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Question 411536: Explain why the product of a number with 3 digits after the decimal point and a number with 4 digits after the decimal point would have 7 digits after the decimal point.
Answer by Theo(13342)   (Show Source):
You can put this solution on YOUR website! .001 * .0001 = (1 * 10^-3) * (1 * 10^-4) = (1 * 1) * (10^-3 * 10^-4) = 1 * 10^-7 = .0000001
.999 * .9999 = (999 * 10^-3) * (9999 * 10^-4) = (999 * 9999) * (10^-3 * 10^-4) = 9989001 * 10^-7 = .9989001
the rules of algebraic and exponential arithmetic dictate that it will be so.
consider:
.999 * 1 = .999
.999 * .1 = .0999
.999 * .01 = .00999
.999 * .001 = .000999
.999 * .0001 = .0000999
the last place, or least significant digit, in the multiplier shifts the answer one more decimal place to the right.
also consider:
.999 * 1 = .999 / 1 = .999
.999 * .1 = .999 / 10 = .0999
.999 * .01 = .999 / 100 = .00999
.999 * .001 = .999 / 1000 = .000999
.999 * .0001 = .999 / 10000 = .0000999
it's a hard and fast rule because that's the way it works.
this will happen regardless of the number of digits after the decimal place, but you have to consider the digits before the decimal place as well.
consider:
1 * .1 = .1
10 * .1 = 1
in the first case, the number of digits in the result is equal to 1 which is equal to 0 decimal places in the multiplicand and 1 decimal place in the multiplier.
in the second case, the number of digits in the result is equal to 0 which is NOT equal to 0 decimal places in the multiplcand and 1 decimal place in the multiplier.
this appears to be so, but the rule, as applied, allows for this.
you take 2 numbers and multiply them together and then you place the number of decimal places in the result based on the number of decimal places in the multiplicand and in the multiplier.
example 1:
1 * .1 = .1
you took 1 * 1 to get 1 and then you shifted 1 decimal place to the left to get .1
example 2:
10 * .1 = 1
you took 10 * 1 to get 10 and then you shifted 1 decimal place to the left to get 1.
example 3:
.999 * .9999 = .9989001
you took 999 * 9999 = 9989001 and then you shifted 7 decimal places to the left to get .9989001
example 4:
999 * .9999 = 998.9001
you took 999 * 9999 = 9989001 and then you shifted 4 decimal places to the left to get 998.9001
I'm not sure if this answers the question to your satisfaction.
The rules are applied because they work based on observations of the results over many trials.
A number with 3 decimal places multiplied by a number with 4 decimal places will always have 7 decimal places in the answer.
The number of decimal places is determined by the fact that the least significant digit to the right of the decimal place can't be 0.
This forces the answer to have the least significant digit in the answer to be the sum of the number of decimal places in the multiplicand plus the number of decimal places in the multiplier.
The scientific notation shows this to be a factor of adding the exponents together.
1 * 10^-3 multiplied by 1 * 10^-4 will yield an answer of 1 * 10^-7 because the exponents are added together as long as the based are the same.
without the use of scientific notation, you can easily see the progression if you work your way out from 0 decimal places in the multiplier to 4 decimal places in the multiplier.
.001 * 1 = .001
.001 * .1 = .0001
.001 * .01 = .00001
.001 * .001 = .000001
.001 * .0001 = .0000001
each time adds the number of decimal places in the multiplicand to the number of places in the multiplier to form the result.
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