Mnemonic as a technique is not a newly conceived idea. Back in 1910, there was a published book with the title, ‘ Magician’s Tricks: How They are Done’ by Henry Hatton and Adrian Plate. This interesting book presents and discusses proofs that the mnemonic idea has already been used by Harry Kellar, a magician who gained popularity during the 1800s. Kellar is believed to have used mnemonic for obtaining the cubes of two-digit numerals at lightning speed. Below is a quote from the book:
While it is not within the province of this book to go into a study of a system of artificial memory, there are certain conjuring tricks frequently presented to the public as “Mental Phenomena,” that have a system of this kind for their groundwork, as, for example, the following which depend, mainly, on numbers, for their effects: “Second Sight” the memorizing of a long list of words at one reacting; the instantaneous raising of any two numbers to the cube or third power…
The Mnemonic system was then advanced by Ron Doerfler. He has included the squares of the said numbers for his extended multiplication mnemonic. Here’s an excerpt from Doerfler’s book:
In this paper, I have extended their scheme to provide squares as well as cubes of two-digit numbers, as these are so important for mental calculation. In addition, I have updated a number of their mnemonic phrases to ones using more modern terms, or to ones I think are an improvement over the original phrases.
So there you go, you need to find out how mnemonic, as confusing as pronunciation it is, can actually become a tool in remembering the square of a number. In fact, you will even realize that this same mnemonic can be a useful tool in solving two a digit number multiplied by another two digit number right in your head. Later we will discuss how you can amaze your friends and family by being able to multiply large numbers at lightning speed.
How Mnemonic Works
According to Wikipedia…
Mnemonic…is any learning technique that aids memory. To improve long term memory, mnemonic systems are used to make memorization easier. They do so by increasing the efficiency of the process of consolidation. This process involves the conversion of short term memory to long term memory.
The Mnemonic Alphabet for Digits
1 t, d, th
6 j, ch, sh, zh, z as in azure, soft g as in genius
7 k, hard c, q, hard g, ing
8 f, v
9 p, b
0 s, z, soft c
To remember the table bear in mind the following cues:
1 = t has one down-stroke
2 = n has two down-strokes
3 = m has three down-strokes
4 = r is the last letter of four
5 = L in Roman notation is fifty
6 = J looks something like a reversed six
7 = k, inverted, is similar to seven
8 = f in script resembles eight
9 = p is similar to a flipped nine
0 = c is the first letter of cipher, which is the word for naught.
Important Points to Remember:
- To be able to use this mnemonic device optimally, you need to keep in mind to consider the sounds of the letters, rather than their alphabet representation.
- Letters w, h and y; along with a, e, i, o, u vowels; can be used to create words with numerical equivalents; chess or chs is equivalent to 60; nail or n-l is equal to 25.
- Letters omitted in the pronunciation (silent) will not be given value. Knife, for example; the silent letter is k. Therefore it will only be n-f that is equal to 28. Another example is lamb or l-m equivalent to 53.
- Double consonants in a word will be considered as a single letter only. An example is mummy. This can be written in mnemonic as m-m and is equal to 33. Another sample is butter: b-t-r equal to 914.
- Double consonants with different articulation or pronunciation will be considered with distinct equivalent. Example is accept: k-s-p-t and you have 7091; bookkeeper: b-k-k-p-r will be equal to 97794.
- Whole numbers do not start with zero, thus the letter “s” can be used at the beginning of the mnemonic to derive a much better phrase for its numerical value.
Basically, you always have to consider using the alphabet as the keyword in deriving the number. Phrases associated with your keywords will then create the mental picture for the squares. Based on that mental image that you can read from left to right, you can already come up with the result.
How To Apply Mnemonic To Get The Squares Magically
Through the book from Hatton and Plate, you will discover that you can create your own mnemonic.
By sounding to one’s self the letters that represent the numbers the word may be easily recalled. It is advisable to prepare such a table for one’s self…
If you are still somewhat confused by the use of mnemonic, you certainly would not want to bother creating your own. Therefore, you can just refer to the list recommended by Doerfler. Here is a link to that useful mnemonics: Mnemonics For Squares And Cubes
Nevertheless, if you need some additional demonstrations, take a little time to go through this:
As an example, you have numbers 83 and 89.
83 Foam (f,m); square of 83 is Shave if happy (6889) (sh,v,f,p)
89 VIP (v,p); square of 89 is Cabinet (7921) (hard c,b,n,t)
To remember 6889 as the square of 83, it is suggested that you associate the two by making a mental picture of “foam” and “shave if happy”. Hatton and Plate found the mental picture method to be more effective for most people. Some experts suggest to use ridiculous images as they create strong impressions in your memory.
Outrageous images? Now don’t let that scare you off and think that math can make you mentally sick.
Memory is developed through associations – not just associations, but through ridiculous associations. The more ridiculous the association is the better. Why? Because it makes a stronger impression upon memory. Logical, “normal” associations will not stand out and hence do not leave strong impressions.
Free memory techniques
So let’s try the unusual thing, you can begin by picturing in your mind a giant foam shaving with the use of a giant shaver if it is happy. You do the same thing with number 89 that is, associate VIP with Cabinet in a mental picture such as a VIP who looks like a cabinet.
Using this technique, when you need to get the square of 89, simply recall VIP in your memory and automatically your memory recalls the cabinet and you get the square quickly.
Now that you know how to use mnemonic to get the square, let’s proceed with the real fun – let’s multiply large numbers.
Multiplying Two-Digit Numbers
With the mnemonic that we learn, let’s use it to multiply 74 and 78.
In algebraic representation, 74 x 78 becomes:
(a+c)(a-c) = a2 – c2
a = is the average of the two numbers
(a+c) = is one of the numbers
(a-c) = is the other number
So the problem becomes
74 x 78 = 762 – 22
“How did that happen?”, you may ask.
Add the two numbers ( 74+78) then divide it by two since there are two numbers involved. That is how you get 76. For some people, it is much easier to do it this way, 78-74 is 4. 4 divided by 2 is 2. 2, which is then added to 74, giving you 76.
The final math problem becomes as simple as…
762 – 22
Now, we need to recall the mnemonics associated with these numbers to get the squares.
76 Cage (k, soft g) = Lock cage(5776) (l, hard c, hard c, soft g)
2 Noah (n) = 4 (this is easy enough that it doesn’t even require mnemonic)
762 – 22 = 5776 – 4
As soon as we get the easy squaring mentally, we are now close to the solution of the problem.
The above illustration shows that by recalling the mental picture of a lock cage, we get the value of 5776 immediately minus 4 which is the square of 2. We arrived at the final answer just as quickly as we glanced at the number problem.
Some Problem With Less Convenient Multiplier
How can we multiply 32 x 35?
Now this one is a different case because 35 – 32 equals 3 and 3 is not divisible by 2. So this requires a slightly different conversion process.
Since we need to get the average of the two numbers and that average should be a positive integer, we need to reduce the multiplier by one and add the other number in the equation.
Therefore, the problem can be translated into:
(32 x 34) + 32 = (332 – 12 ) + 32
= (332 – 12 ) + 32
Then in mnemonic representation:
33 My Home (m,m) = Do save up (1089) (d,s,v,p)
and the square of 1 is 1.
(1089 – 1) + 32 = 1120
1089 is the square of 33
1 is the square of 1
32 is the number you need to add since we reduce the multiplier by one.
1120 is the final answer.
Confusing? It would be a lie to say no but, with practice, you are sure to perfect your math mnemonic. Math can be easy and you can multiply large numbers mentally if you put even just a little part of your heart and mind to it.