
Is there any point to the 12 times table? - kesselvon
http://blog.wolfram.com/2013/06/26/is-there-any-point-to-the-12-times-table/
======
Animats
Only in non-metric countries.

Back in the British era of pounds, shillings, and pence, hardware had to be
built to do arithmetic in that system. This resulted in one of the strangest,
and most complex, purely mechanical computing devices ever built - the McClure
Multiplying Punch [1], from Powers-Samas. This device came out in 1938. The
comparable IBM machine was the IBM 602 Multiplying Punch, but IBM only did
decimal multiplies. The McClure machine had a mechanical multiplier for
pounds, shillings, and pence. It contained a physical multiplication table,
made out of brass plates, and the machinery to use them for multiplication by
table lookup. There's a picture of the "Pence x 7" plate.[2] That's one row of
a multiplication table, and it's 12 wide, from 0 to 11. Sixpence x 7 = 3
shillings 6 pence. The brass column heights reflect that.

[1]
[http://www.computerconservationsociety.org/resurrection/res5...](http://www.computerconservationsociety.org/resurrection/res59.htm)
[2]
[http://www.computerconservationsociety.org/resurrection/imag...](http://www.computerconservationsociety.org/resurrection/images/images59/res59k.jpg)

~~~
deadowl
I disagree.

1\. We have time. Days are 24 hours, a multiple of 12.

2\. We still have time. Hours are 60 minutes, a multiple of 12.

3\. Yet more time. Minutes are 60 seconds, a multiple of 12.

4\. We have circles. There are 360 degrees in a circle, another multiple of
12.

5\. The numbers 1, 2, 3, 4, 6 and 12 itself divide into 12 evenly. The next
smallest number that has more factors than 12 is 24, which manages to be a
multiple of 12.

~~~
beachstartup
you forgot the one i use the most often - there are 12 months in a year.

how many times do you calculate "X per year" in your head? i do it multiple
times per day, and not just at work.

~~~
antod
_> how many times do you calculate "X per year" in your head? i do it multiple
times per day, and not just at work._

Seldom enough that I 'calculate it in my head' (as your comment suggests you
do too) rather than memorise the 12x table.

But then again I grew up in metric country that only bothered with drilling
kids up to the 10x table at school. Never had to worry about feet to inches
conversions very often either.

~~~
beachstartup
i run a business so basically i'm calculating "X per year" all day long.

------
zeveb
Twelve has more divisors than ten (1, 2, 3, 4, 6 & 12 vs 1, 2, 5 & 10). If we
were really smart, we'd switch from base 10 to base 12: many more 'decimals'
(really duodecimals) would be non-repeating. One can very quickly count by
twelves on the joints of one's fingers, using the thumb as an index (0-143 is
a much larger range than 0-10, and it's easier to hold one's hands in the
shape necessary). If we were really smart, we'd switch from base 10 to base
12: many more 'decimals' (really duodecimals) would be non-repeating.

And then there are measures like a gross (144) and a great gross (1,728). Part
of the reason for these traditional measures is that they are more flexible
than base-10 measures: an eighth-gross or a third-gross are both integer
quantities, unlike an eighth-hundred or third-hundred.

~~~
btilly
Easy divisibility is the reason why when we switched to metric, the one thing
that didn't switch is time. Which means that, for example, converting from m/s
to km/h is a mess. (You have to multiply by 3.6, can you easily do that in
your head?)

With a base 12 version of everything you would have 1/12 of a day being 2
hours, 1/144 of a day is 10 minutes, and 1/1728 of a day is 50 seconds. These
units would give us both easy divisibility and are close enough to existing
time units to make sense.

This change would mean we have to memorize a 12 times table rather than a 10
times table. But in base 10 a 10 times table has easy patterns for 1, 2, 5, 9
and 10. A 12 times table has repeating patterns for 1, 2, 3, 4, 6, 8, 9, 11
and 12. The result is that a 12 times table in base 12 is actually less work
to memorize than a 10 times table in base 10.

In the long run this transition would be a clear win. But it isn't enough of
one compared to the transition to ever make sense to initiate.

