Scientific Notation and Significant Figures

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Mole

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I can pretty much do scientific notation, but I'm not sure on whether I'm using significant figures correctly in my answers.

Convert each of the following into scientific notation:

a. 727 -----> 7.27 x 10^2
b. 172000 -----> 1.72 x 10^5
c. 0.000984 -----> 9.84 x 10^-4
d. 200.0 -----> 2 x 10^2 ? How does the extra decimal place affect the answer - is it more certain?
e 0.014 -----> 1.4 x 10^-2
f. 2560000000000000000000000000000000 (use 4 sig. figs) -----> 2.560 x 10^33 Is there a reason for using 4 significant figures, or is that arbitrary?

Convert each into decimal form:

a. 1.56 x 10^4 ----->15600
b. 2.6 x 10^-2 -----> .036
c. 736.9 x 10^5 -----> 73690000
d. 0.0059 x 10^5 -----> 590
e. 0.00059 x 10^-1 -----> .000059

Calculate the following:

a. 2.34 x 10^65 + 9.2 x 10^66 ------> I don't know what to do here. Should I change 2.34 x 10^65 to .234 x 10^66 and add them to get 9.4 x 10^66?
b. 313.0 - 1.2 x 10^3 -----> -.877 x 10^3 ? Same question as in a.

Calculate the following. Give the answer in correct scientific notation with the correct number of significant figures:

a. 8.95 x 10^76/1.25 x 10^56 -----> 7.16 x 10^20
b. (4.5 x 10^29)(2.45 x 10^100) -----> I'm not sure here. Do I multiply the individual components of the terms of the whole terms?
 
Mole said:
I can pretty much do scientific notation, but I'm not sure on whether I'm using significant figures correctly in my answers.

Convert each of the following into scientific notation:

a. 727 -----> 7.27 x 10^2
b. 172000 -----> 1.72 x 10^5
c. 0.000984 -----> 9.84 x 10^-4
d. 200.0 -----> 2 x 10^2 ? How does the extra decimal place affect the answer - is it more certain?

If you're told that a value is 200.0, then that is more precise than saying 200. What if the true value is a little bit more than 200? If I just say 200, then that could mean 200.1 or 200.2 or 200.3 or 200.36 or 200.25 or 200.14 ... it could mean a lot of things. By saying 200.0, I narrow down the error. Now the value is known with more precision -- it could be 200.01 or 200.02 or 200.03 ...

200.01 shows more precision than 200.1

2.000 x 10^2 shows more precision than 2.00 x 10^2

YOU WROTE 2 x 10^2 Note: 2 x 10^2 is not scientific notation

Since you show NO decimal places at all, the resulting precision is so low that the true value could be 221 or 238 or 209 or 241 or ...


e 0.014 -----> 1.4 x 10^-2
f. 2560000000000000000000000000000000 (use 4 sig. figs) -----> 2.560 x 10^33 Is there a reason for using 4 significant figures, or is that arbitrary?

Same situation as above because 2.560 is more precise than 2.56

By seeing the zero in 2.560 x 10^33, I know that the true value has a zero for its fourth digit.

If you write 2.56 x 10^33 instead, then I can't be sure what the fourth digit is.

2564000000000000000000000000000000, for example.

Convert each into decimal form:

a. 1.56 x 10^4 ----->15600
b. 2.6 x 10^-2 -----> .036 This is correct to me only if you intended to type 0.026
c. 736.9 x 10^5 -----> 73690000 Note: 736.9 x 10^5 is not scientific notation, and neither are the next two exercises.
d. 0.0059 x 10^5 -----> 590
e. 0.00059 x 10^-1 -----> .000059

Calculate the following:

a. 2.34 x 10^65 + 9.2 x 10^66 ------> I don't know what to do here. Should I change 2.34 x 10^65 to .234 x 10^66 and add them to get 9.4 x 10^66?

PERFECTION!

b. 313.0 - 1.2 x 10^3 -----> -.877 x 10^3 ? Same question as in a.

Firstly, -0.877 x 10^3 is not scientific notation.

Secondly, we can't write -8.77 x 10^2 either.

We are subtracting something close to 1200 from something close to 313.0

1.2 x 10^3 could be anything from 1200 through 1249, for example.

313.0 could be anything from 313.0 through 313.04, for example.

So, the difference could be anything from about -936 to about -877. We just don't know. Therefore, with respect to precision, we have to settle on -900.

How would you round your answer and express it in scientific notation to indicate the correct number of significant digits? Try again.

Calculate the following. Give the answer in correct scientific notation with the correct number of significant figures:

a. 8.95 x 10^76/1.25 x 10^56 -----> 7.16 x 10^20
b. (4.5 x 10^29)(2.45 x 10^100) -----> I'm not sure here. Do I multiply the individual components of the terms of the whole terms?

Do you remember the Commutative Property of Multiplication?

We are allowed to multiply a string of factors in ANY order we wish.

(A * B) * (C * D) = (A) * (B) * (C) * (D) = (A) * (C) * (B) * (D), if we like ...

