SPECIAL LESSONS: THE SLIDE RULE

[Music] today we have a special lesson on the slide rule the use of the slide rule before taking up the slide rule proper it is essential that we learn the difference between significant figures and non-significant figures and to illustrate this I have on the board a set of numbers the first set of numbers involve three significant figures here is 235 there are three numbers here three figures here is 23.5 this is three significant figures 2.35 or. 235 see the decimal point has just been moved over 1 two three and so on all of these involve three significant figures the decimal point position makes no difference in significant figures here's 0235 or 00235 or 235,000 the decimal point way over here all of these involve three significant figures only here in the 235,000 if one were making a measurement and one could determine 235 something and then you found later that the decimal point had to be over here but all you you could only measure to three figures then this would only be correct to three figures yet you'd write in the zeros because the magnitude of the entire number is 235,000 so the zeros following the 235 are not significant unless you can measure to that accuracy of one part in 200,000 and very few experiments permit such accurate measurements now here again is a set of four numbers four significant figures 3,195 or if you move the decimal point over and get 3195 this is four significant figures or here 3195 or 0.00 3195 four significant figures these are the only significant ones or 3,1 195,000 only four significant figures now suppose the zeros were between two numbers instead of at the right or left then that's a different story here's 2004 the zeros become are between the two numbers the four is significant here because one was able to measure this to determine its number four here or we might read 20.04 as a measurement or 204 and here again three zeros 204 here the zeros are significant here they are not and here again 20, 40,000 these first four figures are significant these numbers are not usually if they're in front or behind they are not significant figures now the slide rule is capable of three significant figures and that's why I Illustrated these by numbers so you would know the difference let me illustrate why three significant figures are of General importance in most measurements suppose one we're going to make a me measure of the area of this rectangle normally you would use a ruler a meter stick and measure the length and the widths or height suppose you measured the length and found it to be 345 mm and the width or height 234 mm then you would multiply these two numbers together to find the area in square millimeters now here I've done that I multiplied 345 by 234 and carrying out the arithmetic as an answer got 8730 8,730 now suppose I had made an error or instead of reading the the length as 345 suppose I had read 344 a difference of only 1 mm now then supp I read this width is 234 and then multiply those two numbers together 2 344 by 234 and get 8,496 look at the difference now that that 1 mm made here's 8,496 and here's 8,730 this is a seven here in the third place and this is well roughly this nine would make that a five this is 805 roughly so 805 and 807 you see makes a difference of two figures in the third significant figure so in this case these last two numbers mean nothing it is only the third figure that is significant and even that is not quite correct it's off by it a little more than the number the original number 345 or 344 now suppose we had read instead of 340 5 346 a millimeter too long then we would multiply 346 * 234 and get 80,0 96 8,964 again you see the third figure is off by about two numbers 807 809 so again our answer is only uh somewhere nearly correct to the third significant figure so the last numbers here are not important one could just as well write this 8070 because we only know these numbers to within one in the third significant figure now the position of the decimal point has no bearing upon this result now we're ready to go to the slide rule proper and the slide rule is enables you to carry out four different processes as a rule there are others but these are the important ones multiplication division square root and squares now slide rules in general consist of three parts they consist of three parts the body of the slide roll the slipstick and the slider the body of the rule the slipstick and the slider now here is a very large slide rule that I'm going to use but I would like to show you some very small ones first the tight that you nor normally buy at the store and those I have picked out first a very inexpensive one about 9 in long this little slide rule will enable you to carry out about 95% about 95% of the operations you normally want to perform multiplication division squares and square roots it cost only 50s so a slide rule comes within the range of everyone's pocketbook and with it came a little instruction book now here is a more expensive slide rule about 10 in long you see they're about the same in length this one has a few more scales on it this rule will cost you anywhere from $35 to $50 but there are very few additional things that you would ever want to do that you would that you would require such a slide rule 90 to 95% of the things you want to do can be done with an inexpensive one but if you want to buy a little better one you can do so and the ACT you see of both of these slide rules is approximately three significant figures now of course the numbers are there too small and that's why we have to use the large one a demonstration one the body of this slide rule the slipstick and the precursor or Runner as it's called this little piece that slides along along with a line on it now to I want to examine the slide rule first by placing the slipstick parallel with the body of the slide of the rule and we'll push the slider out of the way I want you to examine first of all the scales there are four scales here here's the a scale the B scale the C scale and the D scale the A and B scales are alike you see they line up and the C and D scales are alike and they both start at the same point here with one and the numbers run along so that the upper scales are just like the lower ones only they're half as long the lower scales here you see go from 1 to two there whereas the upper scales go from 1 to two in half the distance now for most operations you use the lower scales because they're twice as accurate they're twice as accurate now to see