This is a basic question, but I am revisiting them due to some examinations I need to take that involves mathematics. I want to be nimble with mental arithmetic so have decided to go back and learn my timestable at the age of 25.
I have a number of questions:
Which number do I learn each timetable to? And, what's the most effective method - is it still rote learning as I did back at school?
Thanks
On
Another useful trick is memorizing perfect squares and then using difference of squares to do mental arithmetic.
So $13\times17$ becomes $(15-2)\times(15+2)=15^2-2^2=221$. If the difference is odd (this requires the difference be even), just add or subtract to make the difference even.
On
All the easy to remember numbers are outlined with dashed red border. The table of $5$ that you remember is highlighted with yellow color.
If you can memorise the remaining entries in table of $3$, then all the entries of table of $6$ are twice the corresponding entry in table of $3$. Example, $6*4=24=2(3*4)$
Remembering $9$ table: $9$ times any number is $10$ times that number minus that number. Example, $9*7=70-7=63$
Remembering square helps, because then you know all the diagonal entries in the above figure.
If you can memorise the remaining entries in table of 4, then all the entries of table of 8 are twice the corresponding entry in table of 4. Example, $8*6=48=2(4*6)$
If you know table of 5 then any single digit multiplied by 4 is subtracting that single digit once from what you know is 5 times that single digit and that single digit multiplied by 6 is adding that single digit to what you know is 5 times that single digit.
For example, you know that $7*5=35$, then $7*4=35-7$ and $7*6=35+7$
For adding and subtracting fast/mentally, add/subtract $5$ first and from that result add/subtract $2$. So, $35-7=30-2$ and $35+7=40+2$
Now, for $7*3$ and $7*8$ subtract and add 2 times 7 from what you know is 5 times 7.
$7*3=35-14=25-4=21$ (first subtract 10 and from that subtract 4)
$7*7=35+14=45+4=49$ (first add 10 and to that add 4)
This is decomposition, $7*3=7*(5-2)=7*5-7*2$
Any two numbers can be multiplied this way as follows:Consider $123*678$, $123$ can be written as $100+20+3$ and $678$ can be written as $600+70+8$, now multiply $600,70,8$ with $100,20,3$ separately and write results vertically. Now add vertical results and then add horizontal (bottom row) to get final answer.
Trachtenberg system of mental calculation
For $ \underline{\text{multiplying any number by} 11}$, assume $0$ before that number on it's left side as the starting digit. Now, below each digit write the sum of that digit with it's right side neighbour. For example, $633*11$ .
Under the first 3 from right, write 3+(right side neighbour)=3+nothing=$3$, below the second 3 from right write 3+(right side neighbour)=$3+3=6$, below 6 write 6+(right side neighbour)=$6+3=9$, below 0 write 0+(right side neighbour)=$0+6=6$, you will get the answer as $6963$, if the sum below any digit is more than $10$, then write the units digit of that sum and put a small dot to the left of it as a carry-forward $1$, this 1 is added to the sum below next left digit as show below,
$1754*11$, write $01754$, below 7 we have $7+5=12$, but we write 2 and the carry forward 1 is added to the sum below 1 which is $1+7=8$+(carry forward 1)=9, answer is $19294$
Same is the procedure of $ \underline{\text{multiplication by} 12}$ (carry forward rule is also same)except that each digit is doubled and then added to the right side neighbour,as $0413*12$, below 3 write double of 3=6, below 1 write (double of 1)+3=5, below 4 write (double of 4)+1=9, below 0 write (double of 0)+4=4, answer is $4956$
For $ \underline{\text{multiplying any number by } 6} $, below each digit write the sum of that digit with "half" of right side neighbour and add 5 if the digit is odd (here "half" is integer half, 1's half is 0, 2's half is 1, 3's half is 1, 4's half is 2, 5's half is 2 etc), so $08234*6$, below 4 write 4+(half of nothing)=$4$, below 3 write 3+(half of 4=2)+5(3 is odd number)=10, (write $0$, carry forward 1), below 2 write 2+(half of 3=1)+carry forward 1=$4$, below 8 write 8+(half of 2=1)=$9$, below 0 write 0+(half of 8=4)=$4$
$ \underline{\text{Multiplication by } 7 }$ is same as that of 6 except that we double the digit before adding it to the "half" of right side neighbour (add 5 to the sum if the digit is odd)
For $ \underline{\text{multiplying any number by } 5 } $, write "half" of the right side neighbour below the current digit and if the current digit is odd then add 5, example, $0434*5$, below 4 write "half" of nothing=$0$, below 3 write "half" of 4=$2+5$(add 5 to the "half" if current digit is odd), below 4 write "half" of 3=$1$ and below 0 write "half" of 4=$2$, answer is $2170$
$ \underline{\text{Squaring any two digit number } }$ (Example $36^2$)
Square the right side digit, write the result's unit's digit($6$) on right side, ten's digit is carry forward($3$), multiply first digit with second digit and double it($36$), add the carry forward (36+3=39) and write the unit's digit of this result($9$) on second place from right side, ten's digit is carry forward ($3$), square the left side ($9$) digit and add the carry forward ($9+3=12$), $1296$ is the answer.
$\underline{\text{Multiplying any number by a 2 digit numbers}}$
This is fast if the digits involved are less than 5. Example $34*24$
$4*4=16$, write $6$ and 1 is carry forwarded, $3*4+4*2=12+8=20+1$ (carry forward)=$21$, write $1$, 2 is carry forwarded, 0*4+3*2=6+2(carry forward)=8, answer is $816$.
Answer is correct when $8+1+6=15=1+5=6$ is equal to digit sum of $(3+4)*(2+4)=7*6=42$, $4+2=6$
It is enough to memorize until 9, because this allows you to perform all multiplies of larger numbers by decomposition. (Even though it seems to be an American tradition to learn until 12. 10 is trivial, 11 is easy - by repetition of the digits, 12 is the hardest.)
There isn't so much to remember: multiplies by 0, 1 and even 2 are trivial (for 2 you can add mentally). This leaves you 49 products. But of these, 7 are perfect squares (3.3 = 9, 4.4 = 16, 5.5 = 25 ...) and the remaining 42 appear twice, by symmetry (6.9 = 9.6 = 54).
Your total effort is to remember 7 squares and 21 products.
Extra tricks: multiplying by 9 is the same as multiplying by 10 and subtracting once (e.g. 6.9 = 60 - 6); multiplying by 5 is the same as multiplying by 10 and halving (e.g. 7.5 = 70/2).
You can reconstruct any row of the table by successive additions: e.g. 7, 7+7 = 14, 14+7 = 21, 21+7 = 28...