Turn a number into scientific notation.
The number is expressed as a value between 1 and 10, multiplied by a power of ten.
Form: mantissa × 10^exponent
A scientific notation calculator rewrites a number as a mantissa between 1 and 10 multiplied by a power of ten. The average distance from the Earth to the Moon, about 384,400,000 metres, becomes 3.844 × 10⁸ m. Very small numbers work the same way with a negative exponent.
Science textbooks from Class 8 onwards, and all of physics and chemistry at higher levels, use this form. It makes very large and very small quantities readable, shows their size at a glance through the exponent, and makes multiplication and division easy. It is also how calculators and spreadsheets display numbers that do not fit on the screen, often as 3.844E8.
1. Ignore the sign for now; you will put it back at the end.
2. Move the decimal point until exactly one non-zero digit is to its left. This gives the mantissa m, with 1 ≤ m < 10.
3. Count how many places you moved the point. That count is the exponent n.
4. If you moved the point to the left, as for a large number, n is positive. If you moved it to the right, as for a number below 1, n is negative.
5. Write the number as m × 10ⁿ and restore the sign.
6. Check by multiplying out: m × 10ⁿ should give back the original number.
The exponent is found as the whole-number part of log₁₀ of the number's size. For 384,400,000, log₁₀ is about 8.58, so the exponent is 8, and dividing by 10⁸ gives the mantissa 3.844. The exponent tells you the order of magnitude at once: 10⁸ metres is hundreds of millions of metres. Comparing two numbers in scientific notation starts with the exponents, and only if they match do you compare mantissas.
Multiplying (a × 10ᵐ) by (b × 10ⁿ) gives (a × b) × 10ᵐ⁺ⁿ, so you multiply the small mantissas and add the exponents. Division subtracts exponents. If the new mantissa is 10 or more, shift one place and add 1 to the exponent. Addition needs matching exponents first, much as fractions need a common denominator. These rules are the laws of exponents in action, and they are why physicists can estimate answers mentally with large numbers.
Scientific notation also shows precision. Writing 3.844 × 10⁸ says four significant figures are known, while 3.8 × 10⁸ says only two. Plain 384,400,000 hides this, because the trailing zeros may or may not be meaningful. Engineering notation is a variant where the exponent is always a multiple of three, matching prefixes such as kilo, mega and giga, so 384,400,000 m would be 384.4 × 10⁶ m. The calculator gives standard scientific notation, rounding the mantissa to six decimal places.
For a Class 8 science project on the Chandrayaan missions, Aarav writes down the average Earth to Moon distance as 384,400,000 metres and needs it in scientific notation.
Move the decimal point until one non-zero digit is to its left: Move it 8 places to the left: 3.844
The number of places moved is the power of 10: Moved left, so the power is positive: 10^8
Scientific notation: 3.844 × 10^8
Engineering notation (power is a multiple of 3): 384.4 × 10^6, SI prefix mega (M)
Answer: Scientific notation 3.844 × 10^8; E-notation 3.844e8; Engineering notation 384.4 × 10^6
Leaving the mantissa outside the range 1 to 10, such as writing 38.44 × 10⁷.
Giving a positive exponent to a number smaller than 1.
Adding mantissas in a sum without first matching the exponents.
Reading a calculator display like 3.844E8 as 3.844 × 8.
Writing astronomical distances, such as to the Moon, Sun and planets.
Expressing tiny quantities like the mass of an electron or the size of a virus.
Avogadro's number and mole calculations in chemistry.
Rough estimation and checking answers in physics numericals.
Reading very large or very small values in spreadsheets.
Why use scientific notation?
It makes very large or very small numbers easier to read and compare, and is the normal way of writing numbers in science and engineering.