If you look back at the 'Person 0 to 9' article, you see that I used letters of the alphabet to mean digits from 0 to 9. But I also express double digit numbers by using letters as syllables. Eg. In the "People 00 to 99" article. The syllable approach can empower my ability to do maths with accuracy. See below.

I have played with building syllables out of consonant and vowel to see if there is a nice way to say a sort of word that contains the numbers that need maths done to them followed by the correct answer. I like the idea that consonants can represent one column of numbers and vowels can represent another column of vowels. Since I really like the 00 - 99 system that you have seen in articles such as the Double Digit Acrostics article, I have tried to express maths using that consonant and vowel approach. However, I keep a note of the vowel sounds in the list above in case they become handy one day.

There is a method of adding up that is a system of fives. I show it below. The 00 - 99 system from the earlier articles changes consonant every time 5 is counted. Eg. It starts with BO, BI, BA, BE, BU and then increments the consonant to CH: CO, CI, CA, CE CU. So it is a bit like an abacus where the second number column goes up by 1 and the first column resets its beads to 0. (There is an abacus like that known as a soruban.)

The 1000 Women and 1000 Male cartoon images of people involve logarithm data. Logarithms were very important before computers made difficult calculations easy. They are very weird. For instance, multiplication is achieved by a process only slightly more complicated than a basic addition calculation. Int this way, approximate answers can be calculated that are of sufficient accuracy to be useful to navigators or engineers.

Adding Syllables (System of Fives)

Another way of thinking about summing differently is to apply the syllables form the People 00-99 article:

[O, I, A, E and U are pronounced: 'O like 'pot', I like 'meet' [the 'ee'], A like 'far' [the 'ah' sound], E like 'men', U like 'soon'.]

Let's express 37+39 as a pair of syllables from that article: 37 is KA and 39 is KU; so the word I make is KAKU. If I have memorised the table above then the K and the K mean 35 and 35 which is 70; this is easy to calculate because numbers ending in 5 are easy to add together. The A and U can be looked up in the second table above and, rote, means 6. It is easy to add the 6 to the 70. The answer is 76.

So a small amount of rote learning can make a sum be expressed in a way that is easier to solve; and the syllable pair is also an alternative way of holding an intermediate result in your short term memory if the 37+39 is just a part of a bigger calculation.

I think there is less chance of number errors when syllables are used.

I could also develop a table to state what any two letters add up to. Eg. KN is 85 because it is 35+50 (K is 35, N is 50). If I have a two syllable word like KAKU then the KK would, by rote, give me 70 (K 35 plus K 35) and AU is +5 and +1 to give me 76. KN and NK are going to have the same table answer of 85 because of the similarity in adding 35+50 or 50+35.

Subtracting numbers

A similar table to the one above can be used for subtraction where the answer is going to be a positive number.

37 - 25 is KA -HO or KAHO.

K is 35 according to the maths table above. So the K - H is 35 - 25 = 10. The 7 - 5 part of the subtraction involves the A - O but the K is made of 30 + 5 ; so I need to ask myself if the H is also made of something plus 5. It is: 20 + 5. In the situation where both consonants represent a number ending in 5, I can treat the A - O as 2 . Also, if both consonants involve something + 0 then I can treat A - O as 2. Eg. L - J involves 40 - 30 and there are no 5s; so A - O = 2. But in other cases, you have to follow extra steps.

I can calculate 37 - 25 as KAHO where K and H results in 10; and A and O result in 2. So the answer to 37 - 25 is 10 + 2 = 12 .

I can make a table for subtraction:

Now let's do 27 - 25 = 2 . HAHO is the 27 - 25 where the H ends in 0 and the other H ends in 0; so there is no extra step needed with the units. H - H is 0. A - O is 2. So the answer is 2.

Let's do another example. 78 - 64 is TERU. The T is 75 and ends in 5. The R is 60 and ends in 0. So I know that the units will need an extra step. T - R is 75 - 60 = 15 . E - U = -1; So the 15 has 1 subtracted to give the answer 14

Let's do another one. 73 - 56 is SHEPI. The SH is 70 and ends in 0. The P is 55 and ends in 5 .

SH - P is 70 - 55 = 15 . E - I = 2; so the 15 has 2 added to give the answer 17

Another example: 36 - 8 is KICHE. It is 35+1 minus 5+3. The 1 -3 is 'Minus 2' according to the subtraction table. The consonant maths is 35 - 5 = 30. Now subtract the 2 to get 28.

