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By Murray Bourne, 21 Nov 2004

murray bourneThis site is owned and maintained by Murray Bourne.

I was a teacher in secondary schools in Australia for some years. I was in Broken Hill (outback Australia) and then Grafton (northern NSW).

I then moved to Japan where I taught mathematics (in English) for 4 years in an interesting program that prepared Japanese students for college courses in the USA. The students would spend one year improving their English, followed by one year doing a typical freshman program, then they would go to small mid-West colleges in the US. I also taught English while in Japan.

On returning to Australia, I taught in TAFE (Technical and Further Education), Bond University and Griffith University on the Gold Coast, south of Brisbane.

I then moved to Singapore where I taught engineering mathematics in a polytechnic. Subsequently, I did staff training in the same institution.

I'm now a freelance trainer and consultant, still based in Singapore.

My main site is Interactive Mathematics, which is a series of math tutorials enhanced by interactive javascript- and SVG-based applets. There are hundreds of testimonials about IntMath from happy readers on this page.

To avoid spammers, I hesitate to put my real e-mail address here :-(. If you want to contact me, leave a comment on this post. 🙂

See the 69 Comments below.

69 Comments on “About…”

  1. Tang Kim Seng says:

    Hi Zac,

    Great website ! I can't seem to get enough of it and keep coming back for more. Your topics are interesting and diverse. They stimulated my thinking and I enjoyed our discussions. Hope to see more of other topics being posted and others join in our discussions. Cheers!


  2. Vinod says:


  3. Murray says:

    Vinod's article is an April Fool's type piece from BBC:

    Computers simply cannot divide by zero. Try it on your calculator and you'll get an error message.

    But Dr Anderson has come up with a theory that proposes a new number - 'nullity' - which sits outside the conventional number line.

    What is extraordinary to me is the hundreds of people that have commented on the nonsense.

    I agree with Matt Milford, who says:

    For goodness sake, do NOT teach this to year 10 pupils!

  4. Joice Toh says:

    Hi Zac,

    I chanced upon your blog in regards to your entry about SMART Interactive Whiteboards.

    Would like to touch base with you on that. Do drop me an email when you read this? Thanks heap!

  5. Ivan Chew says:

    Hi Murray,
    I just saw your entry at the sgSocialMediaDir wiki (Individual H-N page). Included additional keywords to describe your blog (you can amend them if you think they are not appropriate).

  6. Murray says:

    Fine with me, Ivan - thanks for the better set of keywords! [I edited your comment - by including the link to the wiki.

  7. Alane Tentoni says:

    I loved your Ten Ways to Beat the Math Blues. I'm going to make it required reading for my high schoolers this year. Thanks!

  8. Murray says:

    Thanks, Alane - glad you found it useful!

  9. Del says:


    I'm teaching Geometry for this few weeks. I've been wondering how I can get my hands on a pair of giant compasses that I can use and demonstrate on the board in class.

    Can you suggest a way for me to buy one locally in Singapore?


  10. Murray says:

    Hi Del. I can see why you had trouble finding a supplier of compasses in Singapore. I've just googled like mad and cannot find a supplier.

    I expect some of the larger Popular bookstores (like at Jurong East) might have them and Bras Basah Complex has a multitude of related shops and a huge Popular.

    The following suppliers (of books, mostly) may know where you can get them:

    If all else fails, you could always contact MOE - they would surely know...?

  11. Murray says:

    Hi Andrey - I will review your product when I get a chance. Good luck with it!

  12. asmat says:

    u r great i am from pakistan,i want to touch with for ever,ur websites and the method help of people is unique,i gain a lot from ur websites,thanksssssss

  13. Alicia says:

    I was just reading your blog and thought you may be interested in checking out a new math resource that my nonprofit organization, Facing the Future, has just released.

    Real World Math: Engaging Students through Global Issues is a supplementary math text that puts math in an interesting context that relates to students' lives. The teacher's guide and corresponding student workbook contain 15 lessons that engage students in learning foundational algebra and geometry through real-world data on global issues such as climate change, population, and financial literacy.

    Each lesson in the teacher's guide includes:
    • A complete lesson plan including an activity based 'hook' to engage students
    • Masters for student lesson handouts
    • Masters for practice worksheets

    All lessons were inspired, researched, designed, reviewed, and pilot tested by educators in the field.

    You can read more about this resource and download sample lessons on our website. If you would like to receive a review a complementary copy of the teacher's guide, please send me your mailing address.

