There is nothing more that The Study Dude wants for you than to become a high-speed human calculator.
Well, in these articles, as The Study Dude, I’ll try to give you the study tips you need to help make your learning easier. I’ll also give you straight and honest opinions and personal anecdotes?even the embarrassing ones that you wouldn’t ever dare read about from any other study tip guru.
Arthur J. Benjamin, PhD and mathemagican, and Michael Shermer write the book The Secrets of Mental Math, this week’s Study Dude hot topic. All you need to slog through this book includes knowledge of a basic one by one multiplication table and the ability to add and subtract (and, yes, divide) simple numbers. If you can do that, you’ve got what it takes to become a spectacular mathemagician.
What’s a mathemagician? A mathemagician computes instant calculations for an eager audience. Present any math question, and the mathemagician will solve it in a matter of seconds. Surely, that gift would aid and abet in getting you top grades.
Multiply by 11
One of the first things I learned when honing the skill of instant multiplication involved calculating math problems from left to right as opposed to the standard backward right to left process.
At one low point in my life, I decided that my identity would include transforming myself into a mental calculator. I started arduously with the two times table. Knowing that multiplying a number by 5 or more carries a 1, I started calculating multiplication by two from left to right. For instance, in multiplying 186 by 2, I would start by multiplying 2X1, noting that the next number is greater than 4, so would involve a one carried over. I would then work my way to the right, following the same process.
Using this system, I became so fast at multiplying by 2’s that I could instantly calculate numbers in the billions multiplied by 2. However, my life’s lull turned around, and I abandoned, temporarily, the quest to groom myself as a human calculator.
Yet, in a computer science course at the university, I could shift digits into binary code in a matter of seconds. This skill of doubling proved handy when it came to computers. The skill paid off.
Arthur Benjamin and Michael Shermer present an easy to learn system for calculating multiplication by 11. So, sit back, learn, and later impress the group with your mathemagic.
– When multiplying a two digit number by 11, say 23X11, take the first two digits of the 23 (the 2 and 3) and insert in between them their sum (2 + 3). In other words 23 X 11 = 253.
– When multiplying two digit numbers by 11, say 84 X 11, where the sum of the two digits (8 + 4) is greater than 9, add one of the left most digit of the solution (8+1). So the answer would be 924, where the 9 is the leftmost digit plus one (8+1), the 4 is the last digit of 84, and the middle digit is the last digit of the sum (8 + 4 = 12, so 2).
Subtraction using complements
I failed to learn a good system for subtraction. Instead, I found myself bogged down when it came to subtracting three digit numbers from three digit numbers.
Fortunately, universities let you rely on calculators, and I discovered that using a calculator that enabled brackets within brackets helped me to make subtraction calculations amidst more complex integral calculations. The calculator was a boon, as the proclivity toward error with subtraction would have magnified without the use of the machine.
The subtraction method used by Arthur Benjamin and Michael Shermer is so simple, I will divulge it by showing you a neat little trick they demonstrate called “subtraction using complements.”
– If you need to subtract a two digit number from a three digit number, say 340 – 78, and the 78 requires you borrow, then take the complement. In other words, do the reverse, 78-40 for the last two digits, to get 38, and then, to acquire the complement, subtract 38 from 100 (100-38=62) and voila! You have the last two digits of the subtraction (62). 340-78= 262
Squaring Two Digits
I never learned the trick to squaring two digits, but one exists, and it will blow your mind away.
The furthest I got with multiplication involved multiplication with any size number times a one digit number. I found ways to record on my fingers the digits that would involve carrying a number, and, again, I calculated the digits from left to right, and not the standard right to left. This method enabled me to call out the digits in the order they would appear on paper, rather than backwards. That way, I just needed to retain in memory the digits long enough to make the calculation and move on to the next digit.
However, Arthur Benjamin and Michael Shermer divulge the secret to squaring two digits. Once you learn this little nifty trick, your life will improve considerably.
– When squaring a two digit number, consider the following formula: a2= (a-b)(a+b)+b2=a2 – b2 + b2
– To make the squaring of a two digit number easier, round up or down the number to the nearest tens place. For instance, to square 86, you would round 86 up to 90. There is a difference of 4 between 86 and 90, so substitute 4 for b in the above formula. For instance, 862=(86+4)(86-4)+42= 90 X 82 + 16= 7380 + 16 = 7396.
– It’s easier to multiply a number ending in zero with another number than it is to square a two digit number not ending in a zero.
– To make the multiplication of 90 X 82 easier, mentally, think (90X80) + (90X2) which equals 90(80+2).
So, there’s nothing to fear. The Study Dude is determined to make right for you all the wrongs I made in grad school?one A+ at a time.
Benjamin, Arthur, & Shermer, Michael. (2006). Secrets of Mental Math: The Mathemagician’s Guide to Lightning Calculation and Amazing Math Tricks. New York, NY: Three Rivers Press.