46 - 18 is MIFE. It is 45+1 minus 15+3. The 45 and the 15 both end in 5. The 1 -3 is 'Minus 2' according to the subtraction table. The consonant maths is 45 - 15 = 30. Now subtract the 2 to get 28.

46 - 13 is MIDE. It is 45+1 minus 10+3. The 1 - 3 is 'Minus 2' according to the subtraction table. 45 - 10 = 35. Now subtract 2 to get 33.

So the maths is kept simple and you can hold it in your head more easily.

I would think about using different vowel sounds for pronouncing the O I A E and U vowels of the subtraction table above: Oe, Ie, Ay, Ee, U [as in 'up'].

Adding up 3 digits

If you pause when you see two digits that need to be summed then maybe you could train yourself not to pause but still be accurate. Eg. 9 + 8 = 17 . It can be learned rote ideally. At school, you might have learned to work it out as steps but you could learn it rote by using flash cards; and start to use rote memory when you are confident that you are remembering well,

How many 3 digits are there when faced with adding three digits together? 1+2+1 will be similar to 1+1+2; so that helps reduce the number of three digit numbers to learn rote the total of.

There are about 220 results that, if rote learned, would speed up the summing of three digits:

More about Multiplication

When I mentioned the big multiplication system, above, to someone in the mental maths arena, he said that the maths can be done mentally without needing mnemonics. I know that; but some people can visualise numbers better than others.

When a 2 digit number is multiplied by a 2 digit number, some people do a lot of adding digits of a column of the answer as part of their technique. By using the 'System of Fives' above, I want to demonstrate something like that approach but also read the section below about multiplying by using squares of numbers!:

You could rote learn that 8 times 2, instead of being 16, is FI from the articles about expressing numbers 00 to 99: FI is 16. You could then work with the 16 by using the 'System of Fives' approach from earlier in this article. But if the 8 x 2 is the rightmost part of a multiplication question then the 6 of 16 will get written down and we just want to work with the 1 expressed as 01 so that it can be worked with: syllable BI from the 00 to 99 articles. So sometimes we want 8 x 2 to be FI (when expressed in the Units column of the diagram above) and sometimes we want it to be BI (when expressing the carrier in the Tens column).

With the multiplication expressed above:

The B cell times D cell is the 8 times 2 part which will be expressed as BI once the 6 has been written down as part of the final answer [using the logic of the previous paragraph].

The A cell times D cell is 7 times 2 = 14. Express the two digits of that answer as a consonant and vowel of the Person 00 to 99 system: DU

The C cell times B cell is 9 times 8 = 72. Express the two digits of that answer as a consonant and vowel: SHA

The A cell times C cell is 7 times 9 = 63. Express the two digits of that answer as a consonant and vowel: RE

With the B times D answer, we can write the 6 of 16 and work with the left side 1 expressed as BI .

BIDU is the addition of BI and DU and I can use the 'Adding Syllables' method to get 10 + 5 = 15 .

I now have to express the 15 as a consonant and vowel of the 00 to 99 system: FO .

Now the SHA needs to be added to FO: FOSHA which is 15+70 plus 2 = 87 .

I can write the 7 as part of the answer. The 8 needs expressing as CHE .

Then the RE can be added: CHERE is 5+60 plus 5 + 1 = 71 . So the final answer is 71 7 6 : 7,176

Multiplication Based on People Attributes

There is an earlier article about people 00 to 99. I can use its syllable system to work out part of the answer to multiplication questions -

but not simpler questions like 'What is 0 x something?'; or 'What is 1 x something?' or 'What is 2 x something?'

But something a bit harder like 'What is 3 x 39?' would be nice to have help with.

So imagine if each person from th 00 to 99 system has attributes memorised:

3 x ... : Favourite animal or pet

4 x ... : Favourite Celebrity

5 x ... : Favourite person

6 x ... : Home town / city [Random example of a number expressed as a country and city: Spain, Madrid is S and M: 6 and 4 is 64.] But for multiples of 6, let's look at 23 x 6. The final unit is easy to work out and does not need memorising: the answer ends in 8 because three sixes are eighteen. So we just want to memorise the left part of the answer. 6 x 39 = 234. We want the 23 to be represented. So:

A = America = 2: G = Georgia (State) = 3; so America, Georgia represents 23. Instead of 'G'eorgia, A city or state beginning with 'E' or 'Th' would suffice also because E/G/Th are part of the same '3' family in the 'Person 0 to 9' article.

