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When I came up with the idea for this video I thought oh, maybe I'll pop down to Office Works,
which is the big stationery shop here in Australia, and I'll buy myself a slide rule to do this video with.
But luckily just before I left, I thought you know what maybe,
just maybe I should check the catalogue first to see if they sell
slide rules at office works and turns out that they don't.
So that's how I've ended up here today with a very makeshift slide rule
and this is just some pieces of paper held together with a piece of plastic
that I actually cut out of something I scrounged out of the recycling, this old croissant packet.
And it's held together with a little bit of Blu Tack at the back there.
I'll put a link to where you can download the template to make one of these yourself if you so desire
but as I'm going to go through and solve some old exam problems with this
you'll probably see that it's not quite recommended.
I've become somewhat of an expert in the last half an hour or so on slide rules.
Before that I knew nothing about them and I'd never used one before.
Now I have mild competence at using one to do things but it's been a bit of a nightmare actually
so if you haven't used one either, I'd probably say you're not missing out on too much
but there's probably a lot of people maybe even watching this video and around that actually remember slide rules
in their hey-day and they were popular and what was used to do multiplication and division
as well as things like trigonometry and logs
square roots things like that before our pocket calculators.
So I think even in like the 50s and even up to the 70s you would see engineers
with one of these to do all their calculations with.
Apparently it's the slide rule - and doing arithmetic with it - that got us to space.
You know, they were used by engineers during like the NASA programs and things like that.
So I guess we owe a lot to the humble slide rule,
and maybe it's a shame that I've never picked one up before and had to play with it.
In a previous video of mine, I looked at an MIT entrance exam from
1869 and I looked at the algebra section. If you've seen that video then know that this
exam I have here and the questions I'm going to go through with my slide rule
are from the arithmetic section of that same exam.
So MIT, 1869.
It was actually on my previous video about some 19th century arithmetic that some people in the comments
pointed out to me the slide rule and reminded me of something, the log table, as well,
which I didn't actually mention at the time but was probably offered to students back then taking exams
and might have made things easier for them since they didn't have the pocket calculator
that we might have in an exam today.
So let's take a closer look at our slide rule here. Most of it we're not going to use today;
we're just going to focus on lines C and D.
C is the one that slides and D is the one here that it meets.
So you can see that C and D are both on a log scale because the distance between
say 1 and 2 is much greater than the distances up here later between like 8 & 9.
This is because this part of the slide rule which is used for multiplication and division
is exploiting properties of logs and specifically the property that
log(x) + log(y) = log(x*y).
So we want to find the product x*y, what we'd do is we can add the log of x to the log of y
which we kind of do by sliding it like this
and then take the antilog of what we get, the product,
and this scale is sort of set up so that it's taking the antilog for you
and all you need to do is worry about sliding it around.
I think these scales up here (A) is to do with squares and square roots,
and K is to do with cubes and cube roots.
So let's have a look at our exam now.
This is it here, these are our MIT arithmetic questions.
If we're looking at question 1:
Find the sum, then the difference, and then the product of the fractions
(3 and 5/9) and (1 and 7/24)
and then divide that first fraction by the second one
So I'm actually going to skip finding the sum and the difference
because I don't think it's worth trying to use this slide rule for that and
instead we'll just look at finding the product of these two fractions.
Now first we want to convert them from being mixed into being like improper fractions.
So let's do 3 times 9 which is 27 plus 5 (ninths)
= 32 / 9.
That's going to be multiplied by 24 plus 7
= 31 / 24
So our first difficult multiplication we need to do is 32 times 31.
Let's do it on our slide rule.
So we're going to take the 1 on the C scale and slide it along to the first number that we're multiplying.
All the numbers need to be less than 10, so we're actually going to slide this one along to 3.2
and then deal with fixing our decimal places later.
So I've got my 1 on 3.2 there.
And then we're going to take this plastic slider
and slide it across to the second number that we're multiplying which is 31.
Make sure everyone stays in place.
31 will be 3.1 so that'll be...
Right in here. Actually it's extremely close to the end
and I'm going to say - that's extremely close to the end of the scale -
I'm going to say that's 9.95 down here on D.
So I'll put 995.
I'm not writing 9 point 95 because I'm aware that
what I'm multiplying would give me a number much bigger than that
so I'm pretty sure it's going to be that one there (995).
And then let's do the bottom multiplication. We're going to do 9 by 24.
