$. By way of comparison, the most voluminous building in the world, the Boeing Everett Factory (used for making airplanes), has a volume of only $0.0133 \hbox{km}^3$. So it would take about 1136 of these buildings to hold all of the necessary hard drives. The cost of all of these hard drives, at$100 each, would be about $1.680 quadrillion. So it’d be considerably cheaper to print this out on paper, which would be about one-seventh the price at$242 trillion.

Of course, a lot of this storage space would be quite repetitive since $2^{26!}$, in binary, would be a one followed by $26!$ zeroes.

Lessons from teaching gifted elementary school students (Part 3a)

Every so often, I’ll informally teach a class of gifted elementary-school students. I greatly enjoy interacting with them, and I especially enjoy the questions they pose. Often these children pose questions that no one else will think about, and answering these questions requires a surprisingly depth of mathematical knowledge.

Here’s a question I once received:

Suppose

$A \times A = B$

$B \times B \times B = C$

$C \times C \times C \times C= D$

If the pattern goes on, and if $A = 2$, what is $Z$?

I leave a thought bubble in case you’d like to think this. (This is significantly more complicated to do mentally than the question posed in yesterday’s post.) One way of answering this question appears after the bubble.

Let’s calculate the first few terms to try to find a pattern:

$B = 2 \times 2 = 2^2$

$C = 2^2 \times 2^2 \times 2^2 = 2^6$

$D = 2^6 \times 2^6 \times 2^6 \times 2^6 = 2^{24}$

etc.

Written another way,

$A = 2^1 = 2^{1!}$

$B = 2^{2!}$

$C = 2^{3!}$

$D = 2^{4!}$

Naturally, elementary school students have no prior knowledge of the factorial function. That said, there’s absolutely no reason why a gifted elementary school student can’t know about the factorial function, as it only consists of repeated multiplication.

Continuing the pattern, we see that $Z = 2^{26!}$. Using a calculator, we find $Z \approx 2^{4.032014611 \times 10^{26}}$.

If you try plugging that number into your calculator, you’ll probably get an error. Fortunately, we can use logarithms to approximate the answer. Since $2 = 10^{\log_{10} 2}$, we have

$Z = \left( 10^{\log_{10} 2} \right)^{4.032014611 \times 10^{26}} = 10^{4.032014611 \times 10^{26} \log_{10} 2}$

Plugging into a calculator, we find that

$Z \approx 10^{1.214028268 \times 10^{26}} = 10^{121.4028628 \times 10^{24}}$

We conclude that the answer has more than 121 septillion digits.

How big is this number? if $Z$ were printed using a microscopic font that placed 100,000 digits on a single line and 100,000 lines on a page, it would take 12.14 quadrillion pieces of paper to write down the answer (6.07 quadrillion if printed double-sided). If a case with 2500 sheets of paper costs $100, the cost of the paper would be$484 trillion ($242 trillion if double-sided), dwarfing the size of the US national debt (at least for now). Indeed, the United States government takes in about$3 trillion in revenue per year. At that rate, it would take the country about 160 years to raise enough money to pay for the paper (80 years if double-sided).

And that doesn’t even count the cost of the ink or the printers that would be worn out by printing the answer!

Lessons from teaching gifted elementary school students (Part 2)

Every so often, I’ll informally teach a class of gifted elementary-school students. I greatly enjoy interacting with them, and I especially enjoy the questions they pose. Often these children pose questions that no one else will think about, and answering these questions requires a surprisingly depth of mathematical knowledge.

Here’s a question I once received:

Suppose

$A \times A = B$

$B \times B = C$

$C \times C = D$

If the pattern goes on, and if $A = 2$, what is $Z$?

I leave a thought bubble in case you’d like to think this. One way of answering this question appears after the bubble.

Let’s calculate the first few terms to try to find a pattern:

$B = 2 \times 2 = 2^2$

$C = 2^2 \times 2^2 = 2^4$

$D = 2^4 \times 2^4 = 2^8$

etc.

Written another way,

$A = 2^1 = 2^{2^0}$

$B = 2^{2^1}$

$C = 2^{2^2}$

$D = 2^{2^3}$

Continuing the pattern, we see that $Z = 2^{2^{25}}$, or $Z = 2^{33,554,432}$.

