Scientific notation: how to write big and small numbers

Your calculator answers a chemistry problem with 6.02E23, and the homework sheet wants the distance to the Sun, about 150,000,000 km, written "in scientific notation". Both raise the same question, what is scientific notation, and the answer is a short number times a power of 10. Scientific notation turns 150,000,000 into 1.5 × 10⁸, and once you see how the decimal point moves, every conversion is quick.

What is scientific notation?

Scientific notation is a way of writing any number as a × 10ⁿ, where a is at least 1 and less than 10, and n is an integer. The OpenStax Prealgebra textbook states the rule as "a≥1 and a<10 and n is an integer". In 4.5 × 10⁴, the front number is 4.5 and the exponent is 4.

The exponent says how many times to multiply or divide by 10. A positive exponent makes the number bigger, so 4.5 × 10⁴ = 45,000, and a negative exponent makes it smaller, so 7.2 × 10⁻⁴ = 0.00072.

If powers of 10 feel shaky, the guide to exponents explained covers 10³ = 1,000 and the negative exponents behind 10⁻⁴.

In the US, scientific notation is a grade 8 topic. Common Core standard 8.EE.A.3 asks students to "use numbers expressed in the form of a single digit times an integer power of 10 to estimate very large or very small quantities", with the example of the US population as 3 × 10⁸ and the world population as 7 × 10⁹.

Why scientists and calculators use it

Scientists and calculators use scientific notation because it keeps the meaningful digits and drops the zeros. NASA gives the Sun's distance as "about 93 million miles (150 million kilometers)", which is 1.5 × 10⁸ km. NIST lists the Avogadro constant as 6.022 140 76 × 10²³ per mole, a number with 24 digits before the decimal point.

The exponent also shows the size of a number at a glance, so 3 × 10⁸ and 7 × 10⁹ differ by a factor of more than 20 without counting zeros. The same habit shows up in metric units: NIST's table of SI prefixes defines kilo as 10³, mega as 10⁶, giga as 10⁹, milli as 10⁻³, micro as 10⁻⁶ and nano as 10⁻⁹.

How to write a number in scientific notation, step by step

Writing a number in scientific notation takes five steps, and they work the same way for big and small numbers.

1. Place the point after the first nonzero digit. In 45,000 the first nonzero digit is 4, so the front number becomes 4.5. In 0.00072 the first nonzero digit is 7, so the front number becomes 7.2.
2. Count the moves. In 45,000 the decimal point sits after the last 0 and moves 4 places left to land between 4 and 5. In 0.00072 it moves 4 places right to land between 7 and 2.
3. Choose the sign of the exponent. A number of 10 or more gets a positive exponent, and a number between 0 and 1 gets a negative one. So 45,000 = 4.5 × 10⁴ and 0.00072 = 7.2 × 10⁻⁴.
4. Drop the trailing zeros. The front number keeps only the digits that matter, so 4.5000 becomes 4.5. Zeros in the middle stay, so 306,000 = 3.06 × 10⁵.
5. Check by converting back. Move the point the other way by the exponent: 4.5 × 10⁴ gives 45,000, and 5.1 × 10⁻³ gives 0.0051. OpenStax gives the same reverse rule: a positive exponent moves the point right and a negative exponent moves it left.

Every hop of the point is one place value, which is why scientific notation is place value in a compact form. The guide to place value for parents shows the columns behind each hop, and the place value game drills them in the browser.

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The part people get wrong: which way the sign goes

The sign of the exponent is the step students get wrong most often, because the point moves left for big numbers and the exponent is still positive. Skip the direction rule and ask one question instead: is the original number bigger than 10 or smaller than 1?

A big number needs a positive exponent to grow 4.5 back into 45,000. A small number needs a negative exponent to shrink 7.2 down to 0.00072. A student who writes 45,000 = 4.5 × 10⁻⁴ can catch the error in seconds, because 10⁻⁴ makes 4.5 smaller than 1.

Standard form vs scientific notation vs E notation

Standard form, scientific notation and E notation are three ways to write the same value, and the names change between countries. In US textbooks, "standard form" means the ordinary numeral: the OpenStax Prealgebra chapter on whole numbers writes a $77 billion budget "in standard form" as $77,000,000,000. In England, "standard form" is the name for a × 10ⁿ itself.

England's national curriculum for mathematics asks Key Stage 3 pupils to "interpret and compare numbers in standard form A × 10ⁿ 1≤A<10, where n is a positive or negative integer or 0". Check which country a worksheet comes from before you read its "standard form".

How to read a calculator's E

A calculator's E means "times 10 to the power of", so 6.02E23 is 6.02 × 10²³ and 7.2E-4 is 7.2 × 10⁻⁴. The display switches to E when the answer is too long for the screen. A spreadsheet can show the same number as 1.5E+08, where the plus sign marks a positive exponent.

To type scientific notation into a calculator, use the key marked EE or EXP, which enters "× 10 to the power of" in one press. Common Core standard 8.EE.A.4 names this skill directly: "Interpret scientific notation that has been generated by technology."