    Please do not hesitate to contact me with any questions about this resource or our organization in general.

  14. Murray says:

    Hi Alicia and thanks for the interesting resource!

  15. Krithika says:

    Hi Sir,

    I was really surprised when I nvigated through your website. Well I have a small work asked by my friend. I have no idea regarding this and I would liek to have your advise and help in order to proceed with this. I’m really curious to get your help!Thanks in advance!

    “Please use Microsoft Excel (Solver add in) to Maximize Z = 3X + Y, under
    these constraints: 12X + 14Y = 85, 3X + 2Y = 18, 0 = Y = 4, X = 0, Z = 0.”

  16. Murray says:

    Hi Krithika

    This is a linear programming problem. You need to add the Solver to Excel and there are plenty of tutorials on the Web how to do that.

    Here's a resource that may help:

    Good luck with it!

  17. Max C. Sandwalk says:

    Hello. First, your site is great (I've been following your post through feeds).
    Now, I was wondering if I can translate your posts (spanish) about math concepts and uses them in my blog. You will be the author I don't pretend to use as my ideas or any.

    Thank you anyway.

  18. Murray says:

    Hi Max. I have responded by mail. Let me know when you translate a post!

  19. Lisa says:

    Hi Zac,
    Happy Wednesday! I recently stumbled upon your website and and wanted to spread the word about our free math help resource and see if you can spread the word out! Let me introduce myself, my name is Lisa Kwan and I am the office manager for Brightstorm. Our company recruits teachers and works with them to create high quality, video based streaming solutions for students and teachers. We recently launched an extremely useful online video-based math solution (feel free to check it out at

    Here's some more info about our FREE math service:

    * Over 2,000 videos covering every topic from Algebra 1 through Calculus

    * FREE: Yes, Brightstorm Math is 100% free to any user!

    We would love to hear what you think about our product and services! =) Take care and have a great week!

    [email protected]

  20. JoAnne says:

    Hi, Murray--
    I am a browser of math sites and your Square-Circle title reminded me of a recent posting on my own blog -- here's the link.


  21. Murray says:

    Thanks for the interesting math poetry, JoAnne.

  22. XuDuo777 says:

    Thank you for such a well-developed and insightful website.

    I'm doing a similar program in China, and your story inspires me. Bookmarked and visited often, you are!

  23. Murray says:

    Hi XuDuo. You are welcome and I'm glad you find it useful.

    Good luck with your Web endeavors!

  24. Murray says:

    Hi Eldhose. I like your puzzles! I've scheduled a Daily Math Tweet about your collection.

    Would you mind if I used a few of them for the IntMath Newsletter? Of course, I would give you full credit and would provide a link to your pages.

  25. C says:

    Hi Zac! A girl from Singapore here. Your posts are pretty interesting and reflective of local culture from what I've read 🙂

  26. Murray says:

    Hi "C" - good to make contact with you.

  27. Cory Poole says:

    Thought you might appreciate this.

    It's time for standardized tests across the nation. I'm a geometry teacher and decided to build a sculpture based on this. It's built out of 80 number 2 pencils and called "Standardized Testing And Reporting" or "STAR" after the California testing program.

    Here's a short blog post on it.

    Cory Poole

  28. Murray says:

    Thanks, Cory - I enjoyed poking around your site. Some very interesting things you're doing there. I liked your Eschers and the 3D fractals.

  29. Aiman Azri says:

    Hi Murray,

    I found your blog and like what you write on Maths, education and society. I'm writing to introduce to you and to solicit your thoughts and opinions on Gnowledge (, an education website that benefits Singaporean educators, students and parents.

    Gnowledge is free for everyone to use. Tests and exercises can be created by anyone and are categorized by title, subject, grade, school and/or country. Once published, these tests are available for everyone to use and share. Students preparing for exams or who wish improve their knowledge and accelerate their academic progress can search Gnowledge for their desired subject. All test results, scores and answers to every question administered is stored on Gnowledge. Registered users can view their own individual test results any time and the test results of everyone who has taken the tests they've created, and these tests and/or results can be shared with others if they choose to do so.

    The best example of how Gnowledge is being used and how it transcends language and cultural barriers and education syllabuses is by a French language teacher in Mexico (a Spanish-speaking country). She uses Gnowledge (an English-language website) to produce simple tests and exercises for her students. In two months she has published 37 tests and these tests have been administered to her (and other) students 1,066 times. Scale of this magnitude does not happen in conventional classrooms or schools. Her stats can be independently verified at:

    I invite you to check out our site, please do let us know your views and opinions. As a token of our appreciation, we would like to send you some souvenirs if you would have them.