~~~
dnautics
decimals cause huge problems for computers. How do you represent 0.6 as a
floating point? In the long run, the correct solution is for everything to be
binary.

~~~
Terr_
Why? No matter what radix you pick, there's _always_ going to be some
rational-value you cannot accurately encode. Moving to binary just gives you a
different distribution of "un-encodeable values". (I'd say a larger/worse set,
but I'm not yet sure how to prove it.)

In the long run, _trinary_ gives you the best radix-economy, being closest to
e:
[https://en.wikipedia.org/wiki/Radix_economy](https://en.wikipedia.org/wiki/Radix_economy)

~~~
Terr_
Making a self-reply here, I _think_ I know how to show binary is "worse" when
it comes to un-representable rational numbers. I'm not a math major, so this
is probably some incredibly obvious textbook stuff to somebody else, but...

In base-N, you can accurately write any fraction of _(1 /y)_ provided that _y_
can be expressed using the same prime-factors found in N.

For example, 10 has the prime factors of 2 and 5, leading to the requirement
that _y=(2^a × 5^b)_ .

This means base-10 can accurately represent _(1 /80)_, because _80 = (2^4 ×
5^1)_. In base-2, you're limited to stuff in the form _(2^a)_.

Finally, the, uh, infinity-of-integers that matches _(2^a)_ is always going to
be "smaller" than the infinity which can be matched by _(2^a × 5^b)_ , since
the latter "contains" the former when b=0.

~~~
dnautics
no. You can accurately represent anything which is k*2^n, where (k,n) are
integers. Simple example: 3 is not 2^n but obviously expressing 3 as a
floating point is not a problem. Similarly 1.5, etc.

In this fashion, primes p seem "better" as they increase, because for a given
size limits on k and n, you get more 'numbers' per unit space on the real
line. Composite numbers are 'even better', but there are a lot of pains in the
ass with composite numbers, stemming from Z/nZ not being a field if n not
prime.

Ultimately, though, your internal representation is binary, using anything not
two produces other inefficiency. That's why IEEE 754 BCD is not base-10 but
base-1000 (since 2^10 = 1024 which is the closest a power of two gets to a
power of 10, for bit-wise representation efficiency). We don't have bi-quinary
computers.

~~~
Terr_
Once you know you can accurately represent _(1 /y)_ in a given base, you've
established do __not __need an infinite number of digits to the right of the
radix-point no matter how many times you multiply it by an integer.

In fact, I think it exemplifies the _upper bound_ for the number of digits
you'd need for "related fractions". For example, we know "one eighth" is
expressible in decimal as 0.125. Even if you pick a really big N value, N *
(1/8) should never need more than three digits to the right of the decimal
point.

------
noobermin
Am I going to be the only one to comment that the article actually was pretty
cool? He put up some really nice plots in a masterful way, which is better
than most of the hand-waving I see here.

The best attempt he put forward to deal with the distribution of numbers one
would see in daily life (a generalization of the "non-metric" argument) is
using Benford's Law[0], which is as good a blog post that doesn't turn to
hardcore statistics (of all numbers used, if any such thing actually exists)
can do.

[0]
[https://en.wikipedia.org/wiki/Benford's_law](https://en.wikipedia.org/wiki/Benford's_law)

------
clarkmoody
This type of post is perfect Hacker News material!

1) Challenge an existing assumption about something we all do

2) Hand-wave some first-order guesses as to why we do it

3) Get nerdy with code and graphs and come to your conclusion, along with
helpful suggestions for bettering the reader

 _With no prospect of the pre-decimal money system returning, I can only
conclude that the logic behind this new priority is simply, “If learning
tables up to 10 is good, then learning them up to 12 is better.” And when you
want to raise standards in math, then who could argue with that? Unless you
actually apply some math to the question!_

~~~
Scarblac
But do we all learn tables up to 12? In the Netherlands it's only up to 10 for
as far back as I know.

~~~
antod
Same in New Zealand. The only times I ever saw anything with 12x tables it had
escaped from the US education market and had exercises involving nickels,
dimes and quarters etc.

~~~
Widdershin
My experience in New Zealand differs, we learned the 12x tables at primary
school and there was nothing vaguely imperial about it. This was around about
a decade ago.

~~~
antod
Interesting - my experience was in the early 80s.

I suspect getting the 12x table added to the NZ maths curriculum was probably
the main deliverable of the 2001 Knowledge Wave Conference.

------
sandworm101
Forget the stats and calculus. Think carpentry, measuring and cutting wood
products. If you are building things in the US/UK/Canada then you are using
feet and inches. 12 inches to a foot. It's a tiny thing to learn and will
serve kids well in any number of professions.

