# 1.2: You will not believe what you can do with math!

You’ve been told many times that meteorology is a math-intensive field. It is. But for this course, you already know much of the math, and what you haven’t seen, you will see in vector calculus. To get ready for the meteorology and atmospheric science in this course, you will need to refresh your ability to solve simple math problems, including solving simple problems in differential and integral calculus. At the same time, we will remind you about the importance of correctly specifying significant figures and units in your answers to the problems. The goal of this first lesson is to boost your confidence in the math you already know.

### How many figures should be in my answer?

Suppose you are asked to solve the following word problem:

In the radar loop, a squall line is oriented in the north-south direction and is heading northeast at 57 km hr-1. In the last frame of the loop, the line is 17 km west of the Penn State campus. You are out running and know that you can make it back to your apartment in 25 minutes. Will you get back to your apartment before you get soaked?

You reason that the line is moving northeast, and thus, at an angle of 45orelative to the east. Therefore, the eastward motion of the squall line is just the velocity times the cosine of 45o. That gives you the eastward speed. You decide to divide the distance by the eastward speed to get the amount of time before the line hits campus. You plug the numbers into your calculator and get the following result:

\begin{align*} \text{time} &= \dfrac{17\, \mathrm{km}}{(57\, \mathrm{km} / \mathrm{h}) \cdot \cos \left(45^{\circ}\right)} \\[4pt] &=0.42178\, \text{hours} \\[4pt] &=25.3070\, \text{minutes} \end{align*}

According to your calculation, you will make it back with 0.3 minutes (18 seconds) to spare. But can you really be sure that the squall line will strike in 25.3070 minutes? Maybe you should figure out how many significant figures your answer really has. To do that, you need to remember the rules:

#### Significant Figures Rules

1. Non-zero numbers (1,2,3,4,5,6,7,8,9) are ALWAYS significant.
2. Zeroes are ALWAYS significant:
1. between non-zero numbers
2. SIMULTANEOUSLY to the right of the decimal point AND at the end of the number
3. to the left of a written decimal point and part of a number ≥ 10
3. In a calculation involving multiplication or division, multiply numbers as you see them. Then the answer should have the same number of significant figures as the number with the fewest significant figures.
4. In a calculation involving addition and subtraction, the number of significant figures in the answer depends on the number of significant figures to the right of the decimal point when all the added or subtracted numbers are put in terms of the same power-of-ten. Add or subtract all the numbers. The answer has the same number of significant figures as the number with the least number of significant figures to the right of the decimal point.
5. The number of significant figures is unchanged by trigonometric functions, logarithms, exponentiation, and other related functions.
6. Exact numbers never limit the number of significant figures in the result of a calculation and therefore can be considered to have an infinite number of significant figures. Common examples of exact numbers are whole numbers and conversion factors. For example, there are exactly 4 sides to a square and exactly 1000 m in a km.
7. For multi-step calculations, any intermediate results should keep at least one extra significant figure to prevent round-off error. Calculators and spreadsheets will typically keep these extra significant figures automatically.
8. When rounding, numbers ending with the last digit > 5 are rounded up; numbers ending with the last digit < 5 are rounded down; numbers ending in 5 are rounded up if the preceding digit is odd and down if it is even.

Examples

Number(s) Answer Number of Significant Figures Reason
25+.3 25 2 25 has only 2 significant figures

25·0.325·0.3
8 1 25·0.3=7.525·0.3=7.5 , round to 8 because 0.3 has only 1 significant figure
$$1.5\left(10^{3}\right)+3.24\left(10^{2}\right)$$ 1.8$$\left(10^{3}\right)$$ 2 $$1.5\left(10^{3}\right)+0.324\left(10^{3}\right)=1.824\left(10^{3}\right)$$,then drop 2 to get 1.8$$\left(10^{3}\right)$$
$$1.5\left(10^{3}\right)+3.86\left(10^{2}\right)$$ 1.9$$\left(10^{3}\right)$$ 2 $$1.5\left(10^{3}\right)+3.86\left(10^{2}\right)=1.886\left(10^{3}\right)$$,round up then drop 2 to get 1.9$$\left(10^{3}\right)$$
$$\frac{(57.3+6.41)}{15.6}$$ 4.08 3 $$\frac{63.71}{15.6}=4.0840,$$ trim to 3 significant,figures to get 4.08
200(3.142) 600 1 200. has 3 significant figures; 200 (no decimal point) has 1 but is ambiguous
152$$\left(e^{-.52}\right)$$ 90 2 number in exponent has only 2 significant figures

Check out this video (11:23): Unit Conversions & Significant Figures for a brief (1 minute) explanation of those rules! Start watching at 9:14 for the most relevant information.

