Last updated: around August 2005

Mathematics is one of my specialties, especially general arithmatic and algebra. I find shortcuts in almost anything involving general arithmatic. Multiplication is my favorite things. In fact, I can multiply 3 or more numbers at once without doing them in pairs. Don't believe it? Here's the proof and how you can do it. In elementary school, almost everyone is taught the old, slow ways. I've noticed many potential shortcuts that could be used never bothered to be taught in school, possibly even college. I see multiplication in a somewhat different light than almost anyone else does. The link above explains how I see multiplication.

Some of the shortcuts I've found are so obvious, I'm surprised it isn't taught. The copy trick, the most powerful one I have, seems more like a rocket engine to a small car engine. I almost never see any references to this technique, and it has so many major advantages. There are, however, less obvious shortcuts available as well that I don't expect schools to teach. The end-five shortcut and transfer shortcuts are two examples.

Multiplication is my favorite form of arithmatic. The two big reasons is that you get big numbers with it, both in the answer and in the stash of scrap piled on top. Multiplying many numbers at once allows for even larger answers or scrap (even double digits). Yet, most don't like multiplication and through the years, I've found many handy shortcuts that I've used since, both in my high-speed mental math skills and with old-fashioned pencil and paper.

Because I see multiplication in a somewhat different light than most everyone else, I need to explain how I see it. The full details are at the beginning of what is covered in the link above.

543210 287921 × 7593 ------

The numbers on the top of this example above are what I call "indexes". A combination of two or more indexes is called a "combo". The indexes of combos are arranged from the top row of the multiplication question to the bottom row in order. When you multiply, you start with the 0-0 combo (which is doing 1×3). The next step is the 1-0 combo (which is doing 2×3). You do the 2-0, 3-0, 4-0, and 5-0 combos after that. Once done with the 5-0 combo, you do the 0-1 combo (which is 1×9). The 1-1, 2-1, 3-1, etc. are done after that. When 5-1 is done, it's the 0-2 combo and when the 5-2 combo is done, it's the 0-3 combo. Once the 5-3 combo is done, you add.

The number of combos done in a multiplication question is easily figured out. Just count the number of digits in each row and multiply these numbers. The example above would have 24 combos (from 6 on the top and 4 on the bottom as 6×4 is 24). The shortcuts I have help reduce the number of neccessary combos as the more combos you have, the longer it takes to process. The exception is the copy trick where combos are skipped rather than omitted.

Combos can have more than two objects in them, but this is only used when multiplying a bunch of numbers at once and is not neccessary for this. They look like this if you're curious: 2-3-1 (for three numbers being multiplied at once).

The biggest advantage of combos is helping to know where you left off if you're doing some long, complex question and got distracted or had to do something. If you stopped on the third row with 6 numbers showing for that third row, you can easily figure out where you left off. The third row is the x-2 combo. X, the unknown is found by subtracting how many numbers you are in from the right on the current row (6 in this case) by the known value which gives you 4. This means, from the example above, you'd be doing the 4-2 combo, which is 8×5. They start on zero rather than one as they are based on the power of 10. 10^0 is one. 10^3, where the combo adds up to 3, is 1000. For further details with combos (and for when more than 2 indexes are in a combo), see the link at the top.

Type: Speed-up Reliability: - Extremely high Ease of learning: - Extremely easy Time saving: - Moderate gain Usefulness: - Extremely useful Difficulty: - Very easy Overall: - Excellent

This interesting shortcut is not recognized by many, however, it applies such basic concepts. Also, you could learn to avoid "zero confusion" and "decimal confusion" as well. Consider multiplying 18 million and 300,000. With all those zeros at the end, it can get confusing.

To explain the benefit of this shortcut, let's take this math question as an example:

180 265 ---

Once worked out the old, slow way, you get this:

1 4 4 180 265 ----- 1 900 10800 36000 ----- 47700

So, why bother having the zero at the end when you could just "attach" them at the end of your result? What I normally do is write a number off to the side to indicate how many dropped digits are involved. Worked out, using the shortcut, you get this:

54 265 18 1 ---- 2120 2650 ---- 47700

Remember, you can work it in any order due to the commutative property of multiplication. Note how you don't have "zero confusion".