Now 11, that's a total mystery. Other than it being between 10 and 12, I see
no reason to memorize 11s.

~~~
zanny
That is only yet another reason in literally infinite ones why the US needs to
actually push metric units. We bleed our stupidity into the UK and Canada
while the rest of the world makes sense.

~~~
ksenzee
No, the US is not exporting its stupidity to the UK in this case. There's a
reason those units are called "imperial."

~~~
to3m
The US doesn't use Imperial units though! They use the US customary system (if
that's quite the right term):
[https://en.wikipedia.org/wiki/United_States_customary_units](https://en.wikipedia.org/wiki/United_States_customary_units)

Though either way, you are quite right that the US has little influence on the
UK in this respect.

------
spyckie2
1-10 you memorize. 13-100 you calculate it out. 100+ you use a calculator.

11-12 is half memorized/half calculated.

I think of them as introducing you to how to calculate with larger numbers you
haven't memorized.

Some of them you know - 12 * 5? 60.

But what about 12 * 7? 10 * 7 + 14. or 12 * 5 + 12 * 2.

Dealing with 11 and 12 in the times tables gives you good practice for those
calculation tricks that you use for numbers greater than 12. It's not worth it
to memorize an additional 44 rules but it is worth it to know how to do math.

------
Amorymeltzer
>Or, as Chris Carlson suggested to me, learn the near reciprocals of 100 (2 x
50 = 100, 3 x 33 = 99, 4 x 25 = 100, 5 x 20 = 100, 6 x 17 = 102, etc.), as
they come up a lot.

This is absolutely worthwhile, easy approximation of 1 or 100 divided by 6 or
8 in particular occurs regularly, and not enough people know it offhand.
Hugely helpful. After 1-10, your energy is probably better spent on cool
patterns and "tricks" within those sets (final digits of 9, division by 7,
etc.); much greater return.

~~~
thaumasiotes
Ok, but what's with the weird phrasing? This is the same set of facts as (1/2
= 0.50, 1/3 = 0.33, 1/4 = 0.25, 1/5 = 0.20, 1/6 = 0.17, 1/7 = 0.14 ...), but
phrased to sound like something you'd have to put effort into.

------
geofft
Even in metric countries, there are still 24 hours in a day, so being able to
have a sense of what 36 hours is or what 96 hours is is useful.

~~~
waxjar
I don't think this comes up that often.

It's easier to speak in terms of days and halves of days. Often this is
accurate enough.

------
switch007
It's moved from "should" to must.

"Every child in England will be expected to know their times tables before
leaving primary school from next year.

Pupils will be tested against the clock on their tables up to 12x12 in new
computer-based exams that the Department of Education (DfE) said were part of
the government’s “war on innumeracy and illiteracy”."
[http://www.theguardian.com/education/2016/jan/03/pupils-
face...](http://www.theguardian.com/education/2016/jan/03/pupils-face-new-
tests-to-ensure-they-know-times-tables-by-age-11)

Gove is no longer the Education Secretary...he is now the Justice Secretary.
God help us.

~~~
TheOtherHobbes
I like the idea of declaring war on innumeracy. That totally works as a
metaphor for improved teaching.

I'm not sure people need to know the 12 x 12 table, so much as knowing how
percentages and compound interest work.

How long will it take someone to pay off a credit card debt if the APR is
39.9% and they can't afford to clear the balance every month? If they don't
understand why the answer is "Probably forever unless they get lucky with an
inheritance" they shouldn't be using a credit card.

------
mserdarsanli
> Take as an example 7,203 x 6,892. If I want to know that exactly, then I
> reach for Mathematica

They are always trying so hard to sell Mathematica in these blog posts, but
this is the lowest I've seen.

~~~
gknoy
I don't know that it's meant to shill the product -- rather, that's the tool
that he has handy. I work all the time with Python, and frequently open an
IPython shell to do some quick math ("what's the average of 4d6 pick 3?" "what
is X% of my paycheck?" "how much do I need to contribute to max out my
$FOO?"), simply because it's easier than a calculator.

I mean, sure, it's advertising Mathmatica, but it doesn't seem to be
unecessarily doing so. It's more that it's ${programing-language-of-choice}
for the author.

~~~
mserdarsanli
I don't work with Python, but when I want to multiply some numbers, I type
`python` into terminal and use it as a calculator, as it allows me to do such
operations quickly.