7 x ... : Career

8 x ... : Sport or pastime

9 x ... : Colour and Vehicle. Eg. 9 x 39 = 351. We want to learn the 35. Colour 3 is a green. 5 needs a spelling prompt of W/N/P [see the 'People 0 to 9' article]. 'Green Prius' or 'Green Porsche' could be used.

The answer can indicate the first two digits of a maths answer.

Person 39 can represent the 39 of "What is 3 x 39?". Kurt Heat is person 39. So I can memorise his 'Attribute 3': his favourite animal or pet.

That attribute must indicate the '11' of the 117: 3 x 39 = 117 and I want to remember the lefthand side of the answer.

So what animal can be 11? One beginning with 'BR' would work since the 'Double Digit Acrostics' article explains that BR can represent 11. But I can not think of a one word animal beginning with 'BR'. So I can think of 11 as two words prompted by the 'Person 0 to 9' article. 'I' or 'Ch' or' J' or 'L' mean 1 in that article: they belong to the '1' family. So a 'Laughing Chimpanzee' can be the L and Ch that make 11. Kurt's favourite animal is a Laughing Chimpanzee.

So now, "What is 3 x 39?" is answered by knowing the '11' and knowing that '3 x 9' ends in 7. The pet is a Laughing Chimpanzee. So that is the 11 left side of the answer. 3 x 9 = 27; so the answer must end in 7.

Note: Numbers ending in zero would not need attribute mnemonics. Eg. 3 x 30 is like '3 x 3' but with a zero stuck on the answer.

Multiplying by Using Square Numbers

In the maths forum that I used to visit, one person said that, rather than memorise all the answers to the one hundered times table, he had a technique involving the squares of numbers. At first, I thought that it was a limited idea. Eg. If you know that 8 x 8 = 64 then, if you are asked what 7 x 9 is, you can think of it as (8-1) times (8 + 1); so the square of 8 becomes part of the working out of the solution; and there is a cancelling out of part of the calculation so that the answer is just the square of eight and then minus 1 from it.

I did not get the wider point that this person was making; and I will use an example to show you the power of it.

But first, imagine that you have:

Learned rote the answer of every square from 2 to 99;

Have a good technique for working out if two digits are odd or even;

Have a good technique for finding the average of those two digits< Eg. 4 and 8 average as 6 becaue 4 + 8 = 12; and halving that answer gives you 6. And you would also need to know when an average involves a remainder of half. Eg. 5 and 8 average to 6.5 . An odd number + an even number gives the average of them a half remainder. An even number plus an odd number also gives an avergae that has a half as a remainder.

You can roughly guess the number that comes between two numbers. Eg. Between 33 and 81, you know that the answer is somewhere around 50 or 60; and nowhere near 33 and nowhere near 81..

You can quickly subtract a smaller number from a bigger number. Eg. 56 - 28 . Do you have a quick method to get to the answer of 28?

You can quickly subtract a big number from a quite big number. Eg. 2304 - 121 = what?/p>

If you can do those behaviours then multiplying the 100 times table becomes very doable.

My example is: 37 x 59. Note: Even 8 x 99 can be expressed as two digits times two digits: 08 x 99.

Step 1: If it was asked as "What is 59 x 37?" then swap it around so that the lower number is on the left and the higher number is on the right.

Step 2: The end digits are 7 and 9. Do they average to a whole number or to a number with remainder half? 7 + 9 = 16. Divide by 2 to get 8. So there is no half remainder. If the remainder did have a half then you need to make a note that the end of all of the following steps requires that 37 is added to your running total. And you'd alter the question to be 37 x 58 rather than 37 x 59. But that's not needed here because the average is even.

Step 3: What is the midpoint between the 37 and the 59? We already know the average of the 7 and 9 is 8. So the mid point must end in 8. A sensible guess is that 48 [which satisfies the need to be a bnumber ending in 8] is that mid point since it looks like it should be between the numbers 37 and 59.