If you try plugging that number into your calculator, you’ll probably get an error. Fortuniately, we can use logarithms to approximate the answer. Since $2 = 10^{\log_{10} 2}$, we have

$Z = \left( 10^{\log_{10} 2} \right)^{33,554,432} = 10^{33,554,432 \log_{10} 2}$

Plugging into a calculator, we find that

$Z \approx 10^{10,100,890.5195}$

$\approx 10^{0.5195} 10^{10,100,890}$

$\approx 3.307 \times 10^{10,100,890}$

When this actually happened to me, it took me about 10 seconds to answer — without a calculator — “I’m not sure, but I do know that the answer has about 10 million digits.” Naturally, my class was amazed. How did I do this so quickly? I saw that the answer was going to be $Z = 2^{2^{25}}$, so I used the approximation $2^{10} \approx 1000$ to estimate

$2^{25} = 2^5 \times 2^{10} \times 2^{10} \approx 32 \times 1000 \times 1000 = 32,000,000$

Next, I had memorized the fact that that $\log_{10} 2 \approx 0.301 \approx 1/3$. So I multiplied $32,000,000$ by $1/3$ to get approximately 10 million. As it turned out, this approximation was a lot more accurate than I had any right to expect.

Engaging students: Powers and exponents

In my capstone class for future secondary math teachers, I ask my students to come up with ideas for engaging their students with different topics in the secondary mathematics curriculum. In other words, the point of the assignment was not to devise a full-blown lesson plan on this topic. Instead, I asked my students to think about three different ways of getting their students interested in the topic in the first place.

I plan to share some of the best of these ideas on this blog (after asking my students’ permission, of course).

This student submission comes from my former student Kayla (Koenig) Lambert. Her topic, from Pre-Algebra: powers and exponents.

A) Applications: What interesting word problems using this topic can your students do now?

I chose the problem below from http://www.purplemath.com because I think that solving a problem that deals with disease would be interesting to my students. People have to deal with sickness and disease everyday and I think that solving a real world problem would entice the students into wanting to learn more.

A biologist is researching a newly-discovered species of bacteria. At time t = 0 hours, he puts one hundred bacteria into what he has determined to be a favorable growth medium. Six hours later, he measures 450 bacteria. Assuming exponential growth, what is the growth constant “k” for the bacteria? (Round k to two decimal places.)

For this exercise, the units on time t will be hours, because the growth is being measured in terms of hours. The beginning amount P is the amount at time t = 0, so, for this problem, P = 100. The ending amount is A = 450 at t = 6. The only variable I don’t have a value for is the growth constant k, which also happens to be what I’m looking for. So I’ll plug in all the known values, and then solve for the growth constant:

$A = Pe^{kt}$

$450 = 100 e^{6k}$

$4.5 = e^{6k}$

$\ln(4.5) = 6k$

$k = \displaystyle \frac{\ln(4.5)}{6} = 0.250679566129\dots$

The growth constant is 0.25/hour.

I think this kind of problem would be beneficial to students because it would help them understand how bacteria grows and how easily they can get catch something and get sick.

C) Culture: How has this topic appeared in pop culture?

Exponents and powers are everywhere around us without the students knowledge. Many movies and video games have ideas related to powers and exponents. Take, for example, the movie Contagion that was released in September 2011. This movie is about “the threat posed by a deadly disease and an international team of doctors contracted by the CDC to deal with the outbreak” (http://www.imdb.com/title/tt1598778). In this movie, there is a scene where the doctors are using mathematical equations with exponents to find out how fast the disease spreads and how much time they have left to save the majority of the population. There are many movies like this that involve powers and exponents, Contagion is just one example. There are also popular video games that deal with the spread of disease. For example, in the video game Call Of Duty: World At War the player is a soldier in WWII and his mission is to kill zombies, and zombie populations grow exponentially. Now, my brother plays this game and I know for a fact that he doesn’t think about the mathematics behind it, but I think talking about pop culture while teaching would really bring some excitement to the classroom and get the students thinking.

D) History: Who were some of the people who contributed to the discovery of this topic?