It sounds weird that someone opens an IDE and waits ten seconds looking at a
splash screen to do a simple multiplication.

~~~
aaron695
Dogfooding

------
kazinator
There is some use in knowing subsets of the time tables beyond 9: specifically
M x N combinations <= 100.

For instance, if you know that 8x12 = 96, you know that 1/12 is approximately
0.08. And since 0.96/12 is exactly 0.08, you know that the remainder is
0.04/12 which gives you the .003333... in 0.0833333...

Essentially, it supports numeric intuition.

These higher times tables can be useful in long division. In long division you
have to form hypotheses about how many times the divisor goes into the partial
dividend, to extract the next digit of the partial quotient. So for instance,
something like this comes up:

    
    
          ________ 
      12 | 980
    

Now 12 doesn't go into 9, so we try 98. How many times does 12 go into 98? If
you know your 12x12 times table by rote, you might realize in a flash that
12x8 is 96, so put down an 8.

If you memorized all times tables up to 99, you could so long multiplication
in base 100:

    
    
           1234
           4567
           ----
    

You would instantly know that 67 x 34 is 2278, so ... put down the 78 and
carry the 22:

    
    
           1234
           4567
           ----
           22
             78
    

and then 67 x 12 is 804. Bring down the 22 into it and we get 826:

    
    
           1234
           4567
           ----
          82678
    

Damn, that was fast! :) Then we keep going with the 45 similarly: and we can
move by two places to the left. 45 x 34 = 1530; put down 30; carry 15:

    
    
           1234
           4567
           ----
          82678
         15
           30
    

and then 45 x 12 = 540, plus carry is 555:

    
    
           1234
           4567
           ----
          82678
        55530
        -------
        5635678
     

Totally awesome, and we only had to memorize 4950 product combinations from
1x1 to 99x99. Contrast that with plodding through it one digit at a time, with
four partial product rows to then add together.

~~~
thaumasiotes
> For instance, if you know that 8x12 = 96, you know that 1/12 is
> approximately 0.08.

You get this same intuition in a much better way by knowing that "since 1/8 is
approximately 0.12, 1/12 is approximately 0.08".

------
lordnacho
There's lots of questions thrown up by this. If people need to know times
tables, do they also need to know powers of 2? They come up quite often if you
deal with computers. What about squares? Anyone doing polynomials later on
will be using them constantly.

My sense is you will end up memorising the things that come up anyway, so why
be so strict? You'll end up teaching kids that math is a memory game.

So for fun, what do I remember, off the top?

2 __10 = 1024. Anything higher I keep doubling.

13 __2 = 169, biggest non derived square in my head. 1 more than 7 * 24 (hours
in a week).

86400 = secs in a day?

Large prime number... 10243 (Useful for demonstrating Diffie Hellman on paper)

Ramanujan's Cab... argh... is it 1729?

3.14159 (Guess I'm not winning any contests)

2.71 (See above)

6E23

Add your own random numbers that are in your head.

~~~
schoen
You're right about 86400 and 1729.

I know up to 2¹⁷=131072 and also the special cases 2²⁰=1048576 and
2²⁴=16777216 (the 7s in the middle make it easier to memorize, and it seems to
come up a lot). I would really like to remember 2³²=4294967296 (I always just
remember "4.2 billion") and should probably get around to that.

I learned a lot of pi in middle school but even then I knew that it wouldn't
be useful for anything, and it hasn't, other than knowing a lot of pi.

The start of e goes 2.718281828, so it's not a whole lot of effort to go a
little beyond your 2.71 if you wanted to.

Here are some Unicode superscripts ¹²³⁴⁵⁶⁷⁸⁹⁰ in case anyone wants to use them
in this thread.

~~~
lordnacho
4.2 Billion is useful, it's the addressable memory in 32 bits.

~~~
schoen
Yes, exactly (and similarly the number of IPv4 addresses) -- I'd like to
memorize the precise value sometime.

------
Bulkington
We're overthinking. Think, instead, as a school marm with real chalk and a
switch in hand, or even when kids used slate tablets: 10x is easy. 11x is easy
up to 10x. 12x requires a little work and pulls all the multiples together,
but 13x just screws with everything we've learned so far. So, stop at 12. Or
is a calculator in hand the default these days? As one who was proud to
finally get to 12 x 12 = 144, I say make 'em suffer.

------
VeilEm
It's really not a big deal for a child to memorize this table. I also think
memorizing the squares up to 25 is worth it.

