Start with a simple example: 2 X 3 = ?
Move the slide until the left index of scale C aligns with 2 on scale D.
Move the cursor until the hairline is on 3 of scale C.
Read 6 at the hairline on scale D.
In slide rule language, we would write
Left index of C to 2 on D
Cursor to 3 on C
Why does this work? What we really did was to take the distance represented by the logarithm of 2 (log 2), and added the distance represented by the logarithm of 3 (log 3). Log 2+ log 3 = log (2 X 3) = log 6, and log 6 is represented by 6 on the D scale.
Even though the numbered locations on the scales are the locations of the logarithms of our numbers, they are labeled with the actual numbers themselves, enabling us to read our answers directly.
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Now try: 6 X 7
lf we slide the left index of C to 6 on D, we see that 7 on scale C is off the end or off scale. So we switch indexes...
Right index of C to 6 on D
Cursor to 7 on C
The slide rule does not tell us the location of the decimal point, but by estimate we know it must be 42, not 4.2, or 420, etc. We recommend this type of rough estimate procedure for determining the location of the decimal point.
Next try: 3.6 X 2.9
Left index of C to 36 on D
Now we try to set the cursor to 29 on C, but it is about a centimeter off the scale.
So, switch indexes.
Right index of C to 36 on D
Cursor to 29 on C
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How can we be confident about the last digit ... 4? Look at where the cursor points on D. It is somewhere between 104 and 105. In the original problem, the last digits of the numbers we are multiplying are 6 and 9.
6 times 9 = 54 ... and the last digit of 54 is 4.
So, 3.6 times 2.9 must end with a 4.
Therefore, the cursor is at 10.44 on D.
Estimate for decimal point: 4 times 3 = 12. So the answer is 10.44.
We cannot always use this technique for finding the last digit in our answer. For instance, if we multiply two numbers that have 3 digits each, then the answer would have 5 or 6 significant digits, which is beyond the ability of slide rule reading.
Solve: 6 divided by 4 (6/4) using the basic C and D scales:
Cursor to 6 on D
Slide to 4 on C at cursor
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You may note that when the problem is set up on your rule this way, the 4 appears on your rule above the 6 ... upside down ... the inverse of how we would write the problem as a fraction ... 6/4. Strange! But this is a traditional way to divide with your slide rule.
There are other ways. If your rule has CF and DF scales, you can use them to divide ... and the fraction will appear on the 2 scales in the normal manner ... the 6 will be over the 4. The answer 1.5 can be found either on DF or D.
You can also use the A and B scales to divide ... and the numbers will appear the same on your rule as they do in the fraction ... 6/4. Any paired set of scales can be used to divide. Some rules have paired K (cube) scales.
Another method is to set up the fraction in the normal fashion ... 6/4 ... on the C and D scales.
You can also divide using the CI and D scales:
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In this case, you are multiplying 6 by the reciprocal or inverse of 4, which is 1/4. 6 X 1/4 = 1.5.
Now try: 6 divided by 4 ... using a different method.
We divided 1 by 4, then multiplied the quotient by 6.
Try: 99 divided by 11 (99/11)
Here are 5 methods for solving this problem.
You can continue alternating the use of C, D and CI (and DI if your rule has it) to easily perform chain calculations of any length, whether all multiplication, all division or both in the same problem.
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Any combination of multiplying and dividing the numbers will solve the problem. Any combination of using the C, D and CI scales will also do the trick! If it appears that you are stuck and need to reset the slide, using the CI scale will often save you. You can also use the CF and DF scales in chain calculations.
Reciprocals
The A and B scales are used for squares and square roots.
Numbers on the A and B scales are squares of numbers on the C and D scales.
Numbers on the C and D scales are square roots of numbers on the A and B scales.
Note that the A and B scales are in 2 sections ... 2 decades... or 2 halves ... 2 scales beginning and ending with the digit one (or on some rules the second section begins with 10 and ends with 100). The number you seek will be on one of these sections only ... and not on the other.
If your number has an odd number of digits before the decimal point, it will be on the first section of the A or B scale. (If it is less than 1 and has an odd number of zeros after the decimal point, also use the first section of the A or B scale).
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If your number has an even number of digits before the decimal point, it will be on the second section of the A or B scale. (If it is less than 1 and has no zeros or an even number of zeros after the decimal point, also use the second section of the A or B scale).
If you prefer not to deal with the zeros after the decimal point, and would prefer to work with whole numbers, simply shift the decimal point to the right 2 places at a time until you have a whole number.
Then use the rules for whole numbers to locate the proper section of the A or B scale to use. In this last example, shifting the decimal point two places to the right twice (a total of 4 places), the whole number would be 62.5, an even number of digits before the decimal point, which tells us to use the second section of the A or B scale.
The square root of 62.5 is then 7.91. Shift the decimal point back to the left 2 places (since we are dealing with the square root, this compensates for our shifting the decimal point 4 places to the right before we started) to obtain the answer 0.0791.
Numbers on the K scale are cubes of numbers on the D scale.
Numbers on the D scale are cube roots of numbers on the K scale.
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The K scale consists of 3 sections, or decades.
The first section is numbers from 1 to 10 ... numbers with one digit before the decimal point ... or 1000 or 1 million, etc. times those numbers (numbers with 4 digits or 7 digits, etc. before the decimal point) ... or one thousandth or one millionth, etc. of those numbers (2 zeros or 5 zeros, etc. after the decimal point).
The second section is numbers from 10 to 100 ... numbers with 2 digits before the decimal point ... or 1000 or 1 million, etc. times those numbers (numbers with 5 digits or 8 digits, etc. before the decimal point) ... or 1 thousandth or 1 millionth, etc. of those numbers (1 zero or 4 zeros, etc. after the decimal point).
The third section is numbers from 100 to 1000 ... numbers with 3 digits before the decimal point ... or 1000 or 1 million, etc. times those numbers (numbers with 6 digits or 9 digits, etc. before the decimal point) ... or 1 thousandth or 1 millionth, etc. of those numbers (no zeros or 3 zeros or 6 zeros after the decimal point).
The number you seek will be on only one of these 3 sections.
Dont like dealing with numbers less than one? We neither. So ... shift the decimal point to the right 3 places at a time until you reach a whole number.
In the last example, one such shift reaches the whole number 6.7
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Using the first section of the K scale per the instructions above, we find the cube root to be 1.89 on D. Now shift the decimal back to the left 1 place (since we are dealing with the cube root, this compensates for our shifting the decimal point 3 places to the right before we started). Our answer is again 0.189.
The CF and DF scales are C and D scales which begin at p instead of 1. They are folded at p and are referred to as folded scales. Thus, all numbers on them are the corresponding numbers on C & D multiplied by p. The CF and DF scales may be used for multiplication and division in the same manner as the C and D scales.
It is no accident that the slide rule is arranged this way. Two good reasons are: Folding at p puts the folded scale indices near the center. This facilitates computations which involve numbers near an index without going off scale so easily. And of course, multiplying by p is frequently used for circle and sphere problems.