Scientific Notation and Large Numbers
Scientific notation writes a number as a coefficient between 1 and 10 multiplied by a power of ten, in the form a × 10n, which makes very large and very small numbers easy to write and compare. In brief: Avogadro’s number is 6.022 × 1023 and the mass of an electron is about 9.109 × 10−31 kg; positive exponents mean large numbers, and negative exponents mean small ones.
Chemistry deals with numbers that range from the number of atoms in a spoonful of water to the diameter of an atom. Writing all the zeros would be impractical and error-prone. This article explains how to convert numbers to scientific notation, how to calculate with them, how this notation shows significant figures, how calculators display it, a table of important constants, worked examples and a short history.
What is scientific notation?
A number in scientific notation has the form a × 10n, where the coefficient a is at least 1 and less than 10, and the exponent n is a whole number. A positive n tells how many places the decimal point moves to the right to get the ordinary number, and a negative n tells how many places it moves to the left.
For example, 4.5 × 103 = 4,500 and 4.5 × 10−3 = 0.0045. The notation also shows the number of significant figures clearly: 3.00 × 108 has three significant figures, but 300,000,000 is ambiguous.
How do you convert to scientific notation?
- Move the decimal point until there is exactly one nonzero digit to its left.
- Count the number of places moved. This is the size of the exponent.
- If you moved the point to the left (a large number), the exponent is positive. If you moved it to the right (a small number), the exponent is negative.
- Write the coefficient followed by × 10 and the exponent.
| Ordinary number | Decimal point moved | Scientific notation |
|---|---|---|
| 93,000,000 | 7 places left | 9.3 × 107 |
| 602,200,000,000,000,000,000,000 | 23 places left | 6.022 × 1023 |
| 0.00056 | 4 places right | 5.6 × 10−4 |
| 0.000000000154 | 10 places right | 1.54 × 10−10 |
| 12.5 | 1 place left | 1.25 × 101 |
How do you calculate with scientific notation?
- Multiplication: multiply the coefficients and add the exponents. (a × 10m)(b × 10n) = ab × 10m+n.
- Division: divide the coefficients and subtract the exponents. (a × 10m) ÷ (b × 10n) = (a ÷ b) × 10m−n.
- Addition and subtraction: first write both numbers with the same exponent, then add or subtract the coefficients.
- Powers: (a × 10n)k = ak × 10nk.
After each calculation, adjust the result so that the coefficient is between 1 and 10.
Important numbers in chemistry
| Quantity | Value | Unit |
|---|---|---|
| Avogadro constant | 6.022 × 1023 | mol−1 |
| Speed of light in a vacuum | 2.998 × 108 | m/s |
| Mass of an electron | 9.109 × 10−31 | kg |
| Mass of a proton | 1.673 × 10−27 | kg |
| Elementary charge | 1.602 × 10−19 | C |
| Typical atomic diameter | about 1 × 10−10 | m |
| Average Earth–Sun distance | 1.496 × 1011 | m |
Worked examples
- Multiplication. (3.0 × 104) × (2.0 × 10−6) = 6.0 × 10−2 = 0.060.
- Division. (8.4 × 105) ÷ (2.1 × 102) = 4.0 × 103.
- Result needing adjustment. (5.0 × 103) × (4.0 × 105) = 20 × 108, which is written as 2.0 × 109.
- Addition. 3.2 × 105 + 4.5 × 104. Rewrite the second number as 0.45 × 105. The sum is 3.65 × 105, which with two significant figures in 3.2 is rounded to 3.7 × 105 (the first number is certain only to the tens of thousands).
- Counting atoms. How many molecules are in 0.050 mol of water? 0.050 × 6.022 × 1023 = 3.0 × 1022.
- Comparing sizes. The electron mass is 9.109 × 10−31 kg and the proton mass is 1.673 × 10−27 kg. The ratio is 1.673 × 10−27 ÷ 9.109 × 10−31 = 1.837 × 103, so a proton is about 1,840 times as heavy as an electron.
- Light travel time. Light covers 1.496 × 1011 m from the Sun at 2.998 × 108 m/s in 1.496 × 1011 ÷ 2.998 × 108 = 499 s, which is about 8.3 minutes.
How do calculators show scientific notation?
Calculators and spreadsheets use E notation, in which 6.022E23 means 6.022 × 1023 and 1.5E−4 means 1.5 × 10−4. On a scientific calculator, use the EXP or ×10x key to enter the exponent. Enter 6.022 × 1023 as 6.022, EXP, 23, and do not type × 10 followed by ^23, since the calculator may treat it differently. Enter negative exponents with the sign-change key (±) or (−), as your calculator requires.
How did large numbers get names?
Archimedes wrote a work called The Sand Reckoner in the 3rd century BCE, in which he developed a system to name numbers larger than the number of grains of sand that would fill the universe, as he imagined it. Modern exponent notation developed over centuries; René Descartes used superscript exponents in the 1600s. The name “googol” for 10100 was invented in 1920 by a nine-year-old, Milton Sirotta, nephew of the mathematician Edward Kasner.
Which mistakes are common?
Mistake 1: Using a coefficient outside the range 1 to 10
12 × 103 is not in standard form. Write 1.2 × 104.
Mistake 2: Getting the sign of the exponent wrong
Numbers smaller than 1 have negative exponents. 0.003 is 3 × 10−3.
Mistake 3: Adding exponents when adding numbers
Exponents are added only when multiplying. For addition, first make the exponents equal.
Mistake 4: Dropping significant figures
Write 2.50 × 103, not 2.5 × 103, if the zero is measured.