I am an electrical engineer by training who is currently working as a hardware development director for a telecommunications company. Over the years, I have become more and more impressed with the ability of relatively small amounts of mathematics to help shed light on significant technical problems.

There are many types of mathematical analyses. They range from very detailed modeling exercises that are impressive, but are really for specialists, to "Fermi problems" that are exercises in gross approximation. This blog will steer a middle course and will look at the simple mathematics that crops up in the daily life of a working engineer.


69 Responses to About

  1. newton says:

    cool stufff! keep posting!

  2. Alias says:

    Where was your photo taken? It reminds me of the Muskokas in Northern Ontario. You keep writing and I'll keep reading.

    • mathscinotes says:

      You are pretty close! The photo is actually from my cabin in far northern Minnesota, which is not far from the border with Ontario. This is what I see in mornings when when I look out from my door. Of course, it is all frozen right now. But this photo reminds me that spring will come and I will soon be there again.

  3. XXXXXX says:

    Hello -
    Congratulations on this great blog - I especially love your tagline "I stumbled upon some math today". As part of my work in XXXXX, I am also helping to manage the XXXXX blog and was wondering, if you would be available for an interview? I would like to introduce your blog to our readers and I am sure they'd love to learn more about you. If interested, please email me at XXXXX.


  4. Barry says:

    "You are also handed two boxes: one box contains an uncharged 1 F capacitor and the other contains 1 million 1 µF capacitors. You can connect the capacitors from one box one at a time across the charged capacitor. Your job is to determine which box will allow you to discharge the voltage on the charged capacitor the most?"

    What charged capacitor? Did I miss something?

  5. Hello, Great blog about a great subject; mathematical modeling. I was attracted to your blog because of the rafters problem and the article on math in baking. I have also looked at baking, architecture, construction, knots, and baking mathematically.
    The thing I struggled with the most was what to call the concept of math in everyday life; Math Encounters is really good.

    I wish I had time to develop all my ideas, but I have chosen to focus on baking. You can check out my website if you want to see more.

    • mathscinotes says:

      I like your blog. Your focus on how to clearly present technical material is near and dear to my heart. I have really struggled with the poor quality of technical presentations in general. As far as baking, I am a neophyte baker and I have a lot to learn. When I was a boy, I loved to watch my mother's dough turn into bread in the oven. Baking is still magic to me.

      The only problem with blogging about stuff is time. I really do have a large amount of material to write about, but there just isn't time to develop it all.

  6. Hello,
    I discovered your blog a few weeks ago, as I was searching for some material on a paper that I am doing in school. The information on your ballistics, ogives and bullet shapes really helped me out and I will definitely urge my colleagues to refer to it. Firstly, thank you for that! 🙂 I would like to ask you how to arrive at equation 3 mentioned on that page. I suppose that it must be rather obvious to the trained eye, but I fail to see it. I understand that the subtraction of the (rho.sin(alpha)) was done to effectively 'pull down' the ogive shape to give the volume of revolution (around x axis) formula. However, how do you arrive at the first part of the equation?

    Well, questions aside, I must say that these past few weeks, I have really taken a liking to your blog. I love the way you quantify everyday problems, or as you call them, 'Fermi Problems'. I am a big fan of 'Gedanken experiments', especially in Physics and this is thoroughly exciting!

    Keep going!

    Thank you,

    • mathscinotes says:

      Hi Abhranil,

      Thanks you for the nice comments. I am sorry that I was not very clear on the derivation of Equation 3 (now Equation 5 after fixing an equation numbering error). I am frequently hurrying to get these blog posts out because I have more math to get done!