    Thank you for your time and I hope to hear from you soon.

    Aiman Azri

  30. Murray says:

    Hi Aiman. Thanks for your kind comments. I checked out Some quick thoughts (which are meant as helpful things to consider - I hope it doesn't come across as negative):

    (1) I'm not sure how this is evidence of "trailblazers and paradigm shifters in education". Yes, there is value in giving students a chance to try past exam papers and to get feedback on their responses, but this is not exactly new. Also, it tends to encourage the view that education is all about passing exams. Why not add some social networking aspects? Discussion on the solutions, maybe? A chance to help each other on difficulties, maybe?

    (2) I'm wondering if you have copyright clearance on the tests in there. Currently your service is free and you can probably claim "fair use", but depending on your business model, at some point you'll want to charge for something and the you may have issues.

    (3) The JS interface is smooth, but the key usability issue I experienced was having to click all over the place to move through questions. Why not place the "Forward" and "Back" buttons right next to the "Submit Question" buttons?

    All the best with your venture.

  31. Dave Fashenpour says:

    Hey Zac!!

    I have officially retired from my third (Math Teacher/Tutor) career.

    I have started Instructional Math Tools, LLC and have a web site:

    What I would like to show you and your readers -- is a VIDEO of two new tools that we have just released, that search Apple's App Store for the right tool for the right job. Please checkout this short highlight video:

    Thanks for everything,
    [email protected]

  32. Murray says:

    Good luck with your new venture, Dave.

  33. Debbie Kimber says:

    Hi Murray, I love your site!

  34. Murray says:

    @Jos: Thanks!

  35. Olumide g. AREO says:

    Kodus, am 4rm nigeria,am a web surfer wit gr8t interest in maths,stumbled on ur website am all i could say was..."amazing" maths made easy...i wish u strength

  36. Michelle Comendul says:


    I enjoy reading your blogs and really admire your passion for teaching math! I am helping to promote math education with the Math Video Challenge! I would really appreciate it if you took sometime to maybe mention and support the MATHCOUNTS Foundation.

    MATHCOUNTS’ third annual “ Math Video Challenge” is a math competition for 6th to 8th graders that encourages student innovation as they create and star in their own math videos, thus exciting them to pursue higher education in math.So far this contest has gathered over 500 submissions and millions of views on the videos. Our goal this year is this year is to do even better.

    For more information on MATHCOUNTS or the Math Video Challenge, visit our webpages at and

    Thanks so much and I look forward to your future posts!

  37. Murray says:

    @Michelle: Happy to support you!

    Suggestion: Make the video submission due date more prominent on the site. I had to look all over the place to find it!

  38. Jake Byrnes says:

    Dear Murrary,
    As a proponent of math education in the United States, we need your help to promote our nationwide math competition by blogging or posting about it on your blog/forum.

    As you probably already know, despite the fact that the US spends the most money on education per capita, our students are ranked 25th globally for math proficiency. The MATHCOUNTS Foundation is a nonprofit dedicated to improving that statistic. MATHCOUNTS’ third annual “ Math Video Challenge” is a math competition for 6th to 8th graders that encourages student innovation as they create and star in their own math videos, thus exciting them to pursue higher education in math.

    As the webmaster of squareCircleZ, we are asking you to help support this effort by mentioning us in your next blog or forum post or promoting our logo with a link to the site. So far this contest has gathered over 500
    submissions and millions of views on the videos. Our goal this year is this year is to do even better. With your help, we are confident we will reach this goal.

    For more information on MATHCOUNTS or the Math Video Challenge, visit our webpages at and

    Jake Byrnes

  39. nigelnigel williams says:

    Hi people very much loving this site, and I have a question about math sighns that are used delta denotes change, but does it sinigfy a number or does it signify an operation?
    Thank you kindly in response.

  40. Murray says:

    @Nigel: "Delta" (Δ) is closer to an operation than to a number. So Δx means "a small change in x" - this can either be a forward, backward or central difference. See Difference Operator.

  41. Alex says:

    Hi Murray

    Lovely site, very engaging, with very clear and quite comprehensive content. Have you considered a topic such as maths for gaming particularly graphics? My not yet teenage son is discovered algebra through voting and is keen to find out how to code the likes of showing rays/projectiles fired at a target, moving at different speeds. - An opportunity to throw in some trig and parabolas heh?