Try again, and keep the multiplication rule for significant digits in mind when rounding your result and writing it in scientific notation.
Click to expand...
 
mmm4444bot said:
Mole said:
I can pretty much do scientific notation, but I'm not sure on whether I'm using significant figures correctly in my answers.

Convert each of the following into scientific notation:

a. 727 -----> 7.27 x 10^2
b. 172000 -----> 1.72 x 10^5
c. 0.000984 -----> 9.84 x 10^-4
d. 200.0 -----> 2 x 10^2 ? How does the extra decimal place affect the answer - is it more certain?

If you're told that a value is 200.0, then that is more precise than saying 200. What if the true value is a little bit more than 200? If I just say 200, then that could mean 200.1 or 200.2 or 200.3 or 200.36 or 200.25 or 200.14 ... it could mean a lot of things. By saying 200.0, I narrow down the error. Now the value is known with more precision -- it could be 200.01 or 200.02 or 200.03 ...

200.01 shows more precision than 200.1

2.000 x 10^2 shows more precision than 2.00 x 10^2

YOU WROTE 2 x 10^2 Note: 2 x 10^2 is not scientific notation

Since you show NO decimal places at all, the resulting precision is so low that the true value could be 221 or 238 or 209 or 241 or ...


e 0.014 -----> 1.4 x 10^-2
f. 2560000000000000000000000000000000 (use 4 sig. figs) -----> 2.560 x 10^33 Is there a reason for using 4 significant figures, or is that arbitrary?

Same situation as above because 2.560 is more precise than 2.56

By seeing the zero in 2.560 x 10^33, I know that the true value has a zero for its fourth digit.

If you write 2.56 x 10^33 instead, then I can't be sure what the fourth digit is.

2564000000000000000000000000000000, for example.

Convert each into decimal form:

a. 1.56 x 10^4 ----->15600
b. 2.6 x 10^-2 -----> .036 This is correct to me only if you intended to type 0.026 3.6 x 10^-2
c. 736.9 x 10^5 -----> 73690000 Note: 736.9 x 10^5 is not scientific notation, and neither are the next two exercises.
d. 0.0059 x 10^5 -----> 590
e. 0.00059 x 10^-1 -----> .000059

Calculate the following:

a. 2.34 x 10^65 + 9.2 x 10^66 ------> I don't know what to do here. Should I change 2.34 x 10^65 to .234 x 10^66 and add them to get 9.4 x 10^66?

PERFECTION!

b. 313.0 - 1.2 x 10^3 -----> -.877 x 10^3 ? Same question as in a.

Firstly, -0.877 x 10^3 is not scientific notation.

Secondly, we can't write -8.77 x 10^2 either.

We are subtracting something close to 1200 from something close to 313.0

1.2 x 10^3 could be anything from 1200 through 1249, for example.

313.0 could be anything from 313.0 through 313.04, for example.

So, the difference could be anything from about -936 to about -877. We just don't know. Therefore, with respect to precision, we have to settle on -900.

How would you round your answer and express it in scientific notation to indicate the correct number of significant digits? Try again. -8.8 x 10^2 ?

Calculate the following. Give the answer in correct scientific notation with the correct number of significant figures:

a. 8.95 x 10^76/1.25 x 10^56 -----> 7.16 x 10^20
b. (4.5 x 10^29)(2.45 x 10^100) -----> I'm not sure here. Do I multiply the individual components of the terms of the whole terms?

Do you remember the Commutative Property of Multiplication?

We are allowed to multiply a string of factors in ANY order we wish.

(A * B) * (C * D) = (A) * (B) * (C) * (D) = (A) * (C) * (B) * (D), if we like ...

Try again, and keep the multiplication rule for significant digits in mind when rounding your result and writing it in scientific notation. 4.5 * 2.45 *10^29 *10^100=11.025 x 10^129=1.1 x 10^130
Click to expand...
Click to expand...
 
Mole said:
b. 2.6 x 10^-2 -----> .036 This is correct to me only if you intended to type 0.026 3.6 x 10^-2

b. 313.0 - 1.2 x 10^3
-8.8 x 10^2 ?

b. (4.5 x 10^29)(2.45 x 10^100)
4.5 * 2.45 *10^29 *10^100=11.025 x 10^129=1.1 x 10^130
Click to expand...

OIC -- 0.036 is correct for the first exercise b in the quote above.

Your answer for the third exercise b in the quote above is also correct.

You did not properly round your result for the second exercise b in the quote above; -887 rounds to -890.

When I see 1.2 x 10^3, then I think that 1200 has only two significant figures. So, I see the difference 313.0 less 1200 as -900 due to the way that my first chemistry professor drilled it into me. (He had different rules for rounding, too, because he saw that continually chopping off little pieces by never rounding up during experiments could eventually lead you astray.)

I would (for myself) write the answer as -9.0 x 10^3; HOWEVER, I found contradictory information when I checked some resources.

If I go strictly by the majority of rules that I saw, then I would indeed need to write -8.9 x 10^3.

Perhaps, one of the eagle eyes at this site will clear this up. Otherwise, please ask your instructor. Let us know what you find out!

Cheers,

~ Mark :)
 
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