Now this time if we were to try and take this 1 and slide it over to 9,
then the rest of our slide was going to be over here and that's of no use to us.
So we're actually going to use the other side, the 1 on the right,
and slide it back over to 9.
Try and get it good...
Okay, and then we're going to take our plastic bit
and slide it over to 2.4 on the C scale.
So 2.4 would be about there.
That corresponds on the D scale to be
2.15 I would say. So just in there.
I don't think it's 2.15 because that wouldn't make sense with these numbers
instead it makes sense for it to be 215.
So that's actually our fraction that we get as an answer.
I'm kind of confident about these first two significant figures,
but the third significant figure I'll be honest was kind of a guess.
So maybe not the most accurate and I feel like most of that's got to do with
The precision here so it's like flimsy pieces of paper and this is barely holding together sensibly
But maybe if you had a more robust slide rule you'd be more confident in your last figure there
So I actually just looked up on my calculator what this would be and it looks like it would be
992 / 216
So we were actually pretty close there.
If we wanted to get some more precision than our slide rule
we could use a log table, which is this crazy table over here,
and we're going to use that in the next question to get a little bit more accuracy.
But I think our sort of approximation here was kind of all right.
I'm going to skip dividing this fraction by this one because it's almost the same as what we've just done.
Instead of dividing I would suggest multiplying
by this fractions reciprocal so that would make a similar calculation.
So I don't think you need to see that again instead we'll do number 2:
Multiply 73 thousandths, which would be 73 / 1000
by 19 hundreds which would be this (19 / 100).
Okay
So if we want to do that
we should exploit the property of logs that the
log(73) + log(19)
will give us the log of the product, 73 * 19.
So if we wanted to find this product here 73 times 19,
then all we need to do is add the logs together and then take the antilog of the answer.
So let's try and do that. We're going to look up in our log table what the log of 73 is
Now in a similar way,
these numbers are all less than 10. So let's have a look for 7.3,
and here it is here. It's a little small, but it corresponds to a decimal number
Which is 0.8633
Now one rule with using this table is that
if there were two significant figures in the number we were taking the log of,
then we do 2 minus 1 and we put a 1 before our decimal point.
It's always going to be n minus 1, I believe.
So it's going to be 1 point, and then our number, 8 6 3 3
Add that to the log of 19. Let's go and find 1.9 which will be here,
that corresponds to 0.2788
So in a similar way that will be 1 (two significant figures) point 2788
If we add those together what we get is 3.1421
That was suspiciously close to pi there for a second.
OK.
That means that if we take the antilog of this number here we should end up with the product that we wanted.
So we'd get another table which is our anti-log table,
you better hope you've got all these handy, and we're going to look up 3.1421
But we're actually just going to look up the part that's after the decimal point in here.
So 0.1421
0.142 and then we're going to add on a 1 which is adding on a 0,
so it's just going to be 0.1387
OK.
So what we're going to get back up here as our answer for the product 73 * 19
will be this value. It will be
0.1387
and then we have a rule that works in reverse,
so remember this 3, you add 1 on to that, becomes 4,
and we times our answer by 10 to the power of 4.
That works out our answer to just be 1,387 as the product between those two numbers.
Now remember what we were actually doing was this up here.
So this is actually equal to 1,387 over
1000 times a 100 will be 1 (counting zeros) one, two, three, four, five...
100,000 and that is our answer (1,387/100,000).
If we look at question 3 now, we're being asked to divide 2880 by 0.0036
Now to do division on our slide rule,
we just do the opposite of what we were doing when we were doing multiplication
So what we want to do is line up
3.6 on our C slider on top of
2.88 on our bottom line, which is our D and we'll see what we get there
So we've got 3.6 lining up with 2.88 roughly there.
And then what we want to do is trace it along and
figure out what our 1 lines up with.
Now, the 1 on the side is off the page. So that's not helpful.
We're going to go to this 1 over here,
and we can see that lines up almost perfectly with our 8.
So our answer to this division question,
Is going to be 8 something, and you maybe need a bit of guess and check on this...
I think that we're going to have 8 (counting zeros) one, two, three, four, five...
...that feels kind of right but I'm gonna go check on my calculator and see.
Yes, verifying on my calculator which is not what you'd be allowed to do in this exam,
looks like this is our right answer here, 800,000. So there you go.
There's three different types of questions solved using either a slide rule or a log table.
We've seen division, multiplication, and multiplication with a bit more accuracy.
The rest of the questions here on this exam I am going to leave as an exercise for the viewer.
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