Exponents and powers have been among humans since the time of the Babylonians in Egypt. “Babylonians already knew the solution to quadratic equations and equations of the second degree with two unknowns and could also handle equations to the third and fourth degree” (Mathematics History). The Egyptians also had a good idea about powers and exponents around 3400 BC. They used their “hieroglyphic numeral system” which was based on the scale of 10. When using their system, the Egyptians expressed any number using their symbols, with each symbol being “repeated the required number of times” (Mathematics History). However, the first actual recorded use of powers and exponents was in a book called “Artihmetica Integra” written by English author and Mathematician Michael Stifel in 1544 (History of Exponents). In the 14th century Nicole Oresme used “numbers to indicate powering”(Jeff Miller Pages). Also, James Hume used Roman Numerals as exponents in the book L’Algebre de Viete d’vne Methode Novelle in 1636. Exponents were used in modern notation be Rene Descartes in 1637. Also, negative integers as exponents were “first used in modern notation” by Issac Newton in 1676 (Jeff Miller Pages).

Works Cited

Ayers, Chuck. “The History of Exponents | eHow.com.” eHow | How to Videos, Articles & More – Discover the expert in you. | eHow.com. N.p., n.d. Web. 25 Jan. 2012. http://www.ehow.com/about_5134780_history-exponents.html.

“Contagion (2011) – IMDb.” The Internet Movie Database (IMDb). N.p., n.d. Web. 25 Jan. 2012. http://www.imdb.com/title/tt1598778/.

“Exponential Word Problems.” Purplemath. N.p., n.d. Web. 25 Jan. 2012. http://www.purplemath.com/modules/expoprob2.htm.

“Mathematics History.” ThinkQuest : Library. N.p., n.d. Web. 25 Jan. 2012. http://library.thinkquest.org/22584/.

juxtaposition.. “Earliest Uses of Symbols of Operation.” Jeff Miller Pages. N.p., n.d. Web. 25 Jan. 2012. http://jeff560.tripod.com/operation.html.

The Scale of the Universe

My former student Matt Wolodzko tipped me off about this excellent website that shows the scale of the universe, from the very large to the very small: http://htwins.net/scale2/. I recommend it highly for engaging students with the concept of scientific notation.

While I’m on the topic, here are two videos that describe the scale of the universe. The first was a childhood favorite of mine — I vividly remember watching it at the Smithsonian National Air and Space Museum when I was a boy — while the second is more modern.

Engaging students: Powers and exponents

In my capstone class for future secondary math teachers, I ask my students to come up with ideas for engaging their students with different topics in the secondary mathematics curriculum. In other words, the point of the assignment was not to devise a full-blown lesson plan on this topic. Instead, I asked my students to think about three different ways of getting their students interested in the topic in the first place.

I plan to share some of the best of these ideas on this blog (after asking my students’ permission, of course).

This student submission comes from my former student Kelsie Teague. Her topic, from Pre-Algebra: powers and exponents.

What interesting word problem using this topic can your students do now?

For the topic of powers and exponents I want to bring in the idea of money, and doubling a salary. The word problem I would give them to start with and to get them thinking would be this:

Two companies were offering you a job. Company A is offering you a salary of $1,000 a day for 30 days and Company B is offering you a salary of$2 the first day and it doubles each day after that for 30 days. Which job is the better offer?

Since this is just my engage problem I’m not expecting them to be able to tell me that the answer is Company B because the answer is $2^{31}-2$, but I am hoping they can get to the point of at least knowing that Company B will be paying the most. I want to get his or her attention and everyone loves money.

How can this topic be used in your students’ future courses in Mathematics or Science?

I believe powers and exponents are important knowledge because students will be using them for the rest of their math career. This comes up when teaching functions, learning the graphs of functions, trig, pre-calculus, Calculus and etc. Powers and exponents are used extensively in algebra and it is important that students have a strong understand of how and why they work before continuing onto those higher classes. For example, when you have $x^3$, and talking about graphing a cubic function or $x^2$ and how it makes a parabola, and also when talking about factoring. If you have $(x-2)^2 = (x-2)(x-2) =(x^2 -4x +4)$, students need to understand what it means to $^2$ something.  Once students get to calculus that also use exponents and powers when doing derivatives and integrals. This isn’t a topic that is only based in math, it is also something used in science, engineering, and physics. Once students start college, no matter their major they will be taking at least one class that require some sort of knowledge with exponents and powers.

What interesting things can you say about the people who contributed to the discovery and/or the development of this topic?

The earliest exponents came from the Babylonians. The number system was extremely different from modern mathematics. The earliest known mention of Babylon was mentioned on a tablet found around 23rd century BC. Even then they were messing with the concept of exponents.