How to multiply, divide, add and subtract in scientific notation

Scientific notation makes multiplying and dividing easy, because the front numbers and the powers of 10 are handled separately. OpenStax puts it in one line: "We use the Properties of Exponents to multiply and divide numbers in scientific notation."

• Multiply: multiply the front numbers and add the exponents. (3 × 10⁴) × (2 × 10⁵) = (3 × 2) × 10⁴⁺⁵ = 6 × 10⁹.
• Divide: divide the front numbers and subtract the exponents. (8 × 10⁶) ÷ (2 × 10²) = (8 ÷ 2) × 10⁶⁻² = 4 × 10⁴.
• Re-normalize: if the new front number is 10 or more, move the point one place left and add 1 to the exponent. (4 × 10²) × (3 × 10¹) = 12 × 10³ = 1.2 × 10⁴.
• Re-normalize down: if the new front number is less than 1, move the point one place right and subtract 1. (2 × 10³) ÷ (5 × 10⁷) = 0.4 × 10⁻⁴ = 4 × 10⁻⁵.

The front numbers are decimals, so the arithmetic is decimal arithmetic. The guide to multiplying and dividing decimals covers 1.5 × 2.4 and 7.5 ÷ 0.25 when the front numbers are less tidy.

Adding and subtracting: match the exponents first

Adding and subtracting in scientific notation work only when both numbers have the same power of 10. For (3.2 × 10⁵) + (4.5 × 10⁴), rewrite 4.5 × 10⁴ as 0.45 × 10⁵. Then add the front numbers: 3.2 + 0.45 = 3.65, so the sum is 3.65 × 10⁵ = 365,000.

Subtraction follows the same rule. (5 × 10⁶) − (2 × 10⁵) becomes (5 × 10⁶) − (0.2 × 10⁶) = 4.8 × 10⁶, which is 4,800,000. Adding the exponents here, as in multiplication, gives a wrong answer like 3 × 10¹¹, a number about 60,000 times too big.

How to check progress with scientific notation

Five quick signals show whether scientific notation has landed, for a grade 8 student or for an adult coming back to it.

• Conversion both ways: Can you write 82,000 as 8.2 × 10⁴ and 3 × 10⁻² as 0.03 without a calculator?
• Sign by sense: Can you say, before counting, whether 0.0009 gets a positive or a negative exponent?
• Normal form: Can you spot that 45 × 10³ and 0.5 × 10³ are not in scientific notation and fix them to 4.5 × 10⁴ and 5 × 10²?
• Calculator reading: Can you read 3.1E-5 aloud as 3.1 × 10⁻⁵ and write it as 0.000031?
• Operations: Can you multiply (4 × 10³) × (5 × 10⁶) and re-normalize 20 × 10⁹ to 2 × 10¹⁰?

A ten-minute test for students and adult learners

A ten-minute test uses one item of each type from the progress list, with new numbers twice a week for two weeks. Mark each answer by converting it back to a full numeral, which catches sign errors.

Adult learners meet scientific notation in lab reports, data-storage sizes, national budgets and spreadsheet cells that suddenly show E+12. The guide on how to get better at math as an adult covers a short daily routine, and the math game for adults page gathers practice for grown-up learners.

Common mistakes and edge cases in scientific notation

Five mistakes cause most wrong answers in scientific notation, and each has a quick fix.

• A front number of 10 or more: 45 × 10³ equals 45,000, but it is not in scientific notation. Move the point one place left and add 1: 4.5 × 10⁴.
• A front number below 1: 0.5 × 10³ equals 500, and the correct form is 5 × 10².
• The wrong sign on the exponent: 0.00072 = 7.2 × 10⁴ is off by a factor of 10⁸. Ask whether the original was big or small.
• Adding exponents in a sum: (2 × 10³) + (3 × 10³) is 5 × 10³, not 5 × 10⁶. Exponents add only in multiplication.
• Counting zeros instead of places: 2.5 × 10³ is 2,500, not 2,500,000. The exponent counts moves of the point, and 2.5 needs only 3 moves to become 2,500.

Negative exponents, negative numbers and 10⁰

A negative exponent does not make a negative number: 7.2 × 10⁻⁴ is 0.00072, a small positive value. A negative number keeps its minus sign in front, so −45,000 = −4.5 × 10⁴. The guide to negative numbers explained covers the minus sign on its own.

The exponent 0 handles numbers from 1 up to 10, because 10⁰ = 1. So 7 = 7 × 10⁰ and 3.9 = 3.9 × 10⁰. The guide to decimals for parents shows why 0.00072 has its 7 in the ten-thousandths place, the fact behind the exponent −4.

Your scientific notation checklist and next steps

• Front number: at least 1 and less than 10, with no trailing zeros.
• Exponent: the number of places the point moved; positive for big numbers, negative for small ones.
• Check: convert back to a full numeral every time.
• Multiply or divide: handle the front numbers, then add or subtract the exponents, then re-normalize.
• Add or subtract: match the powers of 10 first.
• Calculator: read E as "× 10 to the power of".

Scientific notation rests on place value, powers of 10 and quick mental multiplication, and those skills are what MathIt trains. MathIt is a free app for iOS and Android with about 90 game types, adaptive difficulty, daily practice and 2–5 player real-time battles, available in 33 languages.