Significant Digits: Rounding & Exponential form

Question

Round the following numbers to $4$ significant digits and writeteh rsult in exponential form.

1. $102.53070$

2. $656.980$

3. $0.008543210$

4. $0.000257870$

5. $-0.0357202$

$$$$

I have done the following:

1. Rounding the number $102.53070$ to $4$ significant digits we get $102.5$. To write this in exponential form, there are the possibilities: $$10.25\cdot 10^1 = 1.025\cdot 10^2=0.1025\cdot 10^3$$

2. Rounding the number $656.980$ to $4$ significant digits we get $657.0$. To write this in exponential form, there are the possibilities: $$465.70\cdot 10^1 = 6.570\cdot 10^2=0.6570\cdot 10^3$$

3. The significant digits at $0.008543210$ are from $8$ and onwards. So rounding the number to $4$ significant digits we get $0.008543$. To write this in exponential form, there are the possibilities: $$0.08543\cdot 10^{-1} = 0.8543\cdot 10^{-2}=8.543\cdot 10^{-3}=85.43\cdot 10^{-4}=854.3\cdot 10^{-5}=8543\cdot 10^{-6}$$

4. The significant digits at $0.000257870$ are from $2$ and onwards. So rounding the number to $4$ significant digits we get $0.0002579$. To write this in exponential form, there are the possibilities: $$0.002579\cdot 10^{-1} = 0.02579\cdot 10^{-2}=0.2579\cdot 10^{-3}=2.579\cdot 10^{-4}=25.79\cdot 10^{-5}=257.9\cdot 10^{-6}=\ldots$$

5. At $-0.0357202$ is the sign important? The significant digits at $-0.0357202$ are from $0$ and onwards. So rounding the number to $4$ significant digits we get $-0.03572$. To write this in exponential form, there are the possibilities: $$-0.3572\cdot 10^{-1} = -3.572\cdot 10^{-2}=-35.72\cdot 10^{-3}=-357.2\cdot 10^{-4}=-3572\cdot 10^{-5}$$

Is everything correct?

Answer 1

Your rounding is correct.

You made a clercial error in the first exponential form for the second subproblem $\color{red}465.70\ldots$

For the exponential forms, I'm at a loss to understand why you presented exactly the forms you gave (this may have to do with the way you were tought, of course).

For the first rounded number ($102.5$), $102.5\cdot 10^0$ and $0.01025\cdot 10^4$ are also valid exponential forms, and I'm sure you realize you can find a valid form for each integer power of $10$.

Usually when one talks about bringing a number into exponential form, there is one form selected that has a special property. In mathematics, this is usually the form where the mantissa (the factor before the power of $10$) is at least $1$ but less than $10$. That's the same as requiring the mantissa to have only a single, non-zero digit before the decimal point. Each number $\neq 0$ can be represented this way uniquely.

This means the exponential forms I would understand to be requested by the problem would be

1. $102.5 = 1.025\cdot10^2$
2. $657.0 = 6.570\cdot10^2$
3. $0.008543 = 8.543\cdot10^{-3}$
4. $0.0002579 = 2.579\cdot10^{-4}$
5. $-0.03572 = -3.572\cdot10^{-2}$