Question

put the following numbers in order of increasing amount of sig figs (i.e. the number with the least amount of sig figs goes on top of the list (closest to this question) and number with the most sig figs will go at the bottom of the list.) note: if two numbers have the same number of sig figs, put them in order from smallest to largest number. 1. 0.6363 2. 7.57 × 10⁶ 3. 737,290 4. 0.04321400 5. 9.6 × 10³ 6. 7.6136 × 10⁵

Explanation:

Step1: Determine sig figs for each number

- \(0.6363\): All non - zero digits and the digits after the decimal are significant. So, number of sig figs \( = 4\).
- \(7.57\times10^{6}\): In scientific notation, the significant figures are in the coefficient. So, number of sig figs \( = 3\).
- \(737290\): Trailing zero in a whole number without a decimal is ambiguous, but here we consider non - trailing digits. The significant figures are \(7,3,7,2,9\) (the zero may or may not be significant, but assuming it's not, number of sig figs \( = 5\)). Wait, actually, for \(737290\), if we consider the zero as not significant (since there's no decimal), the number of significant figures is 5. But let's re - check:
- \(0.6363\): 4 sig figs.
- \(7.57\times10^{6}\): 3 sig figs (7,5,7).
- \(737290\): Let's count again. The digits are 7,3,7,2,9,0. If the zero is not significant (because it's a trailing zero in a whole number without a decimal), then sig figs are 5. But maybe I made a mistake. Wait, the rule is: trailing zeros in a whole number are not significant unless there is a decimal. So \(737290\) has 5 significant figures (7,3,7,2,9).
- \(0.04321400\): Leading zeros are not significant. The significant figures are 4,3,2,1,4,0,0? Wait, no. The rule is: leading zeros are not significant, zeros between non - zero digits are significant, and trailing zeros after a decimal are significant. So \(0.04321400\): the significant figures are 4,3,2,1,4,0,0? Wait, no. Let's write the number as \(4.321400\times10^{- 2}\). So the coefficient is \(4.321400\), which has 7 significant figures (4,3,2,1,4,0,0).
- \(9.6\times10^{3}\): In scientific notation, the coefficient \(9.6\) has 2 significant figures.
- \(7.6136\times10^{5}\): The coefficient \(7.6136\) has 5 significant figures.

Wait, I think I messed up \(737290\). Let's re - evaluate each number:

1. \(0.6363\): Non - zero digits and digits after decimal. So 4 sig figs.
2. \(7.57\times10^{6}\): Coefficient \(7.57\) has 3 sig figs.
3. \(737290\): Let's count the significant figures. The number is \(737290\). If we assume that the zero is not significant (since there's no decimal), the significant figures are 7,3,7,2,9. So 5 sig figs? Wait, no. Wait, the correct way: for a number without a decimal, trailing zeros are not significant. So \(737290\) has 5 significant figures (7,3,7,2,9).
4. \(0.04321400\): Leading zeros are not significant. The digits \(4,3,2,1,4,0,0\) (the trailing zeros after the 4 are significant because they are after the decimal in the original number? Wait, no. The original number is \(0.04321400\). Let's rewrite it as \(4.321400\times10^{-2}\). So the significant figures are the digits in the coefficient: 4,3,2,1,4,0,0. So 7 sig figs.
5. \(9.6\times10^{3}\): Coefficient \(9.6\) has 2 sig figs.
6. \(7.6136\times10^{5}\): Coefficient \(7.6136\) has 5 sig figs.

Now, let's list the number of sig figs for each:

- \(9.6\times10^{3}\): 2 sig figs.
- \(7.57\times10^{6}\): 3 sig figs.
- \(0.6363\): 4 sig figs.
- \(737290\): 5 sig figs; \(7.6136\times10^{5}\): 5 sig figs. Now, we need to order the numbers with the same sig figs from smallest to largest.
- First, numbers with 2 sig figs: \(9.6\times10^{3}=9600\)
- Numbers with 3 sig figs: \(7.57\times10^{6} = 7570000\)
- Numbers with 4 sig figs: \(0.6363\)
- Numbers with 5 sig figs: \(737290\) and \(7.6136\times10^{5}=761360\). Now, compare \(737290\) and \(761360\). Since \(737290<761360\), so \(737290\) comes before \(7.6136\times10^{5}\)
- Number with 7 sig figs: \(0.04321400\)

Now, order by increasing number of sig figs (and for same sig figs, increasing number):

1. \(9.6\times10^{3}\) (2 sig figs)
2. \(7.57\times10^{6}\) (3 sig figs)
3. \(0.6363\) (4 sig figs)
4. \(737290\) (5 sig figs)
5. \(7.6136\times10^{5}\) (5 sig figs)
6. \(0.04321400\) (7 sig figs)

Wait, let's verify the sig fig counts again:

- \(9.6\times10^{3}\): 2 sig figs (correct, because 9 and 6 are significant)
- \(7.57\times10^{6}\): 3 sig figs (7,5,7 - correct)
- \(0.6363\): 4 sig figs (6,3,6,3 - correct)
- \(737290\): Let's count again. The number is 7 3 7 2 9 0. If the zero is not significant, then 5 sig figs (7,3,7,2,9). Correct.
- \(7.6136\times10^{5}\): 5 sig figs (7,6,1,3,6 - correct)
- \(0.04321400\): Let's see, the number is 0.04321400. Leading zeros (0.0) are not significant. Then 4,3,2,1,4,0,0. The trailing zeros after the 4 are significant because they are after the decimal? Wait, no. The rule is: trailing zeros in a decimal number are significant. So in \(0.04321400\), the digits are 4 (1st sig fig), 3 (2nd), 2 (3rd), 1 (4th), 4 (5th), 0 (6th), 0 (7th). So 7 sig figs. Correct.

Now, for the numbers with 5 sig figs: \(737290 = 7.3729\times10^{5}\) and \(7.6136\times10^{5}\). Comparing \(7.3729\times10^{5}\) and \(7.6136\times10^{5}\), since \(7.3729\times10^{5}<7.6136\times10^{5}\), so \(737290\) (which is \(7.3729\times10^{5}\)) comes before \(7.6136\times10^{5}\).

Now, let's list them in order:

- Least sig figs (2): \(9.6\times10^{3}\)
- Next (3): \(7.57\times10^{6}\)
- Next (4): \(0.6363\)
- Next (5): \(737290\)
- Next (5): \(7.6136\times10^{5}\)
- Most sig figs (7): \(0.04321400\)

Answer:

1. \(9.6\times10^{3}\)
2. \(7.57\times10^{6}\)
3. \(0.6363\)
4. \(737290\)
5. \(7.6136\times10^{5}\)
6. \(0.04321400\)