"Secrets of Mental Math" by Arthur Benjamin & Michael Shermer

• If I were forced to summarize my method in three words, I would say, “Left to right.”
• When you compute the answer from right to left (as you probably do on paper), you generate the answer backward. That’s what makes it so hard to do math in your head.
• To paraphrase an old saying, there are three components to success -- simplify, simplify, simplify.
• You need to know your multiplication tables backward and forward.
• To find the number of years that it will take for your money to double, divide the number 70 by the rate of interest.
• A principle of probability called the Law of Large Numbers shows that an event with a low probability of occurrence in a small number of trails has a high probability of occurrence in a large number of trials.
• A psychological factor called the confirmation bias is where we notice the hits and ignore the misses in support of our favorite beliefs.
• The blind spot bias is one in which subjects can recognize the existence and influence in others of different cognitive biases, but they fail to see those same biases in themselves.
• The Baloney Detector Kit:
• How reliable is the source of the claim?
• Does this source often make similar claims?
• Have the claims been verified by another source?
• How does the claim fit with what we know about how the world works?
• Has anyone gone out of the way to disprove the claim, or has only confirmatory evidence been sought?
• Does the preponderance of evidence converge to the claimant’s conclusion, or a different one?
• Is the claimant employing the accepted rules of reason and tools of research, or have these been abandoned in favor of others that lead to the desired conclusion?
• Has the claimant provided a different explanation for the observed phenomena, or is it strictly a process of denying the existing explanation?
• If the claimant has proffered a new explanation, does it account for as many phenomena as the old explanation?
• Do the claimants’ personal beliefs and biases drive the conclusions, or vice versa?
• You’ve probably heard that math is the language of science, or the language of Nature is mathematics. Well, it’s true. The more we understand the universe, the more we discover its mathematical connections.
• Mathematics, particularly arithmetic, is a powerful and dependable tool for day-to-day use that enables us to handle our complicated lives with more assurance and accuracy.
• How to multiply, in your head, and two-digit number by eleven:
• Simply add the digits.
• Place the sum between the two digits.
• If the sum is greater than 9, carry the tens to the next digit.
• To square a two-digit number that ends in 5, you need to remember only two things:
• The answer begins by multiplying the first digit by the next higher digit.
• The answer ends in 25.
• If I were forced to summarize my method in three words, I would say, “Left to right.”
• To calculate a 15% tip:
• Calculate 10% of the bill (by moving the decimal place).
• Cut the new number in half (this is 5%)
• Sum the 10% and 5% numbers to get 15%.
• When you compute the answer from right to left (as you probably do on paper), you generate the answer backward. That’s what makes it so hard to do math in your head.
• A fundamental principle of mental arithmetic -- namely, to simplify your problem by breaking it into smaller, more manageable parts. This is the key to virtually every method you will learn in this book. To paraphrase an old saying, there are three components to success -- simplify, simplify, simplify.
• Most human memory can hold only about seven or eight digits at a time, this is about as large a problem as you can handle without resorting to artificial memory devices, like fingers, calculators, or mnemonics.
• When subtracting two-digit numbers, your goal is to simplify the problem so that you are reduced to subtracting (or adding) a one-digit number.
• If a two-digit subtraction problem would require borrowing, then round the second number up (to a multiple of ten). Subtract the rounded number, then add back the difference.
• Notice that for each pair of numbers that add up to 100, the first digits (on the left) add to 9 and the last digits (on the right) add to 10.
• Complements are determined from left to right. The first digits add to 9 and the second digits add to 10 (An exception occurs in numbers ending in 0).
• Complements allow you to convert difficult subtraction problems into straightforward addition problems.
• You need to know your multiplication tables backward and forward.
• An especially easy type of mental multiplication problem involves numbers that begin with five. When the five is multiplied by an even digit, the first product will be a multiple of 100, which makes the resulting addition problem a snap.
• You saw how useful rounding up can be when it comes to subtraction. The same goes for multiplication, especially when you are multiplying numbers that end in eight or nine.
• I can assure you from experience that doing mental calculations is just like riding a bicycle or typing. It might seem impossible at first, but once you’ve mastered it, you will never forget how to do it.
• How to square a two-digit number:
• Round the number up & down by the same amount to a multiple of ten.
• Multiply the rounded numbers.
• Add the product and the square of the amount rounded.
• To factor a number means to break it down into one-digit numbers that, when multiplied together, give the original number.
• The cube of a number is that number multiplied by itself twice.
• How to count on your fingers:
• To represent numbers 0 through 5, all you have to do is raise the equivalent number of fingers on your hand.
• To represent numbers 6 through 9, place your thumb on your pinky through index finger.
• With three-digit numbers, hold the hundreds digit on your left hand and the tens digit on your right.