~~~
TheCowboy
If we ignore that it takes "144 facts" compared to "100 facts" for a 10 times
table, and take it as a single item of learning, it may not be a big deal.

The problem is that it's not the only piece of distracting trivia that is
heaped on students as something they should know that likely holds little
consequence to achieving big picture learning goals like critical thinking and
better math skills. A more American example would be memorizing the capital
cities of every US state, along with other state trivia.

Memorizing things like the state capitals just gives the illusion of
education, but is actually a waste of educational resources and learning time.

Time, along with the attention span of a child, is a scarce resource. It's
good to trim the curriculum when we identify information that has weak
relative value, and it's too easy to find something else to replace it.

------
dahart
The reasons given for learning multiplication tables all focus on
multiplication, but one of the best reasons is for division. Doing division on
paper or in your head is made _so much easier_ if you have your single digit
multiplication table memorized.

But why 12 instead of 10? I think:

\- There are loads of common everyday uses that overflow 10 by just a little.

\- Clocks

\- Eggs

\- 11 is almost as trivial as 10, it would be a shame to stop at 10 when 11 is
free.

\- 12 is really super easy, and combined with the above reasons has a good
case.

\- 13s are significantly harder to remember and work with than 11 or 12,
nothing comes in 13s, its the _perfect_ place to stop. :) (edit: course, as
soon as I wrote that I remembered the baker's dozen, which was super common to
hear when I was a kid, not so much anymore for me. But I think the point still
stands, because with a baker's dozen you only pay for 12, you never had to
multiply by 13.)

* edited for formatting

------
contravariant
As someone who never learned the 12 times table, and only uses metric, I'm not
sure if it would have been better if I had learned the 12 times table, but for
some reason I have actually memorized most multiples of 12 (without any
conscious effort). They do seem to turn up a lot for various reasons.

------
panglott
Most of the multiples of eleven are really, really easy, and probably just has
simple pedagogical value.

Multiples of 12 come up a lot because dozens are used a lot in real-world
counting of length and time. Perhaps less in a metric country, but hours and
days are still divisible by 12.

~~~
slavik81
Eggs, donuts and beers also still come in multiples of 6 or 12.

------
twothamendment
Is there any point? Depends on where you live. It doesn't hurt to know that 8'
is 96 inches, 4' is 48, etc. Now if I were in a country on metric system, I
don't know if I could come up with a reason or time that I needed to know
them.

------
stewdio
Jon McLoone really seems to be approaching this topic from a place of
frustration—rather than from an overwhelming sense of beauty. I think both
mathies and visual designers can agree that 12 is a very beautiful number. In
fact the ONLY single thing that makes sense about the imperial system is the
division of feet into 12 inches, something mirrored by picas and points in
typography:
[https://en.wikipedia.org/wiki/Pica_(typography)](https://en.wikipedia.org/wiki/Pica_\(typography\))

This beautiful number is evenly divisible by 1, 2, 3, 4, 6, and itself—a very
high percentage (50%!) of the positive integers leading up to and including
itself. (To be sure 60 is also a beautiful number but its ratio of factors to
non-factors is of course lesser.) I can see you gearing up to argue that 6 is
more beautiful, and I get that. I really do. But the ability to evenly divide
into quarters is just so ... pleasant :)

Does the ability to divide evenly in several ways serve as a soapbox from
which to argue that it is mathematically important to memorize the first 12
numbers of which 12 is a factor? I suppose not. But could we simply do 12 the
honor of learning these two extra columns of the multiplication table because
it is beautiful? (And really you’re just learning ONE extra column because 11
is such a simpleton. Silly 11.)

Of course dividing things into 12 units can SOUND pleasant as well. Western
music is founded on dividing an octave into 12 units—though the exact method
of division can take many forms. Here’s just a sample:
[https://github.com/stewdio/beep.js/blob/master/beep/Beep.Not...](https://github.com/stewdio/beep.js/blob/master/beep/Beep.Note.js#L268)

So I love 12. Join me in loving 12. Together we can praise 12.

Full disclosure, I have been known to previously adore the number 12. On
December 12, 2012 I proposed this “Cleaner Grid of Time” beginning with the
months in a year:
[http://stewd.io/b/2012/12/12/twelve](http://stewd.io/b/2012/12/12/twelve)

------
SixSigma
I recently worked with a person who had to do her three times table using her
hands, starting from 3. She would still struggle at about 6 or 7 times.

We worked behind the counter in a builder's merchant. She is no longer working
for us.

