Scientific notation

Scientific notation is a short way to write very large or very small numbers as a number between 1 and 10 multiplied by a power of 10. For example, 4,500,000 is 4.5 × 10⁶ and 0.00072 is 7.2 × 10⁻⁴. A positive exponent means a big number, and a negative exponent means a number smaller than 1. In the UK it is called standard form.

What is scientific notation?

Every number in scientific notation has the same two parts:

• The coefficient a is at least 1 and less than 10, so it has exactly one nonzero digit before the decimal point.
• The exponent n is a whole number, positive or negative. It says how many places the decimal point moves.

So 3.7 × 10⁵ is in scientific notation, but 37 × 10⁴ and 0.37 × 10⁶ are not, even though all three equal 370,000. Under the Common Core standards, grade 8 students use powers of 10 to estimate very large and very small quantities and do calculations in scientific notation.

Powers of 10

Each step down the table divides by 10. A negative exponent does not make a number negative: 10⁻³ is a small positive number, 1/1,000.

How to write a number in scientific notation

1. Move the decimal point so that exactly one nonzero digit is in front of it. That gives the coefficient.
2. Count how many places you moved the point. That is the size of the exponent.
3. If the original number was 10 or more, the exponent is positive. If it was less than 1, the exponent is negative.

Example 1: a large number

Write 4,500,000 in scientific notation. Move the decimal point from the end to just after the 4: that is 6 places to the left, giving 4.5. The number is large, so the exponent is positive:

Example 2: a small number

Write 0.00072 in scientific notation. Move the decimal point 4 places to the right to get 7.2. The number is less than 1, so the exponent is negative:

Example 3: a number that is not quite right

48 × 10⁵ is not in scientific notation, because 48 is not less than 10. Write 48 as 4.8 × 10¹, then add the exponents: 4.8 × 10¹ × 10⁵ = 4.8 × 10⁶.

Now try one from the Smallest Thing course, which uses scientific notation to compare the sizes of atoms, cells and planets.

Converting from scientific notation to standard notation

Reverse the process. A positive exponent moves the decimal point right, making the number bigger. A negative exponent moves it left, making it smaller. Fill any gaps with zeros.

• 6.02 × 10³: move the point 3 places right to get 6,020.
• 3.9 × 10⁻⁵: move the point 5 places left to get 0.000039.

Multiplying and dividing in scientific notation

Deal with the coefficients and the powers of 10 separately. Multiply or divide the coefficients, then add the exponents to multiply or subtract them to divide, using the exponent rules. Finally, adjust the coefficient if it isn't between 1 and 10.

Example 4: multiplying

3 × 5 = 15 and 10⁴ × 10⁶ = 10¹⁰. But 15 is not less than 10, so write it as 1.5 × 10¹ and add 1 to the exponent.

Example 5: dividing

8.4 ÷ 2.1 = 4 and 10⁹ ÷ 10³ = 10⁶.

Example 6: how many times bigger?

A city of 6 × 10⁶ people sits in a country of 3 × 10⁸. How many times bigger is the country's population? Divide: (3 × 10⁸) ÷ (6 × 10⁶) = 0.5 × 10² = 50 times.

Adding and subtracting in scientific notation

You can only add or subtract coefficients when the powers of 10 match. Rewrite one number so the exponents are the same, then add or subtract.

Example 7: adding

Example 8: subtracting

The Scientific Notation Calculator converts numbers and does all four operations with the working shown, so you can check each step.

Scientific notation in science

Scientists use it because their numbers span huge ranges. Light travels at 3.00 × 10⁸ m/s, and the average distance from the Earth to the Sun is 1.50 × 10¹¹ m. How long does sunlight take to reach us?

That is 500 seconds, a little over 8 minutes. Written out in full, the distance would be 150,000,000,000 m, and it is easy to drop or add a zero.

Scientific notation on a calculator

Calculators and spreadsheets often show scientific notation with an E, for "exponent". 1.5E11 means 1.5 × 10¹¹ and 7.2E-4 means 7.2 × 10⁻⁴. When you write the answer down, change the E back into × 10 with a power.

Check your understanding.

Common mistakes

• A coefficient outside 1 to 10. 12 × 10³ and 0.8 × 10⁴ are not in scientific notation. Adjust them to 1.2 × 10⁴ and 8 × 10³.
• The wrong sign on the exponent. Big numbers get positive exponents, small numbers get negative ones.
• Thinking a negative exponent makes a negative number. 10⁻² is 0.01, not −100.
• Multiplying the exponents when you multiply numbers. Add them: 10⁴ × 10⁶ = 10¹⁰.
• Adding coefficients with different exponents. Make the powers of 10 match first.
• Forgetting to tidy up after multiplying, leaving answers like 15 × 10¹⁰.

Practice problems

1. Write 93,000,000 in scientific notation.
2. Write 0.000000512 in scientific notation.
3. Write 2.06 × 10⁴ in standard notation.
4. Work out (4 × 10⁵)(6 × 10⁻²).
5. Work out (9.6 × 10⁷) ÷ (3.2 × 10²).

Answers: 1) 9.3 × 10⁷. 2) 5.12 × 10⁻⁷. 3) 20,600. 4) 24 × 10³ = 2.4 × 10⁴. 5) 3 × 10⁵.

The daily Geometry, Algebra 2 & SAT Math and Pre-Algebra & Algebra 1 papers are short timed practice tests of 8 questions, with a new one every day.

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Maths · NumberScientific notation calculator: convert and calculateFree scientific notation calculator: convert numbers to and from scientific notation exactly, and add, subtract, multiply or divide them with every step.