# Secrets of Mental Math: Pro Methods Explained ## TL;DR The real secrets of mental math are not tricks - they are a few structural ideas (complements, the distributive property, and built-in checks) that experts apply flexibly. This article explains the pro-level methods - complement subtraction, the close-together multiplication method, fast divisibility tests, and casting out nines - each with a worked example and the reasoning that makes it work, so you can use them on any numbers. ## The Core Idea Watch a stage "mathemagician" like Arthur Benjamin compute a four-digit square in seconds and it looks like memory. It is not. The performers who wrote the books on this - Benjamin and Michael Shermer's [_Secrets of Mental Math_](https://www.penguinrandomhouse.com/books/11339/secrets-of-mental-math-by-arthur-benjamin-and-michael-shermer-with-a-foreword-by-bill-nye/) being the best known - all teach the same thing: speed comes from a small set of structural moves, applied to whatever numbers arrive. The genuine secret is that there is no secret list to memorise. There are about four ideas - measure distance with complements, break products apart with the distributive property, read divisibility off digits, and check work with remainders - and everything else is those four, reshaped. The performers are fast because they understand the structure deeply enough to pick the right move instantly, not because they have a bigger trick collection. That is the difference worth holding onto. A trick is a recipe for one shape of problem. A structural idea is a tool that adapts to every shape. The methods below are presented as the second kind, because that is what makes them transfer beyond the party-piece numbers. ## Secret 1: Subtract With Complements So what is the actual secret behind fast subtraction from round numbers? The fastest subtraction from a power of ten uses **complements** - how far a number sits from 1000. The complement of a number is what you add to reach the next round figure. To compute 1000−638, do not borrow across three zeros. Read the complement directly: subtract each digit of 638 from 999, then the last from 1010. 9−6=3, 9−3=6, 10−8=2; The answer is 362. Check it: 638 + 362 = 1000. The method works because 1000 = 999 + 1, and subtracting from all-nines never borrows - each digit is independent. The final +1 is why the last digit comes from 1010 rather than 999. Complements turn the worst subtractions (round numbers full of zeros) into the easiest, because they replace a borrow chain with a digit-by-digit read. ## Secret 2: The Close-Together Multiplication Method Multiplying two numbers near each other - and especially near a round base - is where the distributive property earns its keep. This is the method behind a lot of "lightning" two-digit multiplication. Take 97 × 98. Both are near 100. Their distances below 100 are 3 and 2. Subtract one number's distance from the other number, then append the product of the distances: 100−(3+2)=95 (the leading part) 3 × 2 = 6 (the trailing part, padded to two digits: 06) 1. **The answer is 9506.** This is not arbitrary. Knowing the algebra is what tells you to pad the 6 to 06 and how to handle bases other than 100 - the trick alone leaves you guessing on the edge cases. ## Secret 3: Read Divisibility Off the Digits How can you tell if a big number is divisible by 3 or 9 without dividing? Experts never trial-divide to check a factor. They read divisibility from the digits, because each test reflects how our base-ten system carries remainders. - **By 3:** the digit sum is divisible by 3. For 4713: 4+7+1+3=15, divisible by 3, so 4713 is too. - **By 9:** the digit sum is divisible by 9. 1515 is not, so 4713 is not divisible by 9. - **By 11:** the alternating digit sum (add, subtract, add) is divisible by 11. For 4713: 4−7+1−3=−5, not divisible, so no. ## Secret 4: Check Your Work by Casting Out Nines How do mental calculators check an answer without redoing the whole sum? The pros do not just calculate fast - they catch their own errors with a remainder check called **casting out nines**. It uses the digit-sum idea from Secret 3 as a verification tool. Suppose you computed 487 + 256 = 743. Take the digit sum of each number, reduced to a single digit: 487→4+8+7=19→1+9=10 256→2+5+6=13→1+3=4 (expected check value) 743→7+4+3=14→5 **The check confirms 743.** It works because a number and its digit sum leave the same remainder on division by 9 - so the remainders must balance across a correct calculation. ## Practice Set Work each with the method named, then check below. 1. 1000−247 (complements) 2. 96×97 (close-together method) 3. Is 5832 divisible by 3? By 9? (digit-sum tests) 4. Check whether 58×7=406 (casting out nines) 5. 752 (the ending-in-5 square) **Answers.** 1. 753 2. 9312 3. Yes to both - digit sum 18. 4. Passes: 58→4, 7→7 5. 5625. ## Common Mistakes With Mental Math ### Mistake 1: Collecting tricks instead of understanding them ### Mistake 2: Trusting a fast answer with no check ### Mistake 3: Misreading the complement's last digit ## How Bhanzu Approaches the Secrets of Mental Math Bhanzu teaches these methods as what they actually are - applications of a few structural ideas - rather than as a trophy cabinet of tricks. A student meets the close-together method through the distributive property, the divisibility tests through how place value carries remainders, and the complement method through the all-nines insight. That framing is intentional: the structure is what transfers. ## Conclusion - **The real secrets of mental math are a few structural ideas**, not a long list of tricks - complements, the distributive property, digit-based divisibility, and remainder checks. - **Complements** turn subtraction from round numbers into a digit-by-digit read with no borrowing. - **The close-together method** is the distributive property in disguise. - **Divisibility tests and casting out nines** both flow from how place value carries remainders. - **Speed plus a check is expertise** - a fast answer with no verification is just fast and risky.