Who'd want to multiply this with all those extra zeros? I wouldn't.

1 1 4800000 220000 ------------- 11 96000000000 960000000000 ------------- 1056000000000

Sometimes, it's rather pointless to write all those zeros. Putting this shortcut to use, you get something like this:

1 1 48 5 22 4 ---- 11 96 960 ---- 1056000000000

You know that a thousand thousands is a million, right? A thousand, 1000, has three zeros at the end. A million, 1,000,000, has six zeros at the end. 3+3, 3 being from each thousand, gives 6. The same concept applies to this shortcut as well. You simply add the number of dropped zeros together and when done, the number of dropped zeros is added to the end of the shorter question.

Interestingly enough, this shortcut also works with decimals in the multiplication question. Decimals can get rather confusing to work with in multiplication and are often left out as the dot is so small. Indeed, you can apply this shortcut with decimals as well. What if you needed to multiply this:

163.8 0.042 -----

It looks a little tricky, however, if you had it like this:

1638 -1 42 -3 ----

It's a lot easier to work with. The negative numbers indicate decimals. However, unlike counting zeros, you count the number of digits to the right of the decimal instead. 0.042 has three digits after the decimal. Since a decimal is used, it's negative. When worked out:

213 1 1 1638 -1 42 -3 ----- 3276 65520 ----- 68796

68,796 is not right though. To figure out where you place the decimal, you need to add the two negative numbers (-1) + (-3) = (-4). -4, being negative, means decimals. The 4 means 4 places. To place the decimal, from the far right of the result, count left 4 places and place a dot. This gives you 6.8796. As a tip, the negative number indicates how many digits are after the decimal.

You can combine the zero elimination and decimal techniques. Take multiplying this:

2400 0.8 ----

You'd think of dropping the zeros and putting a positive 2 off the side and getting rid of the decimal placing a negative 1 to the side. You are correct in this case. It's the result on how to work with the 2 and the -1 that can get confusing. You'd have this, after being worked out most of the way:

3 24 2 8 -1 --- 192

However, what do you do with the 2 and -1? When you have a positive value and negative value together, just find the difference and give the sign of the larger value. Since 2 is larger and it is positive and 2-1 is 1, you'd have a positive 1. This gives 1 as the value and 1, being positive, means you add an extra zero at the end giving the final result of 1920. A different instance is if you had -5 and a 2 off the side. The difference (5 minus 2) is three and since 5 is bigger and it's negative, you have -3. Negative means you place a decimal. -3 means 3 digits are after the decimal.

The graphic at the top of this page shows a more extreme example of this (a negative 6 and a positive 2).

The shortcut works due to powers of 10 and their related combos. 10^0 is one. 10^1 is ten. 10^2 is 100. 10^3 is 1000, 10^4 is 10,000, and so on. 10^-1 is 0.1, a tenth. 10^-2 is 0.01, 10^-3 is 0.001, and so on. Note the relationship between the number set to the side and the 10^x value. 64 is 64 × 10^0. 4,800,000 is 48 × 10^5 as there are 5 zeros at the end. 0.042 is 42 × 10^-3 as there are three digits after the decimal. Note a relationship? A tenth of a hundred, noted as 0.1 × 100, is ten, 10. This is basically 10^-1 × 10^2 which gives 10^1, thus the 1 extra zero. This is why this shortcut works as it does. When multiplying a bunch of numbers at once, you have more to add and subtract, but the principal is still the same.

Type: Speed-up and shortcut Reliability: - Medium-high Ease of learning: - Easy Time saving: - Large gain Usefulness: - Very useful Difficulty: - Easy-medium Overall: - Great

This is a shortcut, most useful in high-speed mental calculations. It's also useful for reducing the number of combos needed for multiplying the question out on paper and fewer combos means less chance of error and it goes faster. It involves multiplication questions ending in a 5 and turning the 5 into a zero so the zero and decimal shortcut can be applied. This makes that an obvious limitation, but the other limitation is that one of the other numbers must be even.

