Scientific Notation

Environmental science is full of numbers that are too big or too small to write comfortably: the mass of carbon dioxide in the air, the water a country uses, a trace of a chemical in a glass of water. Scientific notation and metric prefixes are two ways to write those numbers compactly. On the exam you have only a scientific calculator and no formula sheet, so you need to move between the forms quickly and without losing a zero.

A number in scientific notation has two parts: a front number from 1 up to (but not including) 10, and a power of ten.

$2{,}500{,}000 = 2.5 \times 10^{6}$

To write a number this way, move the decimal point until exactly one nonzero digit is to its left, then count how many places you moved it. Moving it left gives a positive exponent (a big number); moving it right gives a negative exponent (a small number): $0.00042 = 4.2 \times 10^{-4}$ .

Scale words are the same powers of ten in words. Write each number all three ways at least once until it feels automatic:

Scale word	Power of ten	Written out
thousand	$10^{3}$	1,000
million	$10^{6}$	1,000,000
billion	$10^{9}$	1,000,000,000
trillion	$10^{12}$	1,000,000,000,000

So 2.5 million = $2.5 \times 10^{6}$ = 2,500,000, and 40 billion = $4.0 \times 10^{10}$ = 40,000,000,000.

Multiplying and dividing

Powers of ten follow two rules you can check with zeros.

Multiply: multiply the front numbers and add the exponents. $10^{4} \times 10^{3} = 10^{7}$ (10,000 × 1,000 has 4 + 3 = 7 zeros).
Divide: divide the front numbers and subtract the exponents. $10^{9} \div 10^{6} = 10^{3}$ .

If the front number ends up outside 1 to 10, fix it at the end: $0.5 \times 10^{4} = 5.0 \times 10^{3}$ , and $12 \times 10^{6} = 1.2 \times 10^{7}$ . Moving the point one place left adds 1 to the exponent; one place right subtracts 1.

Worked example: per person. A country of $2.4 \times 10^{8}$ people throws away $1.2 \times 10^{11}$ kg of trash in a year. How much is that per person?

Setup: $\frac{1.2 \times 10^{11}\,\text{kg}}{2.4 \times 10^{8}\,\text{people}}$

Fronts: $1.2 \div 2.4 = 0.5$ . Exponents: $11 - 8 = 3$ . So $0.5 \times 10^{3}$ , which is $5.0 \times 10^{2}$ kg per person = 500 kg per person.

On a calculator, type the powers with brackets or the EE key: (1.2 × 10^11) / (2.4 × 10^8). Without the brackets the calculator divides by 2.4 and then multiplies by $10^{8}$ , which is a very different number.

Metric prefixes

A metric prefix in front of a unit is a power of ten in disguise. A kilogram is $10^{3}$ grams; a megaliter is $10^{6}$ liters; a milligram is $10^{-3}$ grams.

To change a prefix, replace it with its power of ten. 750 ML = $750 \times 10^{6}$ L = $7.5 \times 10^{8}$ L. Going to a smaller unit makes the number bigger (2.5 g = 2,500 mg); going to a bigger unit makes it smaller (2,500 g = 2.5 kg).

Worked example: a daily total. A landfill takes in $4.0 \times 10^{5}$ kg of trash a day. How much in 30 days?

Setup: $4.0 \times 10^{5}\,\text{kg/day} \times 30\,\text{days}$ . The days cancel, leaving kg.

$4.0 \times 30 = 120$ , so $120 \times 10^{5}$ kg $= 1.2 \times 10^{7}$ kg = 12 million kg.

Comparing numbers

Before comparing, put every number in the same form and the same unit. $5 \times 10^{8}$ L looks larger than $3 \times 10^{9}$ L because 5 is bigger than 3, but the power of ten matters far more: $3 \times 10^{9}$ is six times larger. And 2 GL (gigaliters) is $2 \times 10^{9}$ L, so compare it only after you rewrite it in liters.

Exam habits

Write the setup with units, then the answer with its unit and its scale: " $6.0 \times 10^{7}$ kg", not "6.0".
If a question says "in millions", give the number of millions with the word: "12 million kg".
Check the size: a city's water use in liters will be billions, not thousands.