      The equation is a bit unusual because of how I setup my coordinate system. I made the x value of point A the Origin in Figure 7. This means that point A has an x value of 0 and x increases to the right. This makes my integration simpler. The equation for a circle is given normally given by the following equation.

      y(x)=sqrt{{{rho }^{2}}-{{x}^{2}}}

      With the origin at the x value of point A, you can think of the circle as having been translated to the right of the new origin by rho cdot cos left( alpha  right). This means that the equation of the circle in my new coordinate system left( {x}',{y}' right) is {y}'(x)=sqrt{{{rho }^{2}}-{{left( x-rho cdot cos left( alpha  right) right)}^{2}}}. In figure 7, I actually define y as the height of the ogive, so I need to subtract off rho cdot sin left( alpha  right). This gives me my final result y(x)=sqrt{{{rho }^{2}}-{{left( x-rho cdot cos left( alpha  right) right)}^{2}}}-rho cdot sin left( alpha  right).

      I hope this helps. If you have any further questions, ask again.


      • Hi,
        Thank you for the explanation! Yes, this definitely helps. So, as I understand it, you formulated an equation for the ogive, so in other words, how the radius, x, changes with a change in ogive height, y. Right?


        • mathscinotes says:

          The radius does not change, but x and y do. Since the ogive is a segment of a circle, I use the standard equation for a circle {{x}^{2}}+{{y}^{2}}={{rho }^{2}} and I limit the range of x to that required for the ogive. I know it seems confusing, but focus on how the equation for a circle works. Once you understand that, look at the range of the x that we are interested in and how you would generate that range of x values. Given the equation of a circle and the range of x values, you can compute the range of y values. These are the values I use in my integral.


  7. Yes, I understand now. Thank you again. 🙂

    Keep it coming!

  8. Don Pridgen says:

    There were several comments on the rafter math post recently but the reply link on my end seems to be broken. Anyway, see if this helps clear things up;

    • mathscinotes says:

      Hi Don,

      Thanks for the note. I pulled the post from the web while I am working with a couple of folks on cleaning it up. I hope to put it back up in a fews days.


  9. Don Pridgen says:

    Great, I appreciate your digging deeper. Shoot me your email if you want scans of FBD's and calcs from a couple of other folks... being a carpenter, it takes at least 2 engineers to pull me out of the ditch. Also click on my website and go down to the rafter thrust and raised tie thrust calcs. The link to my calc on your original post is broken but feel free to link to this one, my site got hacked.

    • mathscinotes says:

      Hi Don,

      I had a number of comments that centered on my FBD, which was incorrect. I have fixed it. Turns out my final result remained unchanged, but I wanted to review everything very carefully.


  10. Robert says:

    Hi mathscinotes,
    i was realy interrestet about your article "Learning How Electronic Parts Work" and want use it for a paper for the university. So can you give me your real name for the reference list?


  11. Michael Dunn says:

    Enjoying your blog! I run a blog/community site - scopejunction.com. If you think you might like to write the occasional blog there, please get in touch.

  12. hammerdallas says:

    Love you blog.... I do have a question for you regarding gravitational fields...by the way...I hate math...until I started reading this blog... 🙂

  13. hammerdallas says:

    Love your blog !!! I have a question for you regarding gravitational fields.
    Thanks, D

  14. Michael Dunn says:

    Hi Mark. I'd still love to discuss blogging a bit at Scope Junction. Hope to hear from you.

    • mathscinotes says:

      Sorry for taking so long. I am just getting around to answering all my blog email this morning. I am interested in blogging at Scope Junction. I will send you a private note to discuss it.


  15. Travis says:

    I love your blog post, "An Analog Circuit Design Review". I love your insights, and want more of them. Thanks so much for your time!

  16. Victor Verma says:

    I can i get in touch with you i have questions.

  17. Just stumbled upon your blog. Fantastic! I also am an EE and have a great interest in mathematics. I used to derive MacLaurin series for different functions for grins. I minored in math when I got my MEE and was fascinated by numerical analysis. Anyway...keep it up, I'm glad to see I'm not the only one!

    • mathscinotes says:

      Numerical analysis has become an important part of my working career. In fact, my very first job was working on modifying Spice for NMOS circuit simulations. It was there that I learned many of the tricks that I still use today.