To get an idea on how this shortcut is useful, let's refer to the question first mentioned:

1 4 4 180 265 ----- 1 900 10800 36000 ----- 47700

Even with the zero and decimal shortcut, you still have 6 combos (9 without).

However, if you multiply the 265 by 2 and divide the 18 by 2, you can seriously reduce the number of combos (from 6 to just 2). A trick to multiplying by 2 is to ignore the 5 and consider the first two. 26×2 is 52 and since it ends in 5, add one more to get 53. Always put a 1 to the side from this (unless other zeros were dropped or the decimal was ignored where you'd add 1 to this). 18÷2 is 9. This makes the question become 53×9 with two ones to the side. Multiplying that is much easier than 265×18 as there's just 2 combos instead of 6, and can be even twice as fast if used properly. Worked out using the new method (and using the zero and decimal shortcut along with it).

2 53 1 9 1 --- 47700

After adding the extra zeros, you get the same exact answer too!

Type: Extreme shortcut and speed-up Reliability: - Extremely high Ease of learning: - Extremely easy Time saving: - Huge gain Usefulness: - Extremely useful Difficulty: - Effortless Overall: - Perfect

The copy shortcut (often called the "copy trick") is the most powerful and the most useful shortcut for all multiplication. This shortcut only applies to multiplication on paper and not mentally. Who'd want to multiply this tough-looking and long question:

51754896381726579139684290473294033814 39571899284652812731224891423844796588 --------------------------------------

I don't think anyone would without using a calculator. However, with the copy trick, multiplying something this complicated is much easier and far faster than it looks, even on paper. In fact, the more digits, the easier the question gets because the more effective the copy trick becomes. This could be done nearly ten times faster, likely even more. The copy trick has several major advantages:

- It reduces the chance of making mistakes on paper.
- Large, complex questions of many digits become much easier.
- Multiplication on paper could be done 10+ times faster, getting faster with more digits.

In fact, you only need to multiply once. When that digit is repeated again on the bottom, just copy what you had from before!

In the old method, when you multiplied this:

376 313 ---

you'd multiply the three in the bottom ones' place normally, maybe copy the one or think of it as 4×1 then 1×1, and 2×1, then you'd multiply by the three again having 4×3 and so on. Why bother doing this when you've already multiplied by three and multiplying by one just keeps the original number? Rather than having to do 9 combos (possibly 6), you actually only have to do 3. 376×3 has the exact same beginning as 376×300, with just two extra zeros on the end. Worked out the old way, on paper, you would have something like this:

21 21 376 313 ------ 1 1128 3760 112800 ------ 117688

If you worked it out with the copy trick put to use for the example question, you'd have this instead (notes follow after //)

21 376 313 ------ 1 1128 // multiplied normally 3760 // multiplying by 1 means copy the top number 112800 // you're multiplying by 3 again - copy it! ------ 117688

If you, on the second step, forgot to carry the 1 or the 2 or forgot to add it in, you make a mistake this way. If you copy the first result, you would avoid the chance of running into this mistake. You could save about 5 seconds on paper when working this out (for those who are fast at it, more around 15 seconds for those who are just learning multiplication), a huge reduction.

However, sometimes, the copy trick doesn't appear to be obvious right away as in this case.

744 627 ---

In school, you learned that you can multiply the numbers in any order (commutative property of multiplication). Rather than multiplying the 6, 2, and 7 individually, make use of the copy trick by multiplying the 7, and two 4's copying one of the results. After doing this, you have:

627 744 ---

You multiply the 4 normally, copy the result into the second line, then multiply the 7 normally. It's much faster than the previous, unflipped version.