~~~
theworstshill
Why not?

~~~
Symbiote
It's exactly the kind of job where someone will phone and ask "do you have 30
widgets I could pick up in an hour?". They come in packs of 4 widgets, and
there are only 7 packs in stock.

------
jheriko
politics is a component too... i wouldn't go looking for purely intelligent
decision making from a government selected by popularity contest.

the perception that school is getting easier, a criticism of the government
that is politically undesirable, can be addressed by a 'return to traditional
methods' in superficial ways... even if there is not really a concrete problem
to address.

politics isn't about doing or achieving anything in terms of education or
quality of life for others - its about making other people happy.

------
tempestn
I would argue that the 11s don't really count, since knowing them is trivial.
The only one that really requires memory is 11x11, and it's probably not
strictly worth knowing, but then kids get the accomplishment of knowing a
whole extra row essentially for free.

I agree that the 12s aren't worth memorizing though. They're pretty easy to
mentally calculate, being a combination of 1 and 2, and after doing that for a
few years, the memorization will often happen naturally. No real need to force
it.

------
legulere
In Germany you usually learn up to 10×10 in school and there are usually 10
eggs in a box (4, 6 and 25 are also common sizes though and there's also a big
set that's 20×20 which you don't usually learn though)

> And when you want to raise standards in math, then who could argue with
> that?

The problem is that you can only spend so and so much time with children in
class. Learning 12×12 is 80% more combinations to learn compared to 9×9.
That's time you could teach them something else that might be more useful.

~~~
kijin
Actually, it's only ~32% more combinations to learn, because nobody really
needs to memorize the 10's and 11's (except 11 x 11).

You memorize all the combinations up to 9 x 9, compute the 10's and 11's on
the fly, and memorize the 12's.

Edit: Corrected the number. This is how you get 32%:

The 9 x 9 multiplication table contains 8 x 8 = 64 combinations to memorize.
The 1's don't count.

The 12 x 12 multiplication table contains an additional 20 combinations to
memorize, namely:

\- 10 combinations from 12 x 2 to 12 x 12, not counting 12 x 10

\- 9 combinations from 2 x 12 to 11 x 12, not counting 10 x 12

\- 11 x 11 = 121

So the total is 84 combinations, which is ~32% more than 64.

------
kitwalker12
he got off lucky. back in India, we had to cram tables upto 20.

------
tempestn
Here's a good example I just came across of why it's useful to be able to do
approximate multiplication in your head (perhaps even while intoxicated):
[http://news.nationalpost.com/full-comment/robyn-urback-
sorry...](http://news.nationalpost.com/full-comment/robyn-urback-sorry-about-
your-1000-uber-bill-but-the-internet-isnt-idiot-proof)

------
greggarious
Can I please point out this is the point where many people get turned off from
math, never to return?

I HATED memorization. I was able to do the math in my head up through 10, but
11x11, 11x12 and the rest of the 12s eluded me, and I just really, really was
not able to sit still and just do flashcards.

I ended up only chancing back on math through programming and a course in
logic in undergrad. Everything else was mostly self taught as needed.

------
Spooky23
To calculate dozens

Lots of commodities are sold in dozen multiples.

------
oska
_> Download this post as a Computable Document Format (CDF) file._

Hmm?

Ah.

 _> CDF files can be read using a proprietary CDF Player with a restrictive
license, which can be downloaded free of charge from Wolfram Research._ [1]

Fuhgeddaboudit.

[1]
[https://en.wikipedia.org/wiki/Computable_Document_Format](https://en.wikipedia.org/wiki/Computable_Document_Format)

------
sageikosa
I suggest learning them up to 16, or at least learn the 16th column-row (some
of the others not so much).

~~~
thomasahle
Yup, that would be great to know, when doing hex.

Really, if I could have come out of school with the 20x20 table memorized, I
wouldn't mind now at all.

------
ck2
They should probably learn HEX at age 9

~~~
spike021
The odd thing I've found in my experience is that hex is thought to be taught
to everyone in grade school. None of the schools I attended from elementary to
high school (in the US) taught hexadecimal. I learned that in my CS courses
once I started college.

~~~
khedoros
I remember doing some radix conversions before college, but with no particular
emphasis on base-16. The same material in CS courses mainly focused on bases
2, 8, 10, and 16, of course.