Unit Conversions and Significant Figures

Click Answer for transcript of the Significant Figures video.

Credit: Crash Course

### What are the typical types of variables?

There are two types of variables – scalars and vectors. Scalars are amount only; vectors also have direction.

#### Dimensions and units are your friends.

Most variables have dimensions. The ones used in meteorology are:

• L, length
• T, time
• Θ, temperature
• M, mass
• I, electric current

Some constants such as $$π$$ have no units, but most do.

The numbers associated with most variables have units. The system of units we will use is the International System (SI, from the French Système International), also known as the MKS (meter-kilogram-second) system, even though English units are used in some parts of meteorology.

We will use the following temperature conversions:

$$K = ^{o} C + 273.15$$

$$\left(\frac{5}{9}\right)\left(^{o} F-32\right)=^{o} C$$

We will use the following variables frequently. Note the dimensions of the variables and the MKS units that go with their numbers.

Variables With Associated Dimensions and MKS Units

$$p=\left(\text { normal force)/area }=(\text { mass } x \text { acceleration }) / \text { area }=M L / T^{2} L^{2}=\right.$$

$$1 \mathrm{Pa}=1 \mathrm{kg} \mathrm{m}^{-1} \mathrm{s}^{-2} ; 1 \mathrm{hPa}=100 \mathrm{Pa}=1 \mathrm{mb}=10^{-3} \mathrm{bar}(\mathrm{hPa}=\mathrm{hecto}$$Pascal)

$$1013.25 \mathrm{hPa}=1.01325 \times 10^{5} \mathrm{Pa}=1$$ standard atmospheric pressure $$=$$1 $$\mathrm{atm}$$

#### Wind speed is another frequently used variable.

The knot (kt) is equal to one nautical mile (approximately one minute of latitude) per hour or exactly 1.852 km/hr. The mile is nominally equal to 5280 ft and has been standardized to be exactly 1,609.344 m.

Thus, 1 m/s = 3.6 km/hr ≈ 1.944 kt and 1 kt ≈ 1.151 mph.

surface winds are typically 10 kts ~ 5 m/s

500 mb winds are ~50 kts ~ 25 m/s

250 mb winds are ~100 kts ~ 50 m/s

#### Temperature is a third frequently used variable.

Kelvin (K) must be used in all physical and dynamical meteorology calculations. Surface temperature is reported in oF or (oC for METARS) and in oC for upper air soundings.

#### Water vapor mixing ratio is another frequently used variable.

$w=\dfrac{\operatorname{mass} \mathrm{H}_{2} \mathrm{O}}{\text { mass dry air }}$

Usually the units for water vapor mixing ratio are $$g \mathrm{kg}^{-1} .$$ In the summer w can be 10 $$\mathrm{g} \mathrm{kg}^{-1}$$ in the winter, it can be 1.2 $$\mathrm{g} \mathrm{kg}^{-1}$$

Dimensions truly are your friend. Let me give you an example. Suppose you have an equation ax + b = cT, and you know the dimension of b, x(a distance), and T (a temperature), but not a and c. You also know that each term in the equation – the two on the left-hand side and the one on the right-hand side must all have the same units. Therefore, if you know b, you know that the dimensions of a must be the same as the dimensions of b divided by L (length) and the dimensions of c must be the same as the dimensions of b divided by Θ.

Also, if you invert a messy equation and you're not sure that you didn’t make a mistake, you can check the dimensions of the individual terms and if they don’t match up, it’s time to look for your mistake. Or, if you have variables multiplied or divided in an exponential or a logarithm, the resulting product must have no units.

Always write units down and always check dimensions if you aren’t sure. That way, you won’t crash your spacecraft on the back side of Mars. View the following video (2:42).

When NASA Lost a Spacecraft Because it Didn't Use Metric

Click Answer for transcript of the NASA video.