CHE 1: Math Prep - Scientific Notation and Significant Figures

Practice flashcards covering scientific notation conversions, exponential operations, rules for counting significant figures, and rounding rules for chemistry calculations.

In scientific notation, how is a number represented, and what is the required numerical range for the coefficient $C$?

A number is expressed in the form $C \times 10^n$, where the coefficient $C$ must satisfy $1 \le C < 10$, and the exponent $n$ is a whole number.

When converting a number to scientific notation, how is the sign of the exponent $n$ determined?

The exponent $n$ is positive when the decimal point is moved to the left (for numbers $10$ or larger) and negative when moved to the right (for numbers smaller than $1$).

How is the number $90065.00$ expressed in scientific notation?

$9.006500 \times 10^4$ (the decimal point moves $4$ digits to the left, and all trailing zeros after the decimal are preserved).

How is the number $0.0000834$ expressed in scientific notation?

$8.34 \times 10^{-5}$ (the decimal point moves $5$ digits to the right).

What is the initial step required to add or subtract two numbers written in scientific notation?

Convert the smaller number to have the same exponential power of $10$ as the larger number before performing addition or subtraction on the coefficients $C$.

In scientific notation, how is the exponential adjusted if the resulting coefficient $C \ge 10$ (e.g., $983.5 \times 10^4$)?

The decimal point is moved to the left and the exponent increases ($983.5 \times 10^4 = 9.835 \times 10^{4+2} = 9.835 \times 10^6$).

In scientific notation, how is the exponential adjusted if the resulting coefficient $C < 1$ (e.g., $0.09835 \times 10^4$)?

The decimal point is moved to the right and the exponent decreases ($0.09835 \times 10^4 = 9.835 \times 10^{4-2} = 9.835 \times 10^2$).

What are the mathematical rules for multiplying numbers written in scientific notation?

Multiply the coefficients $C$ together and add the exponents $n$ ($10^{n_1} \times 10^{n_2} = 10^{n_1 + n_2}$), then adjust the coefficient to ensure $1 \le C < 10$.

What are the mathematical rules for dividing numbers written in scientific notation?

Divide the coefficients $C$ and subtract the denominator exponent from the numerator exponent ($10^{n_1} / 10^{n_2} = 10^{n_1 - n_2}$), then adjust the coefficient to ensure $1 \le C < 10$.

Why is adding a $6.8\text{ ton}$ truck to two drivers weighing $0.1308\text{ ton}$ reported scientifically as $6.9\text{ tons}$ rather than $6.9308\text{ tons}$?

Because of measurement inaccuracy: the scale measures only to the tenths place ($6.8\text{ tons} \ne 6.8000\text{ tons}$), making extra decimal places scientifically meaningless.

According to significant figure rules, are leading zeros (such as those in $0.205$ or $0.00205$) significant?

No; a zero alone in front of a decimal point or to the right of the decimal point before the first non-zero digit is never significant.

How does an explicit decimal point affect whether trailing zeros to the left of the decimal are significant (e.g., $25600.$ vs. $25600$)?

Trailing zeros at the end of a whole number are not significant unless a decimal point is explicitly provided. Therefore, $25600.$ has $5$ significant figures, while $25600$ has only $3$ significant figures.

How many significant figures are present in the values $19628.00$, $0.003416$, and $9 \times 10^{19}$?

$19628.00$ has $7$ significant figures, $0.003416$ has $4$ significant figures, and $9 \times 10^{19}$ has $1$ significant figure.

What rule governs the number of significant figures in addition and subtraction calculations?

Round off the final sum or difference to the first uncertain digit (the position corresponding to the least number of decimal places / least precise measurement).

What is the properly rounded answer for the addition problem $14.668 + 12.2 + 0.1355 = 27.0035$?

$27.0$, because $12.2$ has its first uncertain digit in the tenths place.

What rule governs the number of significant figures in multiplication and division calculations?

The final result must have the same number of significant figures as the least precise number (the value with the fewest significant figures) in the calculation.

What is the properly rounded result for $\frac{(5.5 \times 10^{-8})(4 \times 10^{10})}{6.65 \times 10^{45}} = 3.30827\dots \times 10^{-43}$?

$3 \times 10^{-43}$, because $4 \times 10^{10}$ has only $1$ significant figure.

Earth's equatorial circumference is $40,075\text{ km}$ and an aircraft travels at $925\text{ km/h}$. How many hours does the flight take, rounded to the correct number of significant figures?

$43.3\text{ h}$ (or $4.33 \times 10^1\text{ h}$), rounded to $3$ significant figures to match the least precise given measurement ($925\text{ km/h}$).