~~~
spike021
Yeah, my CS courses covered 2, 8, 10, and 16 as well. I think one class used
some other bases for a project, but I wouldn't say it was a particular focus.

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justinhj
Whenever my son has a new math teacher, I taught him to ask if he can change
the base when reciting the times tables. For example the 12 times table in
base 12 is 10, 20, 30, 40 and so on. It usually gets a laugh or a tut before
he has to do it the right way.

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potatote
Not sure if someone already has pointed this out, but one small caveat I'd add
is that learning the table of 11 is not that difficult. I think the extra
effort required to learn 11 and 12 tables are, therefore, a bit overestimated.

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yason
Memorisation is caching.

If you need 11-times or 12-times tables, you will calculate them enough many
times that your cache will become "hot" and you remember them out of the box.

If you don't, then you don't have enough use for those multipliers.

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protomyth
12 x 12 is 144 and if we're not worried that our children will not have the
mathematical skills to count and divide a gross of bottle rockets, then this
is truly not America.

Plus inches in feet.

On a more serious note, the world does not end with 10.

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cjensen
I grew up in the Bay Area in California. We only memorized up to 9x9. Though
12 is used often in measurements, I can't say as that I've missed having rote
knowledge of the multiples of 12.

~~~
Namrog84
Would you say then that you don't have any x10 memorized and just consider it
trivially easy to compute any number?

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KhalilK
Obligatory Numberphile video:
[https://www.youtube.com/watch?v=U6xJfP7-HCc](https://www.youtube.com/watch?v=U6xJfP7-HCc)

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squozzer
Yes we would not have this cute little ditty.
[http://youtu.be/_uJsoZheTR4](http://youtu.be/_uJsoZheTR4)

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aidenn0
I always assumed it was because we'd forget the largest couple of rows we
memorized, so 20 years later we could still do up to 10

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amelius
Let's just switch to binary, and let them learn the 1-times table.

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mdip
Since I grew up in the United States, learning 12 in our multiplication tables
(which was also required) had a lot of value. Being able to multiply by 12
makes handling inches/feet conversions a mental operation and that's something
an average US citizen encounters in normal life regularly enough. I suspect in
the UK this is partially due to legacy and "everyday life" usage of imperial
units[1], combined with the idea that "it's really just one more number"[2].
"Officially" they are metric, but its usage varies (except temperature, which
nobody can mentally calculate when I tell them it's "90 degrees out" in
Fahrenheit).

In the broader sense, I'd prefer that a lot more time be spent on number
theory than on arithmetic. I am rarely without a smartphone to the point where
its in my hands for even basic math that I could do in my head (but which, due
to infrequent use, I would spend a few seconds questioning whether or not I
really _did_ get the right answer).

The schools start with algebra and other concepts much earlier, now (at least
my district does), but I started teaching my children formulas and basic
algebra from the point they started learning to add and subtract (kindergarten
in my district) along with binary and hexadecimal positional numerical
systems. I remember struggling with that in my teens when I started learning
software development my entire exposure to numbers was in decimal (10 is the
number after 9, always!). In the first grade, my kids didn't have that mental
block and I found they grasped the idea far more quickly than I expected. As
they've learned multiplication and division, I've shown them the same in
binary and hexadecimal and illustrated how they relate to one another.
_Everyone_ struggled being introduced algebra so late when I was growing up.
And it progressed so quickly, going from Algebra (2 yrs), Geometry (1 yr),
Trigonometry (1 yr) and pre-calculus/calculus that students fell mostly into
either "memorize the formulas/algorithms and regurgitate the test answers or
struggle like crazy and never get to Trig/Calculus until college. I left with
an understanding of math but that was mostly due to my taking up programming
at that time and having to learn what it all _really meant_ because I wanted
to do things in code that required me to understand it.

[1] The British Metric Society "The Mess We're In"
[http://www.metric.org.uk/the-mess-we-are-in](http://www.metric.org.uk/the-
mess-we-are-in)

[2] The reality is that you're only asking students to memorize one more
number since 10 is "add a zero and multiply by one" and 11 is an obvious
pattern with single digits.

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djrogers
I can think of about a dozen reasons why this is important...

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myth_buster
@Dang, article from 2013.

