Science, engineering and finance all deal with numbers that are very large or very small, or measured with limited precision. Scientific notation and significant figures give you a clean way to write them and to avoid claiming more accuracy than you have. Here is how both work.

Scientific Notation

Scientific notation writes a number as a value between 1 and 10 multiplied by a power of ten: a × 10^n.

  • 6,020,000,000,000,000,000,000,000 is written as 6.02 × 10^24.
  • 0.00045 is written as 4.5 × 10^−4.
  • 3,500 is written as 3.5 × 10^3.

The exponent tells you how many places to move the decimal point. A positive exponent means moving right, and a negative exponent means moving left. Most calculators show it as 6.02E24.

Calculating With Scientific Notation

  • Multiplication: multiply the numbers and add the exponents. (3 × 10^4) × (2 × 10^3) = 6 × 10^7.
  • Division: divide the numbers and subtract the exponents. (8 × 10^6) ÷ (2 × 10^2) = 4 × 10^4.
  • Addition and subtraction: first make the exponents the same, then add the numbers. 3.0 × 10^3 + 4.0 × 10^2 = 3.0 × 10^3 + 0.40 × 10^3 = 3.4 × 10^3.

If the result is not between 1 and 10, adjust it. For example, 12 × 10^5 becomes 1.2 × 10^6.

What Are Significant Figures?

Significant figures (sig figs) are the digits in a number that carry meaning about its precision. They show how carefully something was measured. The rules are:

  • All non-zero digits are significant. 273 has three.
  • Zeros between non-zero digits are significant. 1,005 has four.
  • Leading zeros are not significant. 0.0042 has two.
  • Trailing zeros after a decimal point are significant. 0.00450 has three, and 2.50 has three.
  • Trailing zeros in a whole number without a decimal point are ambiguous. 1,200 might have two, three or four. Scientific notation removes the doubt: 1.2 × 10^3 has two, and 1.200 × 10^3 has four.

Rounding After Calculations

  • Multiplication and division: the answer should have as many significant figures as the measurement with the fewest. 4.56 × 1.4 = 6.384, which is rounded to 6.4, because 1.4 has two significant figures.
  • Addition and subtraction: the answer should have as many decimal places as the measurement with the fewest. 12.11 + 0.3 = 12.41, which is rounded to 12.4, because 0.3 has one decimal place.

Round only at the end. Rounding at every step can pile up errors, so keep extra digits in your calculator until the final answer.

Why It Matters

Suppose you measure a table with a ruler marked in millimetres and read 1.254 m. Writing the area as 1.254 × 0.7 = 0.8778 suggests a precision the measurement does not have, because 0.7 has only one significant figure. The honest answer is 0.9 m². Reporting too many digits can mislead readers into trusting a result more than they should.

Where You Will See Them

  • Science: Avogadro's number is about 6.022 × 10^23, and the speed of light is about 3.00 × 10^8 metres per second.
  • Computing: file sizes and processor speeds are described with powers of ten (kilo, mega and giga stand for 10^3, 10^6 and 10^9).
  • Finance: very large figures, such as national debt or market sizes, are often reported in billions or trillions, which are the same idea in words.

Our guide to metric and imperial conversions uses the same careful treatment of precision.

Common Mistakes

  • Forgetting to change the sign of the exponent when moving the decimal to the left.
  • Counting leading zeros as significant.
  • Rounding in the middle of a calculation.
  • Mixing up the rules for addition and for multiplication.

Your Next Step

Take a number from your homework or work, write it in scientific notation, count its significant figures and use the rules above to round the result of a calculation. Our Scientific calculator handles the exponent keys for you.