Type: Speed-up Reliability: - Medium-low Ease of learning: - Easy-medium Time saving: - Small gain** Usefulness: - Very useful Difficulty: - Medium-hard Overall: - Good

The transfer shortcut is not all that useful on paper, but it has proven to be quite handy for mental math. It's not suited toward beginners as it can be tricky to make use of the shortcut. This shortcut involves "transferring" factors from one number to another. When you multiply a given number by a second number and divide this result by the second number, you get your original number back which is how this shortcut works. The end-five shortcut applies it with 2's. This shortcut covers everything else, including odd ones like 13 if you notice it. I only recommend using this shortcut when doing mental math or when something is easily and quickly spotted when doing it on paper and most importantly when the number of combos can be reduced (getting the copy trick put to use is also when it comes in handy, my most common use of this shortcut on paper). This shortcut is only useful for small numbers, 3 digits and fewer.

The old method is about as fast as after applying the shortcut with the only exception being if something is noticed right away or many combos can be reduced. Using the old method, you'd have something like this:

22 12 345 154 ----- 111 1380 17250 34500 ----- 53130

In this case, only the copy trick can be used from the one which does help to some extent. But, it can be reduced even more.

The copy trick can only be used with the 1. One of the main things of the transfer shortcut is to eliminate combos if possible. You could try to get rid of the 154 to a double-digit number. Since it's even, you can divide by two. If you divide by two, you have to multiply by two on the other number. 154÷2 gives 77 and 345×2 (think of it as 340×2 and adding 10 to it) gives 690. Drop the zero at the end (from the zero and decimal shortcut) and put a one to the side. Now you have 69×77. Instead of 9 combos (6 due to the copy trick), you have 4. With the 7's repeated on the bottom, it's actually 2 combos from the copy trick. As a result, you'd get this instead:

6 69 1 77 ---- 11 483 4830 ---- 53130

It's a completely different question, but you get the exact same answer (don't forget to apply the finalization of the zero and decimal shortcut).

Sometimes, the transfer shortcut isn't readily noticed. Consider this example:

143 264 ---

Right away, you might be thinking of getting rid of the 143 to getting fewer digits, but 143 doesn't look easy to make use of. 264 is even so you might think of dividing 2 on that and multiplying by 2 on the top, but this gives 286×132 which isn't of any help (except maybe for the next shortcut mentioned). Again, it isn't readily noticed. However, if you know the standards for when a number is divisible by something else, it could help a lot for things like this. Even numbers must end in a 2, 4, 6, 8, or a 0. Multiples of 3 have their digits add up to a multiple of 3. However, only one of these numbers is a multiple of 3 (2+8+6 = 16 and 1+3+2 = 6 - 16 is not a multiple of 3, but 6 is). You could transfer 3 from 132 into 286 which gives 44 and 858. 858×44 is much easier to work with than 143×264. Not only can you use the copy trick, you have fewer combos (6 instead of 9 (3 after the copy trick)). This is why this shortcut really isn't that handy as it can take a while to notice. For best results, use the fewest transfers possible, especially to reduce the number of digits on the shortest number.

Type: Speed-up Reliability: - Low Ease of learning: - Easy-medium Time saving: - Large gain Usefulness: - Very useful Difficulty: - Medium-hard Overall: - Good

The baby-digit shortcut involves the transfer shortcut to get baby digits. A baby digit is a 0, 1, or sometimes a 2 as they're the easiest to work with. This shortcut is most useful when doing mental calculations, but also comes in handy for doing it on paper. Again, like the transfer shortcut, use it only when something is readily and quickly noticed for optimal speed. This shortcut simply involves transfering numbers to get a simpler question to work with (from baby digits).

122 444 ---

Although this may look easy to work with, the copy trick is readily noticable and would be the first (and recommended) thing to do without bothering to apply this shortcut. We'll use this to explain how this shortcut works.

In this case, you can covert the 4 4's into 4 1's by dividing 444 by 4 (giving 111) and multiplying 122 by 4 (giving 488). After working it out, you have this.

488 111 ----- 121 488 4880 48800 ----- 54168

Although not all that much different, the point of this shortcut is to get baby digits, especially on the bottom (the second (or further) index in combos).