  42. Murray says:

    @Alex: Thank you for the positive comments and for the topic suggestion. It's actually been on my list for some time. Here's a kind-of related page (SVG is used a lot in gaming):

  43. Joanne Caniglia says:

    Hello Murray,
    I love your problem on measuring the length of a spherical spiral. I would like to modify it to place in an article on coiling and mathematics (Native American Basket Weaving).
    Thanks so much. Your name and website will be used in the article.


  44. Murray says:

    Hi Joanne

    Glad you find the article useful. You have my permission, as long as it is referenced appropriately as you said.

  45. Simon McCauley says:

    Hey Murray, I was just talking with my 16 year old son and we were discussing the small number of great teachers I had at Broken Hill High School. Your name was top of the list. Naturally I had to track down what you are up to now. As well as maintaining your stupendous work ethic, you seem to have discovered the fountain of youth. You photo looks just like I remember you from the late 70's! Not sure if I ever thanked you for all your efforts teaching maths and music to us ratbags at BHHS. You helped set me up on a path to becoming a pretty good Electrical Engineer and a passable bass player.
    Best wishes, Simon.

  46. Murray says:

    @Simon: Great to hear from you and thank you so much for your kind thoughts. I'm so glad you pursued both engineering and bass playing - it warms my heart!

    Funny thing - just this week I was watching ABC on Australian cable and a short item came up regarding an ex-student of BHHS. They actually showed my teaching room in the background! It brought back a flood of memories...

    All the best to you and your family.

  47. Srijan Biswas says:

    I have developed an app for arithmetic - 'Maths MCQ'.
    Here one has to solve challenging arithmetic problems in 15 seconds. Questions are NEVER repeated. Answer stats are recorded. It has 2 levels on difficulty. This app is good for brain exercise and improving speed of calculation. Kindly take a look at it, and if you find it helpful then please feature it in your blog. Link below:

  48. Levi says:

    Hi Murray,

    I find your this tutorial very helpful. Have you made this into a book?

  49. Murray says:

    @Levi: Glad you find IntMath useful. It's in my plan to produce a book, but at the moment, there's only 24 hours in each day!

  50. Murray says:

    @Srijan: Thanks for sharing. Good luck with your app.

  51. Todd says:

    Hi Murray,

    I was very interested in your Blog post about functional notation and using square brackets instead of parentheses. I had been thinking about similar things which is how I found your blog. You mentioned that you were going to write an article on this. I was wondering if you have done so or no of anyone who has done research on this or written texts using these ideas? I am thinking of writing an OER text exactly because of this issue, but don’t want to reinvent the wheel if one already exists.

    Thanks! Great site!

  52. Murray says:

    @Todd Thanks for your interest in this topic. I think it's a major stumbling block for a lot of students, yet rarely gets talked about.

    I haven't managed to write an article yet (it's on my long list...), but these pages may be of interest:

    Why did Mathematica choose brackets for function arguments over parentheses?

    The Four Kinds of Bracketing in the Wolfram Language

    Makes a lot of sense to me!

    Let me know if you write your text - I'd love to read it.

  53. Patrick LaChapelle says:

    Hi Murray,

    I have an idea to generate interest for the Putnam competition, I'm wondering if I could possible email you about it? Thanks for your time and thanks for the blog, it's more useful than you could know.


  54. Murray says:

    @Patrick: Thanks for your kind comment. I've emailed you separately on this.

  55. filip says:

    Hi, please can you help me? I could't find this program with parabola picture. It doesn't work with parabola on website . Thank you very much in advance for your answer and help.

  56. Zach says:

    Hi Murray,

    I have a question about your "Conic Sections - interactive 3-D graph" page. First of all, I really love this interactive representation of conic sections, it really helps me understand it better!

    But I have one small problem with it. I see how ellipses form when the angle of the intersecting plane is less than the angle of the slant edge of the cones, and I see how hyperbolas form when the angle of the intersecting plane is greater than the slant edge of the cones (I also see how circles are a special case of ellipses that occur when the intersecting plane is normal to the cones).

    My problem is that no matter how I adjust the sliders, I cannot see any parabolas. I know that parabolas only occur when the angle of the intersecting plane is equal to the angle of the slant edge of the cones, so there's only one possible angle at which they could occur. But they never show up on your interactive diagram. When I adjust the angle slider, it just skips right past the parabola, going straight from ellipse to hyperbola and vice versa.