I would show my students this picture and explain to them what the symbols mean and ask them if they feel any better about doing math in modern times rather than working with these symbols to add, subtract, divide, exponents, power and doing equations. This also shows that this concept has been around for many thousands of years and something that is obviously very important if we still use it in modern math. I might also bring up the website least below that talks about modern exponents and works backwards and talks about where they came from to give the students more depth in this knowledge.

Why 0^0 is undefined

Here’s an explanation for why $0^0$ is undefined that should be within the grasp of pre-algebra students:

Part 1.

• What is $0^3$? Of course, it’s $0$.
• What is $0^2$? Again, $0$.
• What is $0^1$? Again, $0$.
• What is $0^{1/2}$, or $\sqrt{0}$? Again, $0$.
• What is $0^{1/3}$, or $\sqrt[3]{0}$? In other words, what number, when cubed, is $0$? Again, $0$.
• What is $0^{1/10}$, or $\sqrt[10]{0}$? In other words, what number, when raised to the 10th power, is $0$. Again, $0$.

So as the exponent gets closer to $0$, the answer remains $0$. So, from this perspective, it looks like $0^0$ ought to be equal to $0$.

Part 2.

• What is $3^0$. Of course, it’s $1$.
• What is $2^0$. Again, $1$.
• What is $1^0$. Again, $1$.
• What is $\left( \displaystyle \frac{1}{2} \right)^0$? Again, $1$
• What is $\left( \displaystyle \frac{1}{3} \right)^0$. Again, $1$
• What is $\left( \displaystyle \frac{1}{10} \right)^0$? Again, $1$

So as the base gets closer to $0$, the answer remains $1$. So, from this perspective, it looks like $0^0$ ought to be equal to $1$.

In conclusion: looking at it one way, $0^0$ should be defined to be $0$. From another perspective, $0^0$ should be defined to be $1$.

Of course, we can’t define a number to be two different things! So we’ll just say that $0^0$ is undefined — just like dividing by $0$ is undefined — rather than pretend that $0^0$ switches between two different values.

Here’s a more technical explanation about why $0^0$ is an indeterminate form, using calculus.

Part 1. As before,

$\displaystyle \lim_{x \to 0^+} 0^x = \lim_{x \to 0^+} 0 = 0$.

The first equality is true because, inside of the limit, $x$ is permitted to get close to $0$ but cannot actually equal $0$, and there’s no ambiguity about $0^x = 0$ if $x >0$. (Naturally, $0^x$ is undefined if $x < 0$.)

The second equality is true because the limit of a constant is the constant.

Part 2. As before,

$\displaystyle \lim_{x \to 0} x^0 = \lim_{x \to 0} 1 = 1$.

Once again, the first equality is true because, inside of the limit, $x$ is permitted to get close to $0$ but cannot actually equal $0$, and there’s no ambiguity about $x^0 = 1$ if $x \ne 0$.

As before, the answers from Parts 1 and 2 are different. But wait, there’s more…

Part 3. Here’s another way that $0^0$ can be considered, just to give us a headache. Let’s evaluate

$\displaystyle \lim_{x \to 0^+} x^{1/\ln x}$

Clearly, the base tends to $0$ as $x \to 0$. Also, $\ln x \to \infty$ as $x \to 0^+$, so that $\displaystyle \frac{1}{\ln x} \to 0$ as $x \to 0^+$. In other words, this limit has the indeterminate form $0^0$.

To evaluate this limit, let’s take a logarithm under the limit:

$\displaystyle \lim_{x \to 0^+} \ln x^{1/\ln x} = \displaystyle \lim_{x \to 0^+} \frac{1}{\ln x} \cdot \ln x$

$\displaystyle \lim_{x \to 0^+} \ln x^{1/\ln x} = \displaystyle \lim_{x \to 0^+} 1$

$\displaystyle \lim_{x \to 0^+} \ln x^{1/\ln x} = 1$

Therefore, without the extra logarithm,

$\displaystyle \lim_{x \to 0^+} x^{1/\ln x} = e^1 = e$

Part 4. It gets even better. Let $k$ be any positive real number. By the same logic as above,

$\displaystyle \lim_{x \to 0^+} x^{\ln k/\ln x} = e^{\ln k} = k$

So, for any $k \ge 0$, we can find a function $f(x)$ of the indeterminate form $0^0$ so that $\displaystyle f(x) = k$.

