Scientific Notation: Rules, Examples And How To Convert

What Is Scientific Notation?

Scientific notation is a way of writing numbers as a × 10ⁿ, where a is at least 1 and less than 10, and n is a whole number. It saves space and makes huge or tiny numbers easy to read and compare.

Compare these two ways of writing Earth's mass:

• Standard form: 5,972,200,000,000,000,000,000,000 kg
• Scientific notation: 5.9722 × 10²⁴ kg

The second version tells you at a glance that the number has 25 digits before the decimal point. You do not have to count zeros.

The Parts Of A Number In Scientific Notation

Every number in scientific notation has three parts.

The exponent is the key. A positive exponent means you multiply by 10 that many times, so the number is large. A negative exponent means you divide by 10 that many times, so the number is small. An exponent of 0 means the number is between 1 and 10, since 10⁰ = 1.

The Rules For Proper Scientific Notation

A number is only in proper scientific notation when it follows three rules.

1. The coefficient is at least 1 and less than 10. So 4.56 × 10⁴ is correct, but 45.6 × 10³ and 0.456 × 10⁵ are not.
2. The base is 10.
3. The exponent is a whole number (positive, negative or zero).

Numbers like 45.6 × 10³ have the same value as 4.56 × 10⁴. They are just not in the standard form, so most teachers and tests will ask you to adjust them.

Why Scientists Use Scientific Notation

Scientists, engineers and students use scientific notation because it makes extreme numbers easier to write, compare and calculate with.

• It is shorter. Avogadro's number, 602,214,076,000,000,000,000,000, becomes 6.02214076 × 10²³.
• It is easier to compare. Which is bigger, 3 × 10⁸ or 7 × 10⁹? Just compare exponents. The one with the larger exponent wins.
• It reduces counting errors. Writing 22 zeros by hand is an easy way to lose or add one.
• It shows precision. Writing 1.50 × 10³ tells the reader the measurement is accurate to three significant figures, which 1,500 does not show clearly.

How To Convert A Number To Scientific Notation

To convert a number to scientific notation, move the decimal point until only one nonzero digit sits to its left. Then count how many places you moved it. That count becomes the exponent.

Converting Large Numbers

For numbers 10 or bigger, the decimal moves left and the exponent is positive.

Example: 45,600

1. Start with 45,600. (The decimal point sits after the last 0.)
2. Move the decimal left until one digit remains in front: 4.5600
3. You moved it 4 places.
4. Drop the extra trailing zeros: 4.56
5. Answer: 4.56 × 10⁴

Converting Small Numbers

For numbers less than 1, the decimal moves right and the exponent is negative.

Example: 0.0000809

1. Move the decimal right until the first nonzero digit is in front: 8.09
2. You moved it 5 places.
3. Since you moved right, the exponent is negative.
4. Answer: 8.09 × 10⁻⁵

A handy memory trick: big number, positive exponent; small number, negative exponent.

How To Convert Scientific Notation To Standard Form

To change scientific notation back to a regular number, move the decimal point the number of places shown in the exponent. Move right for a positive exponent and left for a negative one, adding zeros as needed.

Example: 7.2 × 10⁵

1. The exponent is +5, so move the decimal 5 places right.
2. 7.2 → 72 → 720 → 7,200 → 72,000 → 720,000
3. Answer: 720,000

Example: 3.05 × 10⁻³

1. The exponent is -3, so move the decimal 3 places left.
2. 3.05 → 0.305 → 0.0305 → 0.00305
3. Answer: 0.00305

The scientific notation calculator converts both ways if you want to check your work.

Scientific Notation Examples

These scientific notation examples show the most common conversions, from easy whole numbers to tricky decimals.

Large Number Examples

The last row answers a popular question: 5,970,000,000,000,000,000,000,000 has 25 digits, so its decimal moves 24 places, giving 5.97 × 10²⁴.

Small Number Examples

Numbers Between 1 And 10

A number that already sits between 1 and 10 uses an exponent of 0.

• 7 = 7 × 10⁰
• 3.14 = 3.14 × 10⁰

You will rarely need to write these, but they show the pattern holds for every number.

