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Accounting for Change of Magnitude
To multiply two numbers in scientific notation, you multiply the coefficients and add the exponents. To divide, divide the coefficients and subtract the exponents. This is particularly well suited to multiplying and dividing with a slide rule, except when the coefficient of the result is no longer between one and ten. Numbers on the C and D scales are always between 1 and 10, but the product of two such numbers can be higher than 10, and the quotient less than 1. You must learn to recognize this and adjust the exponent of the result. We'll do some easy examples to demonstrate this.
- a) 2 times 2
- -converted to scientific notation is 2E0 times 2E0.
- -multiply coefficients with slide rule and add exponents in your head: produces 4E0.
- -answer is 4. Correct.
- b) 2 times 6
- -converted to scientific notation is 2E0 times 6E0.
- -multiply coefficients with slide rule and add exponents: produces 1.2E0.
- -answer is 1.2 Not correct.
- You're not likely to make this mistake multiplying 2 x 6, but you might when multiplying 0.000000002 x 600000.
The difference between the two examples is that in a), the coefficient of the answer is 4, which lies between 1 and 10. In b), the coefficient of the answer is 12, which lies outside 1 and 10. You should have written the answer as 12E0, (1.2E1), and got the correct answer of 12.
The fact that you used the 10C side of the slide rule to do this calculation is an indicator that the coefficient of your answer is actually 10 times the value on the scale. Some people use that method to adjust the exponent. But more fundamentally, when you multiply the coefficients on the slide rule, ask youself whether the result coefficient is going to be 10 or greater, and increase the exponent of your answer by 1 if they are.
Most of the time this is very easy even when working with many decimal places. In the times when its not, glancing at the result of the multiplication will tell you whether it is just over or just under 10.
Likewise, when you divide coefficients when performing a division, ask youself if the answer coefficient is going to be less than 1, and reduce the exponent of the answer by 1 if so. This is trivial - the answer will be less than one only if the first coefficient is less than the second coefficient.
Examples:
- 1.34 x 20.5
- convert to scientific notation: 1.34E0 x 2.05E1
- multiply coefficients with slide rule, add exponents in your head:
- answer 2.75E1
- do we adjust the magnitude?
-
- let's see: does 1.34 x 2.05 equal 10 or higher?
- no way, it won't even be 3.
- so we don't change the exponent.
- final answer 2.75E1, 27.5
- 450 x 500
- convert to scientific notation: 4.5E2 x 5E2
- multiply coefficients with slide rule, add exponents in your head:
- answer 2.25E4
- do we adjust the magnitude?
-
- does 4.5 x 5 equal 10 or higher?
- of course, its obviously over twenty.
- so we add one to the result exponent.
- final answer 2.25E5, 225000
- 0.0045 x 0.027
- convert to scientific notation: 4.5E-3 x 2.7E-2
- multiply coefficients with slide rule, add exponents in your head:
- answer 1.215E-5
- do we adjust the magnitude?
-
- does 4.5 x 2.7 equal 10 or higher?
- well, 4 x 2 is 8, and 5 x 3 is 15.
- so its hard to say. So we look at the result coefficient 1.215
- it must be over ten, if it was under ten it would be 8 or more.
- so we add one to the result exponent.
- final answer 1.215E-4, 0.0001215.
- 0.0045 / 0.027
- convert to scientific notation: 4.5E-3 / 2.7E-2
- divide coefficients with slide rule, subtract exponents in your head:
- answer 1.67E-1
- do we adjust the magnitude?
-
- is 4.5 / 2.7 less than 1?
- no, because 4.5 is more than 2.7.
- so we leave the exponent unchanged.
- final answer 1.67E-1, 0.167
- 0.005 / 9000
- convert to scientific notation: 5E-3 / 9E3
- divide coefficients with slide rule, subtract exponents in your head:
- answer 5.56E-6
- do we adjust the magnitude?
-
- is 5 / 9 less than 1?
- yeah. because 5 is less than 9.
- so we reduce the exponent by 1.
- final answer 5.56E-7, 0.000000556
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