Scientific Notation Calculator

Scientific notation calculator at a glance#

What is scientific notation?#

Scientific notation expresses a number as a value from 1 to just under 10 multiplied by a power of 10. The power of 10 indicates how many places the decimal point has shifted. For large numbers, the exponent is positive, and for small numbers, it's negative, so 3,230,000 becomes 3.23 × 10⁶, and 0.00052 becomes 5.2 × 10⁻⁴.

How do you write 3,230,000 in scientific notation?#

Shift the decimal point to the left until a single nonzero digit precedes it. From 3,230,000, this requires 6 moves and gives 3.23, so the answer is 3.23 × 10⁶. By entering 3230000 into the converter above, you will see the same result and the E notation 3.23e+6.

What is 0.00052 in scientific notation?#

Shift the decimal point to the right until it is positioned after the first nonzero digit. This requires 4 moves and gives 5.2, and since moving right makes the exponent negative, the answer is 5.2 × 10⁻⁴. In E notation, it appears as 5.2e-4.

How scientific notation works#

OpenStax Elementary Algebra describes a number in scientific notation as a × 10ⁿ, where the absolute value of a ranges from at least 1 to less than 10, and n is an integer. The initial part is often referred to as the coefficient. A negative number retains its sign on the coefficient, so −0.0000345 is −3.45 × 10⁻⁵.

The exponent acts as a counter. Each upward step multiplies by 10, and each downward step divides by 10. 10³ equals 1,000, 10⁰ equals 1, and 10⁻³ equals 0.001. This explains why a positive exponent signifies a number whose size is 10 or more. An exponent of 0 means a size from 1 to just under 10. A negative exponent signifies a number between −1 and 1, excluding 0.

Converting by hand#

To convert a number to scientific notation:

• Move the decimal point so that the coefficient is at least 1 and less than 10.
• Count the number of places the decimal moved. This count becomes the exponent.
• If the original number was 10 or more, the exponent is positive. From 1 to just under 10, it is 0. If it was between 0 and 1, the exponent is negative.

For 3,230,000, the decimal point moves 6 places to the left, as the diagram below shows. To reverse this process, follow these steps in reverse. A positive exponent will move the decimal point to the right, while a negative exponent will move it to the left, adding zeros as necessary. So, 4.78 × 10⁸ becomes 478,000,000, and 6.5 × 10⁻⁴ becomes 0.00065.

What the converter shows#

When you enter a number into the converter tab, it presents the number in four different forms:

• Scientific notation, for example 1.00904 × 10¹⁰ for 10,090,400,000.
• E notation, which is commonly used by calculators and spreadsheets, where e+10 signifies multiplication by 10 to the power 10: 1.00904e+10.
• Engineering notation, which keeps the exponent a multiple of 3, as shown by 10.0904 × 10⁹. These multiples of 3 align with metric prefixes listed by NIST, like kilo for 10³, mega for 10⁶, and micro for 10⁻⁶.
• The full real number.

You should type the number using only digits, avoiding commas or spaces, or in E notation such as 4.78e8. The real number is written out fully when the exponent ranges from −6 to 20, reverting to E notation outside this range. For example, 1.49e-7 remains as 1.49e-7. The converter also omits zeros at the end of a decimal, so 4.50 becomes 4.5. If these zeros are significant figures, the significant figures calculator rounds to a set count and keeps them.

Calculating with scientific notation#

The calculator tab requires two numbers and one operation: add, subtract, multiply, divide, or raise to a power. You can input each value as 2.3 x 10^5 using the letter x, or in the form 2.3e5. Regular numbers like 46 are also valid. The result will be returned in scientific notation.

Multiplying and dividing#

To multiply, multiply the coefficients and add the exponents. Take 46 times 7.7 × 10⁵, and express 46 as 4.6 × 10¹. From there, calculate 4.6 × 7.7 = 35.42 and 1 + 5 = 6, resulting in 35.42 × 10⁶. If the coefficient is 10 or more, shift the decimal point one place to the left and add 1 to the exponent to get 3.542 × 10⁷. The diagram below illustrates each step clearly.

For division, divide the coefficients and subtract the exponents: (6.4 × 10⁸) ÷ (1.6 × 10³) = 4 × 10⁵. When dealing with powers, the same method applies, so (3 × 10⁴)² = 9 × 10⁸, because squaring 3 gives 9 and the exponent is doubled.

Adding and subtracting#

The exponents must match before adding or subtracting. In (2.3 × 10⁵) + (7.49 × 10⁵) both already have 10⁵ in common, so add the coefficients: 9.79 × 10⁵. For (4 × 10⁵) + (3.5 × 10⁴), adjust the smaller number to 0.35 × 10⁵. The sum is 4.35 × 10⁵. If the new coefficient passes 10, adjust once more: (9.73 × 10⁶) + (8.8 × 10⁶) = 18.53 × 10⁶ = 1.853 × 10⁷.

Scientific notation on a handheld calculator#

Most scientific calculators feature a power of 10 key, which may be labeled as EE, EXP, or × 10^x depending on the brand. Input the coefficient, press the corresponding key, and then enter the exponent. The display often presents answers in E notation, so 3.542E7 means 3.542 × 10⁷. For a negative coefficient or exponent, use the negative sign key rather than the subtraction key.

Where scientific notation is used#

• In science and engineering, it helps deal with measurements ranging from the sizes of atoms to the vast distances between stars.
• Metric units use it too, where prefixes like nano, micro, kilo, and mega each represent a power of 10.
• In computing and spreadsheets, very large and very small results appear in E notation.
• For checking estimates, the exponent quickly reveals the size of a number.

Numbers in scientific notation#

These are numbers that people often convert. Each value is derived from the calculator's converter.

Sources#

• OpenStax, Elementary Algebra 2e, section 6.7: Integer Exponents and Scientific Notation
• NIST, Metric (SI) Prefixes

Frequently asked questions#

What does the e mean in a number like 1.5e+6?#

The e in E notation is shorthand for times 10 to the power that follows. For instance, 1.5e+6 represents 1.5 × 10⁶, or 1,500,000, and 5.2e-4 equals 5.2 × 10⁻⁴, or 0.00052.

Is scientific notation the same as standard form?#

In the UK and many other countries, standard form means the same thing as scientific notation. However, in the US, standard form often refers to the usual way of writing numbers, such as 478,000,000. The standard form calculator converts any number to the a × 10ⁿ form.

What is the difference between scientific and engineering notation?#

Scientific notation keeps a single digit before the decimal point. In contrast, engineering notation ensures the exponent is a multiple of 3, allowing the coefficient to range from 1 to just under 1,000. For example, 0.00052 becomes 5.2 × 10⁻⁴ in scientific notation and 520 × 10⁻⁶ in engineering notation.

Can the coefficient be negative?#

Yes, it can. The sign belongs to the coefficient, while the exponent only determines the size. So −0.0000345 is −3.45 × 10⁻⁵.

How do I round the coefficient?#

To round the coefficient, adjust it to the desired number of significant figures while maintaining the same exponent. For example, 1.568938 × 10⁶ rounded to three significant figures is 1.57 × 10⁶. The rounding calculator handles the decimal part.