Scientific Notation

Introduction

Chemistry deals with numbers that are ridiculously large and ridiculously small. A single drop of water contains about 1 700 000 000 000 000 000 000 molecules, while one water molecule has a mass of about 0.000 000 000 000 000 000 000 03 g. Writing out all those zeros is slow, and it is easy to lose count. Scientific notation solves the problem by writing every number as a short number multiplied by a power of ten.

Core explanation

The format. A number in scientific notation has the form

a × 10ⁿ

where the coefficient a is at least 1 but less than 10, and the exponent n is a whole number (positive, negative or zero). So 3500 becomes 3.5 × 10³, and 0.0042 becomes 4.2 × 10⁻³. In UK schools this is often called standard form .

What the exponent means. A positive exponent tells you how many times the coefficient is multiplied by ten: 10³ = 10 × 10 × 10 = 1000. A negative exponent tells you how many times it is divided by ten: 10⁻³ = 1 ÷ 1000 = 0.001. An exponent of zero means no change, because 10⁰ = 1.

Large numbers. To write 602 000 in scientific notation, move the decimal point to the left until only one non-zero digit remains in front of it: 6.02. The point moved five places, so the number is 6.02 × 10⁵. Big numbers always have positive exponents.

Small numbers. To write 0.000 57, move the decimal point to the right until one non-zero digit sits in front of it: 5.7. The point moved four places, and because the original number was less than 1, the exponent is negative: 5.7 × 10⁻⁴.

Links with prefixes. Scientific notation and SI prefixes are two ways of saying the same thing. A kilo- is 10³, a milli- is 10⁻³, a nano- is 10⁻⁹. So 2.5 × 10⁻⁹ m is 2.5 nm. Being fluent in powers of ten makes prefix conversions straightforward.

Some chemistry numbers in scientific notation:

Quantity Value --- --- Avogadro constant 6.02 × 10²³ mol⁻¹ Diameter of a hydrogen atom about 1 × 10⁻¹⁰ m Charge on an electron 1.60 × 10⁻¹⁹ C Speed of light 3.00 × 10⁸ m/s Mass of a proton 1.67 × 10⁻²⁷ kg

Calculating. When multiplying, multiply the coefficients and add the exponents. When dividing, divide the coefficients and subtract the exponents. Then tidy the answer so the coefficient is between 1 and 10 again.

Formulae

(a × 10ᵐ) × (b × 10ⁿ) = (a × b) × 10ᵐ⁺ⁿ

(a × 10ᵐ) ÷ (b × 10ⁿ) = (a ÷ b) × 10ᵐ⁻ⁿ

Step-by-step reasoning

To convert an ordinary number into scientific notation:

1. Find the first non-zero digit and place the decimal point just after it. 2. Count how many places the decimal point has moved. 3. If the original number was 10 or more, the exponent is positive; if it was less than 1, the exponent is negative. 4. Write the coefficient, then × 10 with the exponent. 5. Check: the coefficient must be at least 1 and less than 10.

Visual explanation

Imagine a number line of powers of ten, each step ten times bigger than the one before: 10⁻³, 10⁻², 10⁻¹, 10⁰, 10¹, 10², 10³. Every number sits somewhere between two of these marks. The exponent tells you which step you are on; the coefficient tells you how far along that step you are.

Real-world analogy

Scientific notation is like giving a postal address: the exponent is the town and the coefficient is the house number. Knowing the town gets you to the right region of the number line immediately; the house number pins down the exact spot within it.

Real-world example

Calculators and spreadsheets use scientific notation all the time. A calculator display showing 6.02E23 means 6.02 × 10²³. Pharmacists and clinical chemists routinely write drug concentrations such as 5 × 10⁻⁶ mol/dm³, because writing six decimal places would invite dangerous mistakes.

Why?

Why insist that the coefficient is between 1 and 10? It makes every number written in one unique way, so numbers can be compared at a glance: the one with the bigger exponent is bigger. Writing 35 × 10² or 0.35 × 10⁴ is not wrong arithmetically, but it hides the size of the number.

Common misconception

"A negative exponent means a negative number." It does not. 3 × 10⁻⁴ is a small positive number, 0.0003. A negative number would have a minus sign in front of the coefficient, as in −3 × 10⁴.

Worked example

Question: Calculate (4.0 × 10³) × (6.0 × 10⁵) and give the answer in scientific notation.

Reasoning: Multiply the coefficients: 4.0 × 6.0 = 24. Add the exponents: 3 + 5 = 8. This gives 24 × 10⁸. The coefficient 24 is not between 1 and 10, so rewrite it as 2.4 × 10¹. Then 2.4 × 10¹ × 10⁸ = 2.4 × 10⁹.

Answer: 2.4 × 10⁹

Quick check

1. Write 0.000 082 in scientific notation. Answer: 8.2 × 10⁻⁵

Exam focus

Examiners check that the coefficient really is between 1 and 10 and that the sign of the exponent is correct. Learn how to enter powers of ten on your calculator using the EXP or ×10ˣ key, and never copy "E" notation straight into a written answer.

Advanced insight

Scientific notation also shows precision. Writing 1.50 × 10³ g tells the reader that three digits are meaningful, whereas 1500 g is ambiguous: the zeros might be measured or might simply fix the size of the number. For this reason scientists often prefer scientific notation when reporting results, a point developed further with significant figures.

Summary

Scientific notation writes numbers as a × 10ⁿ, with 1 ≤ a < 10 and n a whole number. Large numbers have positive exponents and small numbers have negative exponents. To multiply, multiply coefficients and add exponents; to divide, divide coefficients and subtract exponents, then tidy the coefficient. It makes very large and very small quantities easy to write, compare and calculate with.

Practice questions

1. Write 45 000 000 in scientific notation. Answer: 4.5 × 10⁷ 2. Write 3.9 × 10⁻³ as an ordinary number. Answer: 0.0039 3. Calculate (9.0 × 10⁸) ÷ (3.0 × 10²). Answer: 9.0 ÷ 3.0 = 3.0 and 8 − 2 = 6, so 3.0 × 10⁶. 4. Which is larger, 7 × 10⁻⁵ or 2 × 10⁻⁴? Explain. Answer: 2 × 10⁻⁴, because −4 is a larger exponent than −5, so 2 × 10⁻⁴ = 0.0002 is bigger than 0.000 07. 5. Rewrite 52 × 10⁴ correctly in scientific notation. Answer: 5.2 × 10⁵