  18. Richard Harris says:

    Assume you are in the deep backwoods, your electronic gear goes down, but you must solve a simple equation such as X = 111.25^0.514, which would normally be solved as 0.514 * log 111.25; is there a way to approximate a log if you have no electronic gear or log book?

  19. Richard Harris says:

    Thank you very much; well done. Great blog

  20. Richard Harris says:

    Using ogive or logistic curves in future annual volume estimates of production from a limited, depleting resource: Cumulative production of units from the resource beginning in 2005 through 2012 by years is 15, 30, 150, 700, 1200, 2000, 3000, and 4000 units. At lunch, your boss ruins the meal by asking for an estimate or forecast of what the resource will produce each year in the future until it is depleted. He wants it by his 3:00 PM coffee break. Plotting the known annual historical data on log-log suggests it is in the form of an ogive curve. Is there a quick Fermi way using ogive/logistic equations to estimate the future unit volumes year-by-year until the curve becomes asymptotic to the production axis, which would constitute depletion and then equal the resources original recoverable volume of units?


  21. I am a retired geological consultant living outside Calgary in western Canada. The two questions are just things I wonder about. The questions and any answers you might provide are for my own personal use and interest. Thanks!

    • mathscinotes says:

      Greetings from one of your southern neighbors! I don't get to say that to very many people.

      I had to ask about the question source because I get so many homework problems sent my way. It has become so bad that I actually had one twelve-year old boy ask that I write a little bit more like a kid so my work could be copied more easily.

      I will try to take a look at the problem this weekend. I have department budgeting today.


  22. Thanks and there is no hurry from my end. Being retired has given me time to think about stuff but seemed always too busy to pursue.
    I know what you mean by 'homework' because that does occur on other sites like, for example, Dr. Math!

    I spent about 50 years working on jobs in more than 80 countries - in oil, gas, minerals, and gemstones - so I am enjoying kicking back, enjoying a pretty mild winter, and looking at the Rockies out the west windows.

    Have a nice weekend!

  23. coasttal says:

    Just came across your blog and have enjoyed reading almost every post. I am a mechanical engineer, BSMS/ME PE, and got BS in 77. Unfortunately, I have never had the need to use all the math that you use since college. Now that I am retired, I really want to start relearning so much of the math. Again, thank you so much for all these great posts. John

    • mathscinotes says:

      I started writing down my math work when I started to provide help to students. They kept telling me that they had never heard of anyone who uses math as part of their job. That was how this blog was born.


  24. Steven says:

    Thank you so much for putting this site together! Your blog is a fantastic source of inspiration for me! I was wondering if you could recommend any of your favorite or go-to reference books on the subjects of math, science, or general engineering. Thanks!

    • mathscinotes says:

      Thank you for the nice comment. I do not have any specific books to recommend. There are many books that can guide you along your path. The key is to start -- I don't think where you start is particularly important. In my case, it began in the one-room library of my home town of Osseo, Minnesota. Even the encyclopedia was mind-boggling to me! Soon I was trying to learn what a transistor was, and I was trying to build radios. You should see some of the stuff I build now!


  25. Daniel Rocha says:

    Hi mathscinotes!

    I stumbled upon you site when I was thinking about finding planets by radar!

    The idea is "simple". Find planets up to the size of earth up to 0.1 light years with a radar. The volume is huge. I don't think that you could find easily even by other means even if you had a faster than light spaceship, say, like in Star Trek.

    So, even with ultra high tech, I think that exploring the edge of our solar system is possible without some "simpler" tricks. Do you have any idea of how to configure a radar to do that?

    • mathscinotes says:

      Stellar radio communication is tough ... radar would be much tougher. The received power for one-way radio communication varies by the square of the distance. Radar involves reception of the scattered radio power incident on the planet. The received power varies by the fourth power of the range. No good ideas on how to do this from me.

      Also, there are no stars within 0.1 light-years. The closest star is about 4 light-years away. No good ideas from me.


  26. Daniel Rocha says:

    Hi mathscinotes!

    You got it wrong!

    I mean find planets within our own solar system! 0.1 means the edge of our solar system, the outer Oort cloud.