Scientific Notation In Real Life

Scientific notation shows up anywhere numbers get extremely large or small, from astronomy to chemistry to computer storage. These real values show how it is used.

The speed of light and Avogadro's number are exact values set by the International System of Units (SI), and the electron mass comes from the NIST table of physical constants. Earth's mass and distance come from NASA's Earth fact sheet.

Example: how long does sunlight take to reach Earth?

Time = distance ÷ speed

1. Distance: 1.49598 × 10¹¹ m (that is 1.49598 × 10⁸ km converted to meters)
2. Speed: 2.998 × 10⁸ m/s
3. Divide the coefficients: 1.49598 ÷ 2.998 ≈ 0.499
4. Subtract the exponents: 11 – 8 = 3
5. Result: 0.499 × 10³ = 4.99 × 10² seconds, or about 499 seconds (8.3 minutes)

How To Compare And Estimate With Scientific Notation

Scientific notation makes it quick to tell which number is bigger and roughly how many times bigger it is. You mostly look at the exponents.

Comparing Two Numbers

To compare two positive numbers in scientific notation, check the exponents first. The larger exponent means the larger number. Only if the exponents match do you compare the coefficients.

The last row trips people up. With negative exponents, the number closer to zero is the larger exponent. So 2 × 10⁻² (0.02) is bigger than 5 × 10⁻³ (0.005).

Estimating How Many Times Larger

To estimate how many times larger one number is than another, divide them. Round the coefficients to one digit first to keep it quick.

Example: how many kilobytes fit in a gigabyte (using metric units)?

• 1 gigabyte = 1 × 10⁹ bytes and 1 kilobyte = 1 × 10³ bytes
• 10⁹ ÷ 10³ = 10⁶
• About one million kilobytes fit in a gigabyte.

Each step of 1 in the exponent means "10 times bigger." A difference of 3 in the exponents means 1,000 times bigger, and a difference of 6 means a million times bigger.

How To Multiply And Divide In Scientific Notation

Multiplying and dividing are the easiest operations in scientific notation. Work with the coefficients and the exponents separately, then adjust the answer so the coefficient is between 1 and 10.

Multiplying

Multiply the coefficients and add the exponents.

Example: (3 × 10⁴) × (2 × 10⁵)

1. Coefficients: 3 × 2 = 6
2. Exponents: 4 + 5 = 9
3. Answer: 6 × 10⁹

Example: (4 × 10³) × (5 × 10⁶)

1. Coefficients: 4 × 5 = 20
2. Exponents: 3 + 6 = 9
3. That gives 20 × 10⁹, but 20 is not between 1 and 10.
4. Rewrite 20 as 2 × 10¹, then add exponents: 2 × 10¹⁰
5. Answer: 2 × 10¹⁰

Dividing

Divide the coefficients and subtract the exponents (top minus bottom).

Example: (8.4 × 10⁷) ÷ (2.1 × 10³)

1. Coefficients: 8.4 ÷ 2.1 = 4
2. Exponents: 7 – 3 = 4
3. Answer: 4 × 10⁴

Example: (1.5 × 10⁻³) ÷ (6 × 10²)

1. Coefficients: 1.5 ÷ 6 = 0.25
2. Exponents: -3 – 2 = -5
3. That gives 0.25 × 10⁻⁵, but 0.25 is less than 1.
4. Rewrite 0.25 as 2.5 × 10⁻¹, so the exponent becomes -5 + (-1) = -6
5. Answer: 2.5 × 10⁻⁶

These rules come straight from the laws of exponents. An exponent calculator helps if you want to see why 10⁴ × 10⁵ = 10⁹.

How To Add And Subtract In Scientific Notation

Adding and subtracting take one extra step: the exponents must match first. Then you add or subtract the coefficients and keep the shared exponent.

Example: (3.2 × 10⁵) + (4.5 × 10⁴)

1. Rewrite the smaller number with the larger exponent: 4.5 × 10⁴ = 0.45 × 10⁵
2. Add the coefficients: 3.2 + 0.45 = 3.65
3. Answer: 3.65 × 10⁵

Example: (6.8 × 10⁻³) – (2.5 × 10⁻⁴)

1. Match exponents: 2.5 × 10⁻⁴ = 0.25 × 10⁻³
2. Subtract: 6.8 – 0.25 = 6.55
3. Answer: 6.55 × 10⁻³

A quick check: the answer should be close to the larger number, because the smaller one only adds a little. If the exponents differ by more than about 10, the smaller number barely changes the result at all.

