Scientific notation calculator with steps

This scientific notation calculator converts any number to scientific notation and back, and adds, subtracts, multiplies or divides numbers written in it. A number in scientific notation is a value from 1 up to 10 times a power of ten, such as 4.5 × 10⁻⁴. The calculator keeps every digit exactly and shows each step, including how far the decimal point moves.

In the UK the same thing is called standard form. The calculator opens on Convert, with 0.00045 written as 4.5 × 10⁻⁴.

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Scientific Notation Calculator

Convert to and from standard form and calculate with it

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How to use the scientific notation calculator

1. Under What to do, choose Convert to change one number, or Calculate to work with two.
2. Type the number. It can be an ordinary number such as 45000 or 0.0032, E notation such as 6.02e23, or scientific notation such as 4.7 × 10⁻⁶.
3. For Calculate, type the first number, pick +, −, × or ÷, then type the second number.
4. Read the answers: scientific notation, the ordinary number, E notation, and for Convert, engineering notation and the order of magnitude.
5. Follow the working, and watch the visual: in Convert it shows the decimal point hopping along the digits, and in Calculate it places both numbers and the answer on an order of magnitude scale.

Answers are exact, so 6.02e23 converts to 602,000,000,000,000,000,000,000 with nothing rounded off. A division that does not end, such as 1 ÷ 3, is given to 12 significant figures and marked as approximate. For practice, Practice and help sets multiplication and conversion questions that ask for the power of ten, with hints and worked solutions.

Writing a number in scientific notation

Scientific notation writes a number as a × 10ⁿ, where a is at least 1 and less than 10, and n is a whole number. To convert:

1. Put the decimal point just after the first digit that is not zero. That gives a.
2. Count how many places the point moved. That is the size of n.
3. If the original number was 10 or more, n is positive. If it was less than 1, n is negative.

Example 1: a large number

Write 93,000,000 in scientific notation. The point moves from the end of the number to just after the 9, which is 7 places left:

The calculator shows the same, with E notation 9.3e7 and engineering notation 93 × 10⁶.

Example 2: a small number

Write 0.000000072 in scientific notation. The point moves 8 places right to sit after the 7, so the power is negative:

A quick check: a negative power means a number less than 1, and a positive power means a number of 10 or more.

Try one yourself:

Converting back to an ordinary number

To go the other way, move the point n places: right for a positive power, left for a negative one, filling gaps with zeros. So 4.7 × 10⁻⁶ is 0.0000047. In Convert, type 4.7*10^-6 or 4.7e-6 and the calculator gives the ordinary number.

In the UK, standard form means scientific notation. In many US classrooms, though, writing a number "in standard form" means writing it out in full, like 0.0000047. Check which one your teacher means.

Calculating in scientific notation

Choose Calculate to work with two numbers. The method depends on the operation.

Example 3: multiplying

Work out (7.5 × 10⁶) × (4 × 10⁻²). Multiply 7.5 by 4 to get 30, and add the powers: 6 + (−2) = 4. But 30 is not between 1 and 10, so move the point one place left and add 1 to the power:

Example 4: dividing

Work out (6 × 10⁸) ÷ (1.5 × 10³). Divide 6 by 1.5 to get 4, and subtract the powers: 8 − 3 = 5.

Example 5: adding

Work out 4.2 × 10⁵ + 3.1 × 10⁴. You can only add the first parts when the powers match, so rewrite 3.1 × 10⁴ as 0.31 × 10⁵:

Example 6: subtracting small numbers

Work out 2.5 × 10⁻³ − 8 × 10⁻⁵. The larger power is −3, so write 8 × 10⁻⁵ as 0.08 × 10⁻³:

Every one of these can be typed into Calculate to check, and the working matches these steps.

Comparing two numbers in scientific notation

To see which of two numbers is bigger, compare the powers first: the larger power wins, whatever the first parts are. So 3.6 × 10⁻⁵ is bigger than 8.1 × 10⁻⁶, because −5 is greater than −6. Only when the powers match do you compare the first parts. To find how many times bigger, divide: 3.6 × 10⁻⁵ ÷ 8.1 × 10⁻⁶ is about 4.4. In Calculate, the order of magnitude scale places both numbers and the answer, so the gap is easy to see.

E notation, engineering notation and order of magnitude

The calculator gives each answer in several forms, because different places write large and small numbers differently.

Engineering notation keeps the power a multiple of 3, so it lines up with unit prefixes: 450 × 10⁻⁶ m is 450 micrometres.

How to do scientific notation on a calculator

On a handheld scientific or graphing calculator, type the first part, press the key marked EXP, EE or ×10ˣ, then type the power. Use the (−) key for a negative power, not the subtract key. The answer often shows as E notation: 4.5E−4 on the screen means 4.5 × 10⁻⁴. Copy it into your working as 4.5 × 10⁻⁴, not as 4.5E−4: E is calculator shorthand, not written maths.

Common mistakes

• A first part outside 1 to 10. 45 × 10³ and 0.45 × 10⁵ are the right value but are not scientific notation. It should be 4.5 × 10⁴.
• The wrong sign on the power. Numbers less than 1 have a negative power. 0.003 is 3 × 10⁻³, not 3 × 10³.
• Adding the powers when you add numbers. Only multiplying adds the powers. To add, match the powers first.
• Forgetting to adjust after multiplying. 30 × 10⁴ is not finished. Move the point and add 1 to the power: 3 × 10⁵.
• Typing 10 and then EXP. On a calculator, 4.5 EXP 3 already means 4.5 × 10³. Typing 4.5 × 10 EXP 3 gives a number ten times too big.
• Rounding the first part too much. Keep the significant figures the question asks for, then round once at the end.

Practise in a short timed paper

Scientific notation comes up in Pre-Algebra & Algebra 1 practice, and as standard form in GCSE Maths Higher in the UK. It is also used throughout chemistry and physics, from the Moles Calculator to the Gravitational Force Calculator. Each daily paper is a short timed paper of 8 questions worth 25 to 35 marks, and the papers mix topics.

Read next

Maths · NumberScientific notation: how to write, convert and calculateWhat scientific notation is, how to convert large and small numbers into it and back, and how to multiply, divide, add and subtract with it, with examples.