    When I meant faster than light spaceship, I meant, even if we had any kind of Enterprise like spaceship with us, we likely wouldn't find any planet in the rims of our own solar system. It would be much easier to find in other stars, just it is today.

    Look how many planets we found in around other stars and how little (dwarf planets up to now) we found in own solar system.

    Take as an example Sedna:


    We only found it because it is close to perihelium (76 AU), at its discovery 89.7 UA. Its aphelion is at 937 AU. Now, imagine how much we are missing, since this is a detached object, that is, formed outside the inner cloud of our sollar system.

    Maybe we are missing thousands or millions of planets. So, I am proposing using a radar to find planets withing our own solar system.

  27. Daniel Rocha says:

    Notice that 1 light year = 63,000 UA. So, 6,300UA is about 7 times the aphelion of Sedna, so, it's not an absurd number.

  28. Daniel Rocha says:

    Also, from the protoplanetary page:

    "Protoplanetary disks around T Tauri stars differ from the disks surrounding the primary components of close binary systems with respect to their size and temperature. Protoplanetary disks have radii up to 1000 AU"


    So, an eccentric orbit of an object formed at the rim of a protoplanetary disc, with the excentricity of Sedna, would indeed require 0.1 light years.

    • mathscinotes says:

      Sorry for the misunderstanding. Radar is still tough out to 0.1 light-years. You do see some techniques that are being used out to a couple of AU. Here is an example of a radar analysis project to gather data on Ceres, Pallas, and Vesta.


      Radar is being used to study near-Earth asteroids (Link). To use radar out to the distance of Saturn (~8-10 AU), you would only be able to detect large objects (Link). The required target size increases as you go out. Pluto is out about ~30 AU and I do not see how we could detect it using the radar systems we have today.

      In theory, you could use very narrow radar beams, but that requires very large antennas. It also requires very accurate beam steering and would take a lot of time.

      At least with what I know about radar (it has been over 10 years since I worked on a radar project), I do not know how to use it out to 0.1 light-year. It is tough to apply beyond 10 or so AU.


      • Daniel Rocha says:

        I looked at those pages. But there's a difference. What I am looking for here is a a beep from a very slow movement object, nearly static. Whereas in that case they are trying to find much faster objects. Not only that, in the case of the big asteroids, to figure out properties of those asteroids.

        • mathscinotes says:

          I agree there is a difference, but your problem is still tougher. In general, they know where there target is and can generate a very narrow beam to image the target. You are trying to acquire a target, which generally requires a much wider beam and there is much less power returned.

          Kuiper belt objects can be almost anywhere (e.g. Eris is 44 ° off of the ecliptic). It would take forever to search this volume of space with a narrow beam. I still do not see how you could get a radar system that could search out to 0.1 light-year for an object of say 2 km diameter.


  29. Daniel Rocha says:

    I meant dwarf planets, with >~500km or more, this is why I mentioned Sedna, which has a diameter of ~1000km. Besides, maybe an object with over 10,000km could be found. Say, http://en.wikipedia.org/wiki/Tyche_(hypothetical_planet) .

  30. Giorgio says:

    Hi Mark, I know you have an interest in naval matters. I thought of sharing with you an article appeared on the May 1978 issue of "NAVY International" (I was a sophomore at the Naval Academy). The title is "The mathematics of convoy"; quite esoteric, given the fact that the last convoys operated almost 70 years ago! It's a small pdf file. Please let me know if you are interested. Regards

  31. tony says:

    You're an amazing find!

    I'm working now on building a private multiage classroom. I just retired from the Army part of my life, and went and got a Masters in education. I continue to operate my flight school in South Saint Paul and teach several firearms courses while I work on the classroom concepts. One of my goals is to be able to help credibly prepare students for STEM careers--and especially to inspire those who aren't from high-academics families to success.