how to use the slide rule we first have to learn how to read a number when we find a position on it so we're going to examine the different parts of the scale in in parts one at a time now let's start with a section from 1 to two in the A and B scales from here to here we see that interval is divided into five parts 1 2 3 4 5 6 7 8 nine I should say 10 parts the 10 parts and each of the tenth divisions are divided into five parts from here to here there are five intervals in each one of these so you can look upon this as being one and these is T and these little intervals here to give you hundreds but each interval here will be 200s because there are five of them in there suppose we had a reading here on the scale this we would read as 1.4 1.4 because it's over 4/10 or if we had a reading here it would be 1.6 or if we had a reading here it would be 1.62 this is two hunds of the way from here to here or two t0 from there there since these are T that will be hundreds 162 1.62 or if we had a reading here that would be 1 9 6 1 n is here and this is six 1 n six now this interval then can represent one in a decimal or it can represent a larger number a smaller number the decimal point makes no difference this for example could be 14 or 140 or 1 and 4/10 as I said before now suppose we had a reading here of 6 4 this could be 16 6 4 it could be 16 and 410 or 1.64 now let's examine the next change in the scale from two up to five here instead of having each tenth divided into five equal parts it's only divided into two equal parts so these represent 500s here from here to here a reading here would be 2.4 or 24 or 240 a reading here would be 2.7 or 270 or 27 or here would be 38 3.8 or 380 notice here is a little line for pi because often you have to multiply something by pi to get the circumference of a circle or the area of a circle so they tell you where 3.1417 would be now let's go up to the next section the rule between five and one it changes again and now we have only our tenth divisions between numbers between five and six you see they're only the tenth divisions and we have to learn to interpolate now this position would be 5.2 this position 5.8 this position 7.4 and this position 8.4 and so on now when we get to one we see that the rule repeats it itself between one and two is just the same as it was at the left hand end of the rule and so on with the rest of the scale it repeats itself all the way down to the end so we have two scales one following the other both alike now let's look at the C and D scales they're the same as the upper a little little like them anyway and only spread out twice as far here's the one and two interval from here to here and notice now the tenth divisions are not only marked with numbers 1/1 2 T 310 4/10 and so on but between you get in 10 lines so they're hundreds now this position would be read 1.1 or 11 or 110 this would be 1.2 12 or 120 or 1200 or 12,000 now a reading here if I get the right on there would be 1.26 or 12.6 or 126 let's go over to another figure here this would be 1.55 or one uh 155 this would be 161 or 1.6 excuse me 166 would be there wouldn't it 166 at that point or or 1.66 now that from two up to 4 the slider out of the way from 2 to four the scale changes again and here's 2 1 2 two 2 3 4 5 6 7 8 9 3 now this position would be 2 4 and this would be 25 27 this would be 27 4 but it's 4/10 away from there there 274 or 27.4 2.74 and so on up we would read that scale now from four all the way up to one this scale is changed again that's because they couldn't get more lines in here and still read them here this position would be 4.1 because there's four here and five there 4.1 this would be 415 this would be 525 525 and so on up to the end of the scale now that we have seen this part of the scale and seeing how to read them we'll carry out some processes let's start with multiplication and for a simple number to begin with we'll take 2 * 3 just to illustrate you all know the answer of course but we'll take the simple example so that you can see how to use the rule and if when you've learned you've forgotten after a few days always start with simple numbers you know the answer to and will be easy to pick it up again now we're going to multiply 2 * 3 and that is equal to 6 now what we do we start with this index one on the ccale and put it Opposite our first number which is two so we put this in index on the scale opposite two now line it up exactly right there the index on the ccale opposite our first number then we follow along this ccale until we come to our second number and our second number is right there 2 * 3 is equal to 6 now usually one would put the slider here on two on the three and read below it the six but I'm using this pointer because it's a little easier for you to see so 2 * 3 is 6 now let's take a little harder product 25 * 3 which you all can see gives 75 25 * three how would we do that our first number is 25 so we come up here to our D scale and let this represent 20 and this represent 30 and 25 was here so we put our index C right on on the 25 and then we multiply that by three so we go along the scale to three and come down and we find 75 this would be 70 80 75 all right now let's take a little harder number 2.4 * 35 now on the slide rule you forget the decimal point until you get your answer so this is the same as multiplying 24 * 35 two significant figures times two 2 four * 35 now we come up to our Rule and here's 25 here would be 24 so 24 is right here time 35 carries us over here here's 35 at this point and right down here is 84 now in the answer we don't know whether it's 84 or 8.4 or 840 so you write your number down here and then you examine these figures and you round them out to the E even even numbers this is approximately two and this is approximately three uh 30 2 * 30 would be 60 that tells us the decimal point is here 84 is not far from 60 the decimal point goes there certainly 2 * 30 is not going to give six nor is it going to give 600 2 * 30 is 60 so it tells us the decimal point is there not there nor there so you get the decimal point by inspection now let's take 2.6 * 5.5 2.6 * 5.5 now we go up to our scale and we set this index on 2.6 and then we come along the scale and we see we run off we can't read our answer so what do we do we slip the stick back in the other direction we come back to our 2o what did I have here I set that on 27 27 didn't I instead of 26 well anyway we're off the scale so we come back with our index X here to 26 in the other direction see if I get