Type: Speed-up and shortcut Reliability: - Extremely high Ease of learning: - Extremely easy Time saving: - Large gain Usefulness: - Extremely useful Difficulty: - Effortless Overall: - Superior

Many actually know this shortcut, but there are some who don't. This shortcut involves having a zero in the center on the bottom. It is most useful on paper and has little use for mental math.

The old method of doing this is something like this:

53 74 695 608 ------ 1 5560 0000 417000 ------ 422560

Having that line of zeros is actually not neccessary as, if you add zero to something, well, you still have your original value. The shortcut simply is dumping this extra row of zeros. After applying the end-five shortcut (When the zero-in-the-center shortcut can be used, only use the end-five shortcut when the digit to the left is even.), and working it out, you'd get something like this:

12 13 139 1 304 ----- 1 556 41700 ----- 422560

Again, you got exactly the same answer.

Type: Mental math shortcut Reliability: - Medium-high Ease of learning: - Medium Time saving: - Small gain Usefulness: - Somewhat useful Difficulty: - Medium Overall: - Good

This shortcut involves using a pair of double-digit numbers. This shortcut is meant for mental math only but does work on paper as well. Since squaring numbers (taking a number times itself) is quite common (and for using the next shortcut after this one), this is a handy shortcut to know.

The old, slow way is working it out normally, applying any other shortcuts available if possible. If you multiplied 73×73 normally, you'd get this:

2 73 73 ---- 219 5110 ---- 5329

I discovered this shortcut when dealing repetitively with squaring numbers and I happened to notice a pattern. Here are the squares of the first 50 numbers:

Num | SQ | Num | SQ | Num | SQ | Num | SQ | Num | SQ |

1 | 1 | 11 | 121 | 21 | 441 | 31 | 961 | 41 | 1681 |

2 | 4 | 12 | 144 | 22 | 484 | 32 | 1024 | 42 | 1764 |

3 | 9 | 13 | 169 | 23 | 529 | 33 | 1089 | 43 | 1849 |

4 | 16 | 14 | 196 | 24 | 576 | 34 | 1156 | 44 | 1936 |

5 | 25 | 15 | 225 | 25 | 625 | 35 | 1225 | 45 | 2025 |

6 | 36 | 16 | 256 | 26 | 676 | 36 | 1296 | 46 | 2116 |

7 | 49 | 17 | 289 | 27 | 729 | 37 | 1369 | 47 | 2209 |

8 | 64 | 18 | 324 | 28 | 784 | 38 | 1444 | 48 | 2304 |

9 | 81 | 19 | 361 | 29 | 841 | 39 | 1521 | 49 | 2401 |

10 | 100 | 20 | 400 | 30 | 900 | 40 | 1600 | 50 | 2500 |

Look carefully at the table. Ever notice that everything ending in 1 like 31 and 11 have the answer ending in a 1 and everything ending in a 4 ends in a 6? Also, note the pattern in the square value's result. It goes 1, 4, 9, 6, 5, 6, 9, 4, 1, 0 and continually repeats? Then, take note of the tens digit. Once the ones digit of the base (the number being squared) passes 3 or 7, the tens digit counts by one more than previous and keeps counting by that same number until the ones digit passes by a 3 or a 7 again? Then, look more closely at the square values. Each number below adds two more than what was added previously. From 1 to 4 is 3, from 4 to 9 is 5, from 9 to 16 is 7, from 16 to 25 is 9, and so on. This is how I discovered the shortcut.

So, how do you apply it? Applying the shortcut is easy to do. Here's the process:

- Round your number (the base number) to the nearest ten.
- Square the rounded number.
- Double the rounded number then:
- If the base number was rounded down, multiply the difference between the rounded and base numbers afterwards. Then add this result.
- If the base number was rounded up, multiply the difference between the rounded and base numbers afterwards. Then subtract this result.

- Add the square of the difference between the base number and the rounded number.

