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Website last updated 03 August, 2010

Multiplication, Division, and Reciprocals with the C and D scales

These instructions cover:
  1. Multiplication using the C index method
  2. Multiplication using the D index method
  3. Division using the reciprocal method
  4. Division using the fraction method
  5. A note on reciprocals
  6. One-step divide and multiply
These are the first and most important techniques to master, as well as perhaps the most difficult.

Terminology

In our discussion, we will always assume the C and D scales run from 1 to 10. We will refer to numeric positions as decimal numbers between 1 and 10. The term "C index" means the 1 or the 10 on C. Most of the time we will say 1C or 10C explicitly.

Multiplication using the C index

z = x times y

  • Set 1C or 10C at xD
  • Read the coefficient of z at yC, on D
  • Assign the decimal point
Follow along on your slide rule for the following examples:
2 times 3
Set 1C over 2D, the answer is at 3C, 6 on D. Decimal point is correct.
12 times 0.45
Set 1C at 1.2D, the answer is at 4.5C, 5.4 on D. Decimal point is correct.
5 times 3
Set 1C at 5C, the answer is at 3C, ... Oh oh.
3C is off the scale, off the screen even!
This is the sort of case where you have to use 10C instead of 1C.
Set 10C at 5C. The answer is at 3C, 1.5 on D.
Shift decimal point one right because of Cambios de Magnitud
So 5 times 3 is 15.
How do you know whether to set 1C or 10C? Ignore the decimal places and assume both numbers lie between 1 and 10. If the product of those two numbers will be ten or greater, use 10C. Otherwise use 1C. This is the same as the change of magnitude rule. Most of the time this is easy:
240 times -0.023
think: "2.4 x 2.3 - that's going to be less than 10. I'll use 1C."
-57 times 7.94E9
think: "5.7 x 7.9 - way over 10. I'll use 10C."
3.6 times 2.81
think: "a tough call. I'll just use 1C and reset to 10C if it doesn't work."
alternately think: "a tough call. But I'm using extended scales so I'll be likely be fine."
alternately think: "I don't care because that's what Escalas Dobladas are for."
And what about cases where it's not obvious where the decimal point should go? It's good to think about this quite clearly because if its not in the right place, your answer will be amazingly wrong. Part of the advantage of using a slide rule is it forces you to continually keep track of order of magnitude, and this ability will develop with practice. Read the tutorial Setting the Decimal Point"

Multiplication using the D index

z = x times y

  • Set xC at 1D or 10D
  • Read the coefficient of z at yD on C
  • Assign the decimal point
Since the C and D scales are identical, we can switch their roles with no effect on the result. Isn't it confusing to learn two different methods? Yes, but it is necessary when you have an intermediate result on the C scale instead of the D scale. For instance, when your trig scales are on the Slider, computing values like 5.3sin(45) can be done with the most efficiency using this second method. Here are the same examples as above, reworked using 1D.
2 times 3
Set 2C at 1D, the answer is at 3D: 6 on C. Decimal point is correct.
12 times 0.45
Set 1.2C at 1D the answer is at 4.5D: 5.4 on C. Decimal point is correct.
5 times 3
Set 5C at 1D, find 3D and notice it is off the C scale.
So we have to use 10D instead of 1D.
Set 5C at 10D. The answer is at 3D, 1.5 on C.
Shift decimal point one right because of Cambios de Magnitud
So 5 times 3 is 15.

Division, reciprocal method

This type of division is a little easier than multiplication. First, you can't go off scale, and second, both the answer and the reciprocal of the answer (ie y/x) can be found without resetting the scales.

z = x / y

  • Set yC at xD
  • Read coefficient of z on D at 1C or 10C, whichever is on scale
  • Assign the decimal point
  • ( Also, reciprocal is found on C at 1D or 10D, at the other end. )
We call this the "reciprocal method" because when yC is sitting overtop xD, it looks like the fraction y over x, when in fact we are calculating x over y. There is a "fraction method" described below which sets xC over yD. We describe this one first because it is the most important. This type of division can be combined with C index multiplication to create efficient chains of operations which have minimal resetting and the least accumulated error.

Example

3 / 4
Set 4C over 3D, the answer is 7.5 on D at 10C, because 1C is offscale.
Shift decimal point one left because of Cambios de Magnitud
Final answer 0.75.
Also, reciprocal (4/3) is 1.333 on C at 1D, because 10D is offscale. Decimal point is fine.

Division, fraction method

This is identical the reciprocal method, but the fraction is right side up, and you read the answer off C instead of D.

z = x / y

  • Set xC at yD
  • Read coefficient of z on C at 1D or 10D, whichever is on scale
  • Assign the decimal point
  • ( Also, reciprocal is found on D at 1C or 10C, whichever is on scale )
Example

2 / 3
Set 2C over 3D, the answer is 6.7 on C at 10D, because 1D is offscale.
Shift decimal point one left because of Cambios de Magnitud
Final answer 0.67.
Also, reciprocal (3/2) is 1.5 on D at 1C, because 10C is offscale. Decimal point is fine.

Reciprocals

The reciprocal of y is 1/y. Use any of the division methods on this page to find the value.

One-step divide and multiply

to find z = (w*x)/y

  • set yC at wD (reciprocal method division)
  • Read coefficient of z on D at xC (C index multiplication)
  • Accumulate decimal point at both steps.
This is a gateway to a lot of important techniques. It performs a divide then a multiply with with one motion, improving speed and reducing error.

This technique can fail if just the C and D scales are used. The division is performed first, and if the quotient is too large or small, the multiplication will be off-scale. But you can use the CF and DF scales in that case.

Examples

2 x 6 / 3
Do the division 2/3 first:
Set 3C over 2D, answer 6.67 on D at 10C. Intermediate result 0.667 after shifing decimal.
Now you can read the multiplication by 6 without changing anything:
coeffiecient of answer is 4 on D, at 6C.
Set decimal: 0.6667 * 6 is going to be 4E0, just 4.
but notice from the image that you couldn't multiply by 14 rather than 6.
(2/3) x (6/7) x 8
Do the division 2/3 first:
Set 3C over 2D, answer 6.67 on D at 10C. Intermediate result 0.667 after shifing decimal.
Read the multiplication by 6 without changing anything:
Intermediate result is 4 on D, at 6C.
Divide by 7:
Set cursor on 6C
Move 7C to the cursor, intermediate result .571 at 10C on D.
now read the multiplication by 8 without changing anything:
final result 4.57 on D, at 8C.
The second example is the same as the first example, except we continue on and divide by 7 then multiply by 8. Notice you only move the slider twice for all that calculation, saving an intermediate result with one motion of the cursor. You could then go on to divide and multiply again following the same steps. This is an example of a chain. It pays to organize a list of operations into alternating divides then multiplies as much as possible.