Saturday, December 11, 2004 - 03:04 PM

You may have heard the recent news of Dr. Gert Mittring, who correctly extracted the 13th root of a 100-digit number in less than 12 seconds...in his head. This article shows you how to accomplish the same feat in the same amount of time using an ordinary calculator.

This math article was inspired by the news article posted by our site administrator, Clay, who was kind enough to encourage me to resubmit my post as this present article for the interest of our readers.

The posted article reported how Dr. Gert Mittring, who has mastered mental calculations to a unimaginable level, managed to give the correct answer to extracting the 13th root of a 100 digit number in front of an audience and 2 umpires who selected the number at random. It was also reported that: "Spectators using electronic calculators were left minutes behind. "

If you read on, you will find out how you could outperform the spectators, calculator in hand! Just imagine the time it takes to input a 100 digit number is already more than 12 seconds!

If we were to be requested to count the number of grains of sand at a beach, most of us would stare on the white sand and say, how would I know. This is perfectly normal. For those of us who believe that it is possible, we will TRY and develop a method. Perhaps not the exact number, perhaps not even nearly exact, but it will be an approximation. From the approximation, we refine our method, and get even closer. This is how geniuses like Dr. Mittring work.

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Now let us look at how we can solve the problem, with or without a calculator.

First, looking at the 13th root, by very simple calculations, we understand that:
ANY NUMBER RAISED TO THE 13TH POWER WILL HAVE THE LAST DIGIT OF THE RESULT EQUAL TO THAT OF THE ORIGINAL NUMBER.

That settles the last digit of the answer, simply by copying the last digit of the number to be extracted 13th root as the last digit of our answer!

Next, we would like to find out within what range the answers should be, given that the original number has to be a 100 digit number. Therefore the logarithm (to base 10) of the original number must be between 99.000 and 99.9999999...

Using an ordinary calculator, we calculate the numbers

99/13=7.61538462.. and 7.69230769230...

Raising 10 to these powers gives us the lower and upper limits as:
41246264 and 49238826.

In fact, the answers are as follows:

41246264 ** 13=1000000053891531265062238747077907560690804374410559722392499871913587301621661365915461814157574144

49238826 ** 13 =9999999162880950507644688918524814522347242627239243189376115437857450758590978444820365074927230976

So we know that the answers are 8 digit figures, starting with 4, in fact, between 41246264 and 49238826, with the last digit equal to the last digit of the given number!

We are already getting somewhere!

With this information in mind, now we are going to find the 13th root of a given 100-digit number using a calculator:

2345813452573965450990121027421105301917006585675372430552395152492628208920201161613694616524029952

If you are a mathematician, you probably have access to superior software such as Mathematica that will let you get the 13th root in a flash. If you are like the rest of us, with at best a good calculator in hand, this is how we could proceed:

We will find the logarithm (to base 10) of the number using the calculator. We won't be able to enter the complete number into most calculators, but we can do it approximately by noting that the number is nothing more than

2.3458134525739..*10^99 (that makes a number of 100 digits)

The logarithm is simply 99+log10(2.3458134525739)=99.3702934725

Divide the number by 13 to get 7.643868729

Calculate 10^(7.643868729) to get 44042172, which is the correct answer, noting that the last digit corresponds with the last digit of the original number!

So here it is, if you were with Dr. Mittring at the performance, with a calculator in hand, you could have beaten him with your answer in less than 11.8 seconds! For those of you who can preprogramme the calculator to do the task, the time it takes is what it takes to enter 10 digits.

44042172 ** 13 =2345813452573965450990121027421105301917006585675372430552395152492628208920201161613694616524029952

Happy Mental Calculations!


Note: The above calculations for raising to the power of 13 are thanks to a multi-digit calculator available at the following site:

http://www.mathpath.net
look under Arbitrary Precision Arithmetic.

That link takes you to a calculator that can do calculations of thousands of digits. The number of digits of precision is YOUR choice.

Editor's Note: mathmate is so modest. MathPath.net is actually mathmate's web site! I suggest you visit and bookmark. Wonderful site.

If you want more practice (to compete with Dr. Mittring!), you could raise numbers between 41246264 and 49238826 to the 13th power and try your hand with the above method.

Now the big challenge: What does it take to do the calculations MENTALLY?

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