    I think it would be really insightful to make it so that the parabola appears when the angle slider is at just the right spot (I think it's about 45 degrees in yours). That way users can see all four (seven if you count degenerate) conic sections that are possible and are listed on that website.

    Thanks a lot, all the best,

  57. Adithya Sabarish Saravanan says:

    Hello Murray,

    I used the pictures as screen shots in my project report. The pictures taken from following link:

    The angle of inclination illustration picture and the slope calculation formula. Please let me know if there are any concerns.

    Thanks and regards,

  58. José Luis Acebal says:

    Dear Professor

    Congratulations on the site. During the Covid pandemic, I intended to use some animations in my virtual classroom. However, it involves making videos with limited access to around one hundred students. So, in addition to formal citations, author announcements, links, etc., are there any copyright limitations on the use of the content on your site?

    Thank you
    Best regards

  59. Nick La Maina says:

    Thank you Jose. In summary, if someone wants to make videos of animations on the site, and they give full credit to the author and a link to the site, there is no problem at all, especially as they are not deriving any commercial gain. The problem comes (and it happens a lot) when people just copy slabs of content and put it on their own site (thus generating extra content on their site and then earning advertising revenue.)

  60. ho says:

    Dear Murray,

    I have a question about the Singapore toto probability page @(

    I apply the formula and manage to get the probability for all. However, when i try to compute the probaility of getting 2 number + 1 additional number out of 49

    =1/(COMBIN(6,2)*COMBIN(43,4)/COMBIN(49,6)*4/43)=81 which is bigger than if i were to get only 3 number + 0 additional number which is 1 in 61. How can it be? where is wrong.

    Same problem if i were to choose 1 number + 1 additional number using the formula, the result seem funny (1 in 16).

  61. Michael pAUL says:

    Hello Murry.
    I recently came across your solution to the length of an Archimedeans spiral. I looked at it but for practical reasons it was useless for the ordinary reader, because of the complex solution to the integral. If you and some and your math geeks would look at this problem further by analyzing that integral you would come up with a far simpler solution. Here it is:
    By the way I am not mathematician, my field is a self taught electronics engineer
    Michael Paul

  62. Matt says:

    Thank you for creating this math resource and it's great to see it featured on HN!

    One tiny typo: Brenton should be Bretton. 🙂

  63. Nick La Maina says:

    Updated, thank you 🙂

  64. Kayvon says:

    I think there's a typo on example 3. You mistakenly show b = 1.4 / 2 * pi. Should be b = 1.4/(2*pi).

    I'm trying to follow where that final integral comes from. In particular, I'm trying to find an equation that will give me coordinates for a given length. I.e., I know the total curve, but I want to find polar coordinates for a position in the curve 20 units after the start.

  65. Sue Holland says:

    Hey Murray,

    I've written you before. Still loving your website it is absolutely invaluable! Thanks.
    I noticed a problem on the interactive about Reimann sums on this page:
    Choosing a trapezoidal sum type shows a left sum type. Not a huge deal, but... thought you might like to know.

    Happy Mathing,

  66. Nick La Maina says:

    Yes, you're right. In an earlier version of the math notation setup, it correctly formatted such an expression, but looks like that is longer the case. It has now been updated 🙂

    To answer your other question, since the formula for the curve is r = 5+0.22282 θ, then 20 units after the start would be

    r = 5+0.22282 (20) = 9.4564

    The polar coordinates would be

    (9.4564, 20) ≡ (9.4564, 1.1504)

  67. Nick La Maina says:

    Hi Sue - thanks for reporting! This should now be fixed 🙂

  68. jishnu says:

    wow this jewel of an website is really old. I wish i had found it sooner....anyways thanks a lot to Mr murray for all the wonderful math explanations which has helped me a lot!

  69. Martin says:

    The piece you wrote on the rope covered sphere was excellent, very insightful.

    I have an idea about quantum mechanics that explains what the number 1/137 represents in nature, and why it is so important.
    The solution is derived from the rope-covered sphere from this conjecture

    Assume 137 is degrees, expressed as 1/3600 arcseconds (or better expressed in radians?) and represents potential time. The arc length at radius pi m is 7.513 m

    A helical covered sphere has a radius of pi m
    A rope of consistent thickness is wound around and is not touching itself anywhere - there is a consistent gap equal to the thickness of the rope throughout all rotations
    There are 12 rotations from pole to pole.
    What is the length of the rope?

    I wonder if there is a relationship?


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