In other words, we could justify defining $0^0$ to be any nonnegative number. Clearly, it’s better instead to simply say that $0^0$ is undefined.

P.S. I don’t know if it’s possible to have an indeterminate form of $0^0$ where the answer is either negative or infinite. I tend to doubt it, but I’m not sure.

Engaging students: Laws of Exponents

In my capstone class for future secondary math teachers, I ask my students to come up with ideas for engaging their students with different topics in the secondary mathematics curriculum. In other words, the point of the assignment was not to devise a full-blown lesson plan on this topic. Instead, I asked my students to think about three different ways of getting their students interested in the topic in the first place.

I plan to share some of the best of these ideas on this blog (after asking my students’ permission, of course).

This student submission comes from my former student Claire McMahon. Her topic, from Pre-Algebra: the Laws of Exponents (with integer exponents)

These laws are essential not only in math classes but in science classes as well.  The laws of exponents are essential when learning scientific notation and important facts like Avogadro’s constant.  This is just one of the important facts that students will encounter as they enter the world of exponents.  There is a really awesome lesson plan devoted to finding this enormous number at the following website here.  I implemented this in a classroom that called for an interdisciplinary lesson plan and had great success with it.

There are some really cool videos that deal with the laws of exponents and I love to incorporate music wherever I can in my math classes.  This is one of my favorite videos that I came across as I was trying to reach for things to help engage my students in the middle of math class.  Watch this YouTube video and see if you think you would enjoy showing this to your class.  Even better for your class would be to create a video like this in a project.

I love to also find some great online activities that I can give to my students that are not too intensive but give them some great confidence in understanding.  There are a few different websites that I have found to be very useful and somewhat cute!!  I do want my students to have a basic understanding on how the laws of exponents work but we all get better at math by DOING math.  This website gives you some great practice on laws of exponents with the same base and has a cute little monster to cheer you on along the activity!  I am also a big fan of foldables and have found a great one on the internet to utilize for your class.  It’s always fun to create something in math class that you would normally do in kindergarten!!  Cutting and folding and making something your own is an awesome way to drive a topic and even to make a homework assignment fun.  A foldable for the laws of exponents can be found here.

Why does x^0 = 1 and x^(-n) = 1/x^n? (Part 2)

I distinctly remember when, in my second year as a college professor, a really good college student — with an SAT Math score over 650 — asked me why $x^0 = 1$ and $x^{-n} = \displaystyle \frac{1}{x^n}$. Of course, he knew that these rules were true and he could apply them in complex problems, but he didn’t know why they were true. And he wanted to have this deeper knowledge of mathematics beyond the ability to solve routine algebra problems.

He also related that he had asked his math teachers in high school why these rules worked, but he never got a satisfactory response. So he asked his college professor.

Looking back on it, I see that this was one of the incidents that sparked my interest in teacher education. As always, I never hold a grudge against a student for asking a question. Indeed, I respected my student for posing a really good question, and I was upset for him that he had not received a satisfactory answer to his question.

This is the second of two posts where I give two answers to this question from two different points of view.

Answer #2. This explanation relies on one of the laws of exponents:

$x^n \cdot x^m = x^{n+m}$

For positive integers $n$ and $m$, this can be proven by repeated multiplication:

$x^n x^m = (x \cdot x \dots \cdot x) \cdot (x \cdot x \dots \cdot x)$       repeated $n$ times and $m$ times

$x^n x^m = x \cdot x \cdot \dots \cdot x \cdot x \cdot \dots \cdot x$       repeated $n+m$ times

$x^n \cdot x^m = x^{n+m}$

Ideally, $x^0$ and $x^{-n}$ should be defined so that this rule still holds even if one (or both) of $n$ and $m$ is either zero or a negative integer. In particular, we should define $x^0$ so that the following rule holds:

$x^n \cdot x^0 = x^{n+0}$

$x^n \cdot x^0 = x^n$

In other words, the product of something with $x^0$ should be the original something. Clearly, the only way to make this work is if we define $x^0 = 1$.

In the same way, we should define $x^{-n}$ so that the following rule holds:

$x^n \cdot x^{-n} = x^{n + (-n)}$

$x^n \cdot x^{-n} = x^0$

$x^n \cdot x^{-n} = 1$

Dividing, we see that

$x^{-n} = \displaystyle \frac{1}{x^n}$