that right 25 26 2.6 * 5.5 now we go along the upper scale to our second number and it comes right here 5.5 so we come down here and our answer is 14 and now here we have to guess how far it is between there if we want to interpolate our answer would be 143 see one four is to there 1 two 3 1 4 3 and we could say two but the three figures it would be 143 so we write down 1 143 and then to find the decimal point we examine these two numbers you see this is round two or three and this is around five now 3 * 5 or 2 * 5 would be 10 or 15 it tells us the answer has a decimal point here it couldn't be one 2 * 5 is certainly greater than one and it certainly is not 140 or 143 so there is a decimal point now let's take a three significant figure number like the one of the area of our rectangle 345 You Remember by 234 345 by 234 now we slide this along here would be 300 and 40 and here would be five 345 we'd put our Index right there 301 2 4 and here's halfway between 345 time 234 this would be 200 10 20 30 and this would be four here now our answer is eight here I better put this slider on here now 234 is right there now here's 8 and there's 85 there's 81 so it's not quite up to one we'd have to interpolate here 80 this would be 81 this would be 80 7 or8 looks merely seven so our answer would be and we could write it down here 807 now to find the decimal point we round off the numbers 200 roughly * 300 200 * 300 would give us six and then four zeros so we see this should be 8,700 and you remember the answer we got was 807 something in here these are not significant figures because this is only correct or nearly correct in the third significant figure now let's carry out division processes see how they are done here again we use the lower scales and to take a simple example Let's Take 6 / 3 = 2 you can all do it in your head of course but let's try it on the slide rule now this is the reverse process of multiplication what you do is set these two numbers up on your C and D scales upside down the six on the D scale and three opposite on the C scale so we'll find six the numerator and that is here and then we'll bring along opposite it our denominator three and then over on our index here we find two because you see 2 * 3 is 6 6 / 3 is 2 now let's take a little more complicated number 28 ided 4 which mentally you can see gives 7 28 divided by four I we go up on this D scale and find 28 that would be here here's 30 that would be 28 divided by four bring along four opposit it and then look to the index that is on the rule and the answer is seven now if you'd gone the other direction to the other index you'd find it off the rule you couldn't read it now let's take a little more complicated number uh 675 / 121 both have three significant figures 675 by 121 always set the top one up on the bottom scale 600 and 70 would be there 675 would be there now I put the slider there this is the way you'd normally do it 675 would be there and then your other number 121 is brought directly over it now here is 100 and 200 121 there's 120 there 121 is right under this slider Mark and the answer is right here and that will be five five and about eight seven or eight in there you have to interpolate this is five this is one this 5 five 56 5 five seven or eight and the answer actually is 558 now you get the decimal point here by looking again at the numbers this is roughly 600 divided by 100 100 goes into 600 six times so that tells us the decimal point goes here and the answer is really 5.58 now here's a process using a little larger number 9528 / by 695 9528 / 695 is 13.71% 9528 to do that we'll go over the far right hand side and pick up 95 there's 96 9528 would be about here I better put the slider over that position and divided by 695 so we find six and here's 869 695 is right under the mark now this index is off so we have to go way over to the left and find the answer here as 1 3 7 13 7 we can't get the fourth significant figure because the slide rule in general is only good to three significant figures so we get as an answer 137 and the examination of the figure shows its decimal point is here now let's take square roots of some numbers suppose we start with a square root of 25 which you all know is five now for this purpose you put the scales and the sliding scale directly opposite each other you don't have to do this but you make you're less AP to make an error if you slide them even with each other and leave them there then what you do is take your number find it on the upper scale we go way over here to the left and examine this now you don't know whether to go to the right hand scale of the left so we look first at some simple numbers we know here's a four and a two the square OT of four is two if you go over here to nine you find the square root of 9 is three if you go along here there's 15 this can be 16 the square Ro TK of 16 is four now we wanted the square root of 25 so here is 20 30 there's 25 and the square root of it is five now suppose you want to the square root of 400 you don't know on the upper scales whether to come to the four here or whether you go to the four on the other side now the rule is this if the numbers between 1 and 10 you use the left hand upper scale between 10 and 100 you use the right hand scale between 100 and a th000 between in the left hand scale and you can do that by examination now the square root of 400 says we should use this scale and the square root of 400 is 20 so you take the square roots by finding your number on the top scale and the square root of it will come out below to square numbers you just do the reverse process you can either find your number here and read it off on the upper scale which gives you the square like square of two is square of two is four and the square of three is 9 or you can multiply 3 * 3 or 4 * 4 whatever your number is by using just the C and D scales alone using the regular regular multiplication process now a little practice with the slide rule should enable you to carry out these simple operations multiplication division square root and squares lots of practice will save you a great deal of time so get yourself a slide rule you can all afford one cheapest ones are perfectly good and we will'll give you three significant figures examine them carefully when you go to the store to buy one line the scales up at the left and see that they agree with each other all along but one of them isn't hasn't shrunk a little but get yourself a slide rule learn how to use it it will save you a lot of time and energy