• How to check divisibility:
• 2 -- check if the last digit is even
• 4 -- check if the two-digit number at the end is divisible by 4
• 8 -- check if the last three digits are divisible by 8
• 3 -- check if the sum of the digits is divisible by 3
• 9 -- check if the sum of the digits is divisible by 9
• 6 -- check if the number is divisible by 3 and even
• 5 -- check if the number ends in 0 or 5
• To find the number of years that it will take for your money to double, divide the number 70 by the rate of interest.
• To check computations, add the numbers again in the opposite direction and check for a match. By adding the numbers in a different order, you are less likely to make the same mistake twice.
• You can check sums with the mod sums method.
• Sum the individual digits of each number in the sum.
• Repeat, until you have a single digit for each number.
• Sum the new digits, until you have a single digit, this is the mod sum.
• Compute the mod sum of the answer.
• The mod sums should be the same.
• Mnemonics work by converting incomprehensible data (such as digit sequences) to something more meaningful.
• Phonetic code for memorizing digits:
• 1 -- t or d sound
• 2 -- n sound
• 3 -- m sound
• 4 -- r sound
• 5 -- l sound
• 6 -- j, ch, or sh sound
• 7 -- k or hard g sound
• 8 -- f or v sound
• 9 -- p or b sound
• 0 -- z or s sound
• Aside from improving your ability to memorize long sequences of numbers, mnemonics can be used to store partial results in the middle of a difficult mental calculation.
• Without using mnemonics, the average human memory can hold only about seven or eight digits at a time.
• A principle of probability called the Law of Large Numbers shows that an event with a low probability of occurrence in a small number of trails has a high probability of occurrence in a large number of trials.
• A psychological factor called the confirmation bias is where we notice the hits and ignore the misses in support of our favorite beliefs.
• The confirmation bias explains how conspiracy theories work. People who adhere to a particular conspiracy theory will look for and find little factoids here and there that seem to indicate that it might be true, while ignoring the vast body of evidence that points to another more likely explanation.
• The confirmation bias also helps explain how astrologers, tarot-card readers, and psychics seem so successful as “reading” people.
• People typically invoke the term miracle to describe really unusual events, events whose odds of occurring are a “million to one.”
• This is a short primer on how science works. In our quest to understand how the world works, we need to determine what is real and what is not, what happens by chance and what happens because of some particular predictable cause. The problem we face is that the human brain was designed by evolution to pay attention to the really unusual events and ignore the vast body of data flowing by; as such, thinking statistically and with probabilities does not come naturally. Science, to that extent, does not come naturally. It takes some training and practice.
• The data do not just speak for themselves. Data are filtered through very subjective and biased brains. The self-serving bias dictates that we tend to see ourselves in a more positive light than others see us.
• The blind spot bias is one in which subjects can recognize the existence and influence in others of different cognitive biases, but they fail to see those same biases in themselves.
• How does science deal with such subjective biases? How do we know when a claim is bogus or real? We want to be open-minded enough to accept radical new ideas when they occasionally come along, but we don’t want to be so open-minded that our brains fall out. This problem led us at the Skeptics Society to create an educational tool called the Baloney Detector Kit. In the Baloney Detector Kit, we suggest ten questions to ask when encountering any claim that can help us decide if we are being too open-minded in accepting it or too closed-minded in rejecting it.
• How reliable is the source of the claim?
• Pseudo-scientists have a habit of going well beyond the facts, so when individuals make numerous extraordinary claims, they may be more than just iconoclasts.
• Science is messier than most people realize.
• Does this source often make similar claims?
• What we are looking for here is a pattern of fringe thinking that consistently ignores or distorts data.
• Have the claims been verified by another source?
• Typically pseudo-scientists will make statements that are unverified, or verified by a source within their own belief circle.
• How does the claim fit with what we know about how the world works?
• An extraordinary claim must be placed into a larger context to see how it fits.
• Has anyone gone out of the way to disprove the claim, or has only confirmatory evidence been sought?
• This is the confirmation bias, or the tendency to seek confirmatory evidence and reject or ignore disconfirmatory evidence. The confirmation bias is powerful and pervasive and is almost impossible for any of us to avoid. It is why the methods of science that emphasize checking and rechecking, verification and replication, and especially attempts to falsify a claim are so critical.
• Does the preponderance of evidence converge to the claimant’s conclusion, or a different one?
• The theory of evolution, for example, is proven through a convergence of evidence from a number of independent lines of inquiry.
• Is the claimant employing the accepted rules of reason and tools of research, or have these been abandoned in favor of others that lead to the desired conclusion?
• Has the claimant provided a different explanation for the observed phenomena, or is it strictly a process of denying the existing explanation?
• This is a classic debate strategy -- criticize your opponent and never affirm what you believe in order to avoid criticism.
• If the claimant has proffered a new explanation, does it account for as many phenomena as the old explanation?
• Do the claimants’ personal beliefs and biases drive the conclusions, or vice versa?