    A couple of your ballistics articles landed me on your blog, and I'm hooked. I'm trying to bridge my knowledge gap between the fun things I get to do with airplanes and firearms and the fact that I never dug into higher math most of my life. Working to fix this, slowly but steadily.
    So I have the great activities for my students, but want to link them to developing kids into mathematical thinkers.

    So, thanks for everything you've put out here. If you'd ever be open to discussing your views on Math and Science teaching and learning, I'd welcome the opportunity.


    • mathscinotes says:

      I applaud your efforts at education. Developing mathematical thinking skills is an interesting problem. I hate doing arithmetic and algebra -- so I don't. Yet I still do a lot of math. Fortunately, I have tools today that allow me to bypass all the things I don't like.

      I was raised in a VERY working class family: single parent (my mother was a secretary for an insurance agent), poor, small-town, rural. Yet I was able to find people to help me develop my interests. What you are doing is similar to those folks who helped me. You are doing good work and it will be appreciated.

      Contact me any time. I will send you my email.


  32. raju says:

    hello sir ,
    i want know some application on conformal mapping will u send to me

  33. Tom Lawrence says:

    I notice that in previous parts, V was a function of F0 while D was a function of Fm. In "Pejsa Trajectory Midpoint Formula Given a Maximum Projectile Height", they are both functions of F(undefined). Are you assuming that F0 = Fm? I notice that this would be approximately true for sufficiently small X.

  34. Douglas Ruisaard says:

    I am hoping you can assist me in a project a friend and I have undertaken. It involves, amongst other things, employing digital potentiometers to emulate analog potentiometers. From your posted article “Potentiometer Math” … Posted on December 22, 2011 by mathscinotes” (http://mathscinotes.com/2011/12/potentiometer-math/), I am specifically interested in the various “taper” descriptions you discussed and I am “discovering”.

    In a related article, “The Secret life of Pots” by R. G. Keen (http://www.geofex.com/article_folders/potsecrets/potscret.htm), I derived and have employed his formula for a pseudo-logarithmic curve… which does, as he describes, fit the “idealized” curve quite well. I can generate and emulate the LOG, REVERSE LOG and LINEAR tapers quite well, mathematically.

    I know I could actually “gang” the digital pots together in ways to emulate various tapers but my goal is to try to derive the expected values of a variety of tapers mathematically and employ one digital pot to emulate a given taper.

    However, I see from your article and ones like it, in particular the following site: http://www.alps.com/prod/info/E/PDF/Potentiometer/TAPER.PDF
    that a considerable number of other “common” types of tapers are available and in use. I am wondering if you could provide or lead me to a resource which could specify the mathematical formulae for the curves you describe … in as “simple” a math formulation, as possible (I’m not a mathematician… only a programmer) … if possible. I am particularly interested in the “S” taper… sometimes referred to as “log-antilog” referenced, for example, here:
    and here:

    I cannot seem to find any information on a formula regarding these tapers elsewhere.

    Also in your article about “Potentiometer Math”, in the formula:
    Eq. 1
     R0 is a curve fitting parameter
     R1 is a curve fitting parameter

    Exactly what are the actual values for R0 and R1 … “curve-fitting parameters?

    If you could reply directly back to my email address below and any assistance you could provide would be greatly appreciated. I look forward to hearing back from you in reply.

    Douglas Ruisaard
    Kamloops, British Columbia, Canada

  35. Ken Connor says:

    Could you comment on your experience with WordPress? I am about to start a blog on engineering education and, since I have enjoyed yours for some time, I thought your experiences would be helpful. Thanks and keep up the great work!

    • mathscinotes says:

      Hi Ken,

      I experimented with a number of other blogging (e.g. Blogger) platforms before I settled on WordPress. I am not a professional tech pubs person, and it was easy to learn basic WordPress editor and develop my skills over time. I like the fact that I can customize WordPress to suite my needs and that there are many developers working on new features. There is also quite a bit of useful advice in various online forums.

      The one BIG mistake I made was to start with the free WordPress.com site. I should have just paid the money right from the start and created a self-hosted site. I was three years into blogging before I realized that I (a) wanted to get rid of the site advertising, and (2) I wanted to customize things. Moving from WordPress.com to a WordPress.org self-hosted site was a huge pain. Fortunately, I found a guru at Godaddy who told me how to do it. His support was the main reason that I use Godaddy to host my site. I am happy with the transition, but it was a huge amount of work that I should have avoided.

      When you get your site up, drop me a line. I hire new engineers, and I am very interested in how they are trained. If there is anything I can do to help, just ask.


  36. Ken Connor says:

    Thanks. This was really helpful. I already own a bunch of URLs through GoDaddy, so I thought I would transition to a self-hosted site as soon as I see how things work with the free site. Having one more site or changing over one of my other sites for a blog is no big deal for me, but I see that waiting too long could be an issue. I will find someone at GoDaddy to help unless you have someone to recommend. I would definitely like your feedback once I get some content on my blog.

  37. Saif says:

    Hello mathscinotes.

    Your posted about Thermistor Mathematics is very helpful for my final year project. Your blog posted about the linearization of thermistor using series resistor. I am still working on it. I need used different approach. One of the is used this series resistor.

    On the other hand, I have done with parallel resistor, can I email you my circuit design so that you have any thought for improvement that I can do. I am struggling doing the signal conditioning circuit for thermistor. Hope you can help me.

    Middlesbrough, UK

  38. Gene Mirro says:

    I have been using Mathcad 11 for ages, and have roughly 150 Mathcad files that I use over and over. I just replaced my hard drive, and I can't get Mathcad to run anymore, and of course PTC won't talk to me about a new activation file. I have Mathcad 7 which does not require activation, but I'm kind of scared to try to load it onto my Windows 7 machine. PTC Mathcad costs $1660, a little steep for a self-employed consultant approaching retirement. What would you do? Thanks.

    • mathscinotes says:

      Sigh … I wish I had a good answer for you.

      I do not use Mathcad at home because of the cost – math is a hobby for me, and it is hard to justify that cost. I did talk to my wife about it, but $1660, and high annual maintenance fees are tough to swallow. I have used every computer algebra system out there and, for general use, Mathcad beats them all. I have a home version of Mathematica, which cost me about $400, and meets my needs there. Mathematica has more advanced capabilities, but for general use nothing is quicker than Mathcad. Maple is okay but again not as good for quick calculations. Some of the open-source stuff also is good, but does not allow the production of quality documentation.

      Ah Mathcad 11. Many power users feel that Mathcad 11 is the best version of Mathcad ever released (e.g. no static unit checking) – me included. I know folks who claimed they were able to get it to run on recent Windows releases, but I am not one of them. The vast bulk of my work is done with Mathcad 15, with some Prime. Neither is as good as 11, but I feel it is still better than the competition for general use.

      Wish I had a better answer …


  39. Joe Bush says:


    Wow, my productivity has taken a hit as I discovered your blog an can't stop reading! I too worked for HP for close to twenty years and still work for Agilent as we were shed and as we continue to shed pieces of the once venerated company. I could talk for hours on what has become of HP. I too use MathCad. I use version 14... my work copy. I could use Prime as the company has floating licenses but have chosen to stick with 14. I too like to fool with math for recreation but I am not as prolific as you. I am constantly searching for pre-worked MathCad files. Speaking of that, I like that you offer a link to download some of your MathCad models. Is there any way to get all of them? Perhaps a zip file? I love just exploring ideas and your work gives me a great jump start in a lot of areas.


    • mathscinotes says:

      Great to hear from an HP person! I worked there in the old days when it was a first-class organization – I have no idea what it is like now. I still remember being in awe of Bill and Dave when they came around for division reviews.

      I am now including source with my posts because of demand – this whole thing started as searchable place to keep my notes and then folks wanted to use some of my routines for training. I am slowly going back through my old work and adding source as I go. There are nearly 600 posts, so it will take some time. The new posts will have source. If there are specific routines you need, just leave me a comment here, and I will update the specific posts you are interested in.



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