Step 4: What is that midpoint number minus the low number from the original question? ie. What is 48 minus 37? The 40 [of 48] minus the 30 [of 37] is 20. Looking at the lower number;'s right digit [the 7 of 37], do you need to count down from 7 to reach the 8 [of 48]? If yes then add 1 to the left digit of that lower number to get 4 rather than the original 3 [of 37]; otherwise, stick with the 3 [of 37]. Since you count up from 7 [of 37] to get to 8 [of 48], you can just stick with the '3'; then/p>

Subtract the right digits: the 8 of 48 minus the 7 of 37. If you had to add 1 to the 3 of 37 earlier then you need to subtract the 7 of 37 from 10; and add it to the 8 of 48; but that wasn't the case; so no need to do that step. [In contrast, if it had been 38 as the lowest number and 47 as ther mid point number then 8 descends to arrive at 7; so the 38 would become 48; and, when looking at the units, the "subtraction from 10" rule must happen: 10 minus 8 is 2; so you would add the 2 and the mid point 48's 7 to get 9.]

But anyway, back to our 37 x 59 . We have worked out that 48-37 = 11. We have worked out a mid point number of 48. And we have rote learned the square of both of those numbers separately: 11 x 11 = 121; and 48 x 48 = 2304 .

Step 5: The answer to 37 x 59 will involve us subtracting 121 from 2304.

Step 6: At the beginning, we asked of there is a remainder half when we average two digits. If the answer is yes then the 59 needs to be added to whatever the answer is that we have reached. Let's call that the "Additive Step".

Answering "What is 37 x 59?", steps need to include the following subtraction of squared numbers: 2304 - 121 = 2183. That could be hard. Let's make the digits line up by expressing 121 as 0121; then both numbers are four digits in length.

I do have a proposal for quickly working that out. It involves learning a lookup table of syllables and matching digits.

It involves expressing 2304 - 0121 as 20 31 02 41 [by paitring off the left side 2 with the left side 0, etc.]

It involves expressing 20 31 02 41 as syllables from the People for 00 to 99 article.

The word you make from 20 = GO, 31 = JI, 02 = BA, 41 = LI is.... GOJIBALI.

If the "Additive Step" mentioned aove is needed then now is a good chance to add 37 to GOJIBALI; you would use the adding numbers technique that I introduced at the start of this article.

It involves having a rote memory of what each syllable leads to when you do subtraction. Eg. With the 20, you want 2-0 = 2 to be the result; with 31, you want 2 to be the result. With 02, you want -8 to be the result; with 41, you want 3 to be the result

You work from left to right across GOJIBALI and replace the GO with your rote memory of 2: say, "Two JIBALI"

You work from left to right across Two JIBALI and replace the JI with your rote memory of 2: say, "Two Two BALI"

You work from left to right across Two Two BALI and replace the BA with your rote memory of -8: say, "Two One Eight LI"

[The -8 means that the part of the answer preceding it needs to be adjusted; so the "Two" became "One"]

When the LI is replaced with "Three", you can say out loud, "Two One Eight Three". That is correct.

Maybe I should insert a new step 1 that asks, "Is there a shortcut? Sometimes an alternative method reaches the answer significantly faster.

I also wanted to mention that the subtraction of 1 (at the column to the left of the maths column you are working on) can then lead to columns further to the left needing a subtraction of 1. Eg. If 1000 has 1 subtracted then you can work on the tens column and put a 9 there but then the answer is 1099 rather than 0099. And that makes me thik that I should add an end step: "Check your answer."

Ten Times Table

The 00 to 99 people from an earlier article is a way to represent the ten times table. Each person can be imagined stood at a table. Person 67 can represent "What is 6 x 7?". Person 48 can represent "What is 4 x 8?". Then what occurs on the table represents the answer. I would use the colours 0 to 9 from Person 0 to 9 article [or the Colours article] to represent the first digit of a 2 digit answer. So 4 x 8 = 32 needs colour 3 and an object that is spelled with a letter from the Person 0 to 9 article. In that article, 2 is A / H / Sh as a spelling prompt. So I need a colour 3 [Green] object like a 'Hat' or an 'Apple' or a 'SHoe'. A Green hat = 32 .

It is much faster to just know the answer. The '2 times table' seems oo easy to apply this method to. But if you mislearned a times table answer when you were younger, this could be an improvement.

Memorising Squares up to 100 squared

The square of 32 is 1024. What if, at person 32 from the 00 to 99 cartoon people article, I imagine person 32 carrying a square tray; and on the tray is something that means 1032? In that way, I am memorising that 32 x 32 = 1024.