Powers And Roots In Scientific Notation

To raise a number in scientific notation to a power, apply the power to the coefficient and multiply the exponent. For a square root, take the root of the coefficient and halve the exponent.

Example: (2 × 10³)²

• 2² = 4 and 3 × 2 = 6
• Answer: 4 × 10⁶

Example: √(9 × 10⁸)

• √9 = 3 and 8 ÷ 2 = 4
• Answer: 3 × 10⁴

If the exponent is odd, rewrite the number first so it becomes even. For √(4 × 10⁵), use 40 × 10⁴ instead, so √40 ≈ 6.32 and the answer is about 6.32 × 10². A scientific calculator handles the decimal root.

E Notation And Engineering Notation

Calculators and computers often show scientific notation in a shorter form called E notation. Engineering notation is a close relative that uses exponents in steps of three.

E Notation On Calculators

E notation replaces "× 10^" with the letter E. The number after E is the exponent.

Most scientific and graphing calculators use an EE or EXP key to enter these numbers. To type 6.02 × 10²³, press 6.02, then EE, then 23. Do not type "× 10 ^ 23" with the EE key as well, or you will multiply by 10 twice. Spreadsheets like Excel and Google Sheets also show very large results in E notation, such as 1.23E+15.

Engineering Notation

Engineering notation is like scientific notation, but the exponent is always a multiple of 3 and the coefficient can be from 1 up to 1,000. It lines up with metric prefixes, which makes it popular in electronics and engineering.

The common metric prefixes and their powers of 10 are: kilo (10³), mega (10⁶), giga (10⁹), tera (10¹²), milli (10⁻³), micro (10⁻⁶) and nano (10⁻⁹). The unit converter uses these when you switch between units like grams and kilograms.

Scientific Notation And Significant Figures

The coefficient in scientific notation shows exactly how many significant figures a measurement has. This removes the guesswork that trailing zeros cause in standard form.

All three equal 1,500, but they tell the reader something different about how precise the measurement was. In science class, round your coefficient to match the least precise value in the problem.

When Students Learn Scientific Notation

In schools that follow the Common Core State Standards, scientific notation is a grade 8 topic. Students use it to estimate and compare large and small quantities, then perform operations with it.

The 8.EE.A.3 standard uses a real example: estimating the US population as 3 × 10⁸ and the world population as 7 × 10⁹, then showing the world population is more than 20 times larger. Those are the rough values used in the standard, not current counts.

Common Mistakes To Avoid

Most scientific notation mistakes come from a coefficient outside the 1 to 10 range or an exponent with the wrong sign.

Practice Problems With Answers

Try these, then check below.

1. Write 380,000 in scientific notation.
2. Write 0.000047 in scientific notation.
3. Write 9.1 × 10⁶ in standard form.
4. Write 6.4 × 10⁻⁵ in standard form.
5. (2.5 × 10⁻⁶) × (4 × 10²)
6. (9 × 10⁸) ÷ (3 × 10⁻²)
7. (5.1 × 10⁴) + (3 × 10³)

Answers

1. 3.8 × 10⁵
2. 4.7 × 10⁻⁵
3. 9,100,000
4. 0.000064
5. 10 × 10⁻⁴ = 1 × 10⁻³
6. 3 × 10⁸⁻⁽⁻²⁾ = 3 × 10¹⁰
7. 5.1 × 10⁴ + 0.3 × 10⁴ = 5.4 × 10⁴

Summary

Scientific notation writes a number as a × 10ⁿ, where a is at least 1 and less than 10. Large numbers get positive exponents, and small numbers get negative ones. To convert, count how many places the decimal moves.

To multiply, multiply the coefficients and add the exponents. To divide, divide and subtract. To add or subtract, match the exponents first. Always finish by adjusting the coefficient back into the 1 to 10 range.

Next step: work through the practice problems, then check your answers with the scientific notation calculator.

Frequently Asked Questions