To apply this process, let's take 12 as an example. 12 rounds to 10. The square of 10 is 100. Double the rounded number is 20 and since I rounded down, I multiply this by the difference, or 2. This gives 40. I add the result which gives 140. Adding the square of the difference (which is 4) gives the final result of 144. 12

Let's take 47 as another example. 47 rounds to 50. The square of 50 is 2500. Double 50 is 100 and since I rounded up, I multiply this by the difference (the difference is 3) which gives 300. Subtracting this gives 2200. Adding the square of the difference, 9, gives 2209, which, also is on that chart.

For 73, the original case we used, 73 rounds to 70. The square of 70 is 4900. Double 70 is 140 and since I rounded down, I multiply this difference by 3 to get 420. Adding this gives 5320. Adding in the square of the difference, 9, gives the final result of 5329 and look, they match!

Of course, if you wanted, you could round 47 down to 40 which gives 1600 as the square. Doulbing it gives 80 and 80 times the difference of 7 is 560. This gives 2160. I then add the square of 7, or 49, and it, too, gives me 2209. The version I've given is the recommended method, unless you are more comfortable with adding than subtracting.

Type: Mental math shortcut Reliability: - High Ease of learning: - Hard Time saving: - Small gain Usefulness: - Moderately useful Difficulty: - Medium-hard Overall: - Good

The shortcut is almost exactly the same thing as the same-number shortcut, however, it involves a much broader range of possibilities, but with an extra step. Instead of being restricted to numbers being identical, the numbers used for this shortcut can be difference, but they must have an equal distance from the mid-point (a difference that is even). This is the most powerful mental math shortcut, but doesn't have as much use on paper, although it still works on paper. This is my favorite shortcut for mental math as I use it very often.

Let's multiply 28×34. You can't really make much use of the other shortcuts with this one. Worked out the old way:

2 3 28 34 --- 112 840 --- 952

The discovery of this shortcut came from studying more patterns. Take 8×8. This is 64. 9×7, one more and one less on each, is 63. 10×6 is 60. 11×5 is 55, 12×4 is 48, 13×3 is 39, 14×2 is 28, 15×1 is 15, and 16×0 is 0. Note the patterns again. The first number is going up one step at a time and the second is going down one step at a time. The mid-point between the two numbers being multiplied is always 8. Then, take a look at the answers. You have 64, 63, 60, 55, 48, 39, 28, 15, then 0. The differences from the square of the mid-point, 8×8 (or 64) is 1, 4, 9, 16, 25, 36, 49, and 64. These are the square numbers in counting form. From this, and since I already knew the same-number shortcut at the time, this new shortcut was discovered very shortly after. However, unlike adding the square of the difference, it's always subtracted instead.

To use the shortcut, you need to know the difference between the two numbers being multiplied. The difference between 34 and 28 is 6. 6 is even so this shortcut can be used for this. The mid-point is half of the difference, or 3. This gives 31 as the mid-point. From the 31, you apply the same-number shortcut. 31 rounds to 30. 30 squared is 900. Double 30 is 60 and multiplying this by the difference of 1 remains at 60 giving 960 and adding the square of the difference, 1, gives 961. However, from 961, how do you get 952? The difference from the mid-point is 3. From my discovery of even more patterns, you have a 9 involved. When doing the difference between mid-points, you subtract the square of the difference. 961-9 just happens to be 952.

Let's consider one more example. What's 73×63? The difference is 10 and the mid-point is half the distance from these giving 68. 68 rounds to 70, 70 squared is 4900, double 70 is 140 and the difference from the rounded number is 2 so I have 280. 4900-280 gives 4620. Adding the square of the difference, 4, gives 4624. The mid-point's difference from the numbers is 5 and the square of 5 is 25. I subtract 25 from 4624 to get 4599.

Footnotes:

* This has a small gain, but it's also based almost entirely on your skill. An expert would get about 8 stars, but a newbie will have 1 or 2 stars.

** This speed-up requires great skill. Those who use it wisely can get 8 or 9 stars, but, even someone who is intermediate on math would get only 2 stars! This is indeterminate for this reason.

*** This is of almost no use from beginners to those who are advanced in general arithmatic, however, masters would find this of high use, about 7 stars. The limitation is your skill and it's very strict, thus reliability gets a very low rating.

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