I already know easy squares like 10 x 10 = 100; so I would start at 11 squared and go on up to 99 squared. I know that 100 squared is 10,000. So I would not neeed a mnemonic for that.

Note: I will use 11 x 11 as an example. 11 is an interesting number to multiply with; and in a way, it would be nicer to solve 11 x 11 by knowing a bit about numbers; but this section is just about mnemonics and squares.

11 x 11 = 121 .

Mainly, the approach here is to revisit the 'Double Digit Acrostics' article where the first letters of a word match up with a two-digit number. I want a word spelling that means the 12 of 121

I want the 1 and the end of 121 to be a digit person from the 0 to 9 article early in this course. In the double digit acrostics article, there is a way of expressing 12 as any word between 'Bush' and 'Cannon' [so 'Bush' would be OK but not 'Cannon' because 'Cannon' is where the words meaning 13 begin from]. I have a visual way to represent the 1 at the end of 121. In the 'Person 0 to 9'-related article, 1 is depicted by an imaginary orange comic book character. My word between 'Bush' and 'Cannon' will be 'Butter'. I imagine the Person 1 comic book character small enough to be wading through a container of butter; and that whole visual imagery is imagined on a tray being carried by person 11.

Here are scenes to imagine on each tray [they all involve the double digit acrostics approach:

Reciprocals

The Super Hero Letter Pairs article contains some of the answers to reciprocals maths; and the 100 History People aticle contains reciprocal maths as well.

Rough Results

Sometimes, an approximate answer to a maths problem is useful. Eg. Logarithms.

Regarding division, if I know (approximately) what 5/7 is (0.714...) and what 6/7 is (0.857..) ten I can guess what 5.5/7 is because it has to be somewhere between those two answers. Or 5000/75 can be guessed at by using the 5/7 and 5/8 results and moving the decimal point a little: 5/8 is 0.625; so somewhere between 0.714 and 0.625 is the answer to 5/7.5; so let's say 0.65. 5000/75 is like 5.000 / 7.5; so the answer is close to 0.65 but with the decimal point moved twice: 65. The answer is really 66.666 recurring. So it is not too bad.

An item 57 could have a clue in its image to the answer to 'what is 5/7'.

An item 58 could have a clue in its image to the answer to 'what is 5/8'.

In the article about Hero and Robot images (and in the article about '100 History People'), I presented how characteristics can imply a division results lookup table.

Note: Another way to approximate an answer is by using logarithms. Eg. To start off the answer to a division maths question.

Conversion Tables

Another area of numbers which would be nicely represented by mnemonics is conversion tables. Eg. Converting from kilograms to stones or vice versa.

Being in Synch with the Cards article

I have a slightly different letter : number system for the Cards article. It's hard to explain my reasoning but it's partly based on the undesirability of some letters sounding very similar; with the cards, J, CH and SH sound very similar but it's not a problem there; and it lets me use L to mean the Queen [Lady].

Patterns in numbers

Another approach to memorising maths results is to notice a pattern.

In the 8 times table, in most cases, a digit multiplied by a digit that is 2 less than it will have a typical first digit of the answer. Eg. 8 x 6 has a gap of two between 8 and 6 [and a further gap of 2 below the 6 is a '4']. A further gap of 2 is 4 since 4 is two away from the 6. The answer 48 begins with a 4. The pattern does not work for 9 x 7. The pattern does not work for 3 x 1 nor 2 x 0; but 4 x 2 is 08; so it works as far as 4 x 2.

In the 8 times table, in most cases, a digit multiplied by a digit that is 1 less than it will have a typical first digit of the answer. Eg. 8 x 7 has a gap of 1 between 8 and 7. A further gap of 2 is 5 since 5 is two away from the 7. The answer 56 begins with a 5. The pattern does not work for 9 x 8. The pattern does not work for 2 x 1 nor 1 x 0; but 3 x 2 is 06; so it works as far as 3 x 2.

Maths Carriers

5 minus 7 is a bit like 7 minus 5 but it is a negative answer: -2; but if there is a tens column then the maths works differently. Eg. 23 minus 14. The 3 minus 4 leads to a 9 being written down. It would be good to rote learn carriers for addition and for that type of subtraction. Maybe a flash card routine would be best for this: