Pick a mode: convert a decimal into scientific notation, expand scientific notation back into a decimal, or multiply and divide two numbers written in scientific notation. Every answer shows the decimal point move or the coefficient and exponent arithmetic with your own numbers.
What scientific notation is
Scientific notation writes a number as a coefficient between 1 and 10 multiplied by a power of ten, so astronomers and chemists can stop counting zeros. Instead of writing Avogadro's number as 602,000,000,000,000,000,000,000, they write 6.022 x 10^23, and instead of writing an electron's mass as 0.00000000000000000000000000091 kilograms, they write 9.1 x 10^-31. Same numbers, far fewer chances to lose count of a zero.
Converting to scientific notation: the decimal point walk
Slide the decimal point until exactly one nonzero digit sits to its left, then count how many places it moved. Moving the point left makes the exponent positive, because the original number is 10 or larger. Moving the point right makes the exponent negative, because the original number is smaller than 1. The digit count itself never changes, only where you place the point and what power of ten you multiply by to make up the difference.
45,800,000 = 4.58 × 107 (point moved left 7 places)
0.0000032 = 3.2 × 10-6 (point moved right 6 places)
Converting from scientific notation back to a decimal
Run the walk in reverse. For a positive exponent, move the decimal point right that many places, filling in zeros as you run out of digits. For a negative exponent, move it left the same way. 6.02 x 10^23 moves the point right 23 places from "6.02," landing on 602,000,000,000,000,000,000,000, a 24 digit number built entirely from placeholder zeros after the first three digits.
Why the significant figures matter
Plain decimal notation cannot always tell you how precisely a number was measured. Scientific notation makes significant figures unambiguous: 3.10 x 10^2 states three significant figures on purpose, while plain 310 could mean two, three, or that the last digit was just a rounded guess. Every digit you write in the coefficient is treated as meaningful, so trailing zeros after the decimal point carry real information instead of just taking up space.
Worked examples
Convert 45,800,000 to scientific notation: the decimal point starts after the last zero. Move it left until one digit remains to its left: 4.58, moved 7 places. Since we moved left, the exponent is positive: 4.58 x 10^7.
Convert 0.0000032 to scientific notation: the point starts right after the leading zero. Move it right until it sits after the first nonzero digit, 3: that took 6 places. Since we moved right, the exponent is negative: 3.2 x 10^-6.
Multiply (3 x 10^8) by (2 x 10^-3): multiply the coefficients, 3 x 2 = 6, and add the exponents, 8 + (-3) = 5. The coefficient 6 is already between 1 and 10, so no renormalizing is needed: 6 x 10^5.
Engineering notation: the metric-friendly cousin
Engineering notation is scientific notation with one extra rule: the exponent must be a multiple of 3, so the coefficient can run from 1 up to just under 1,000 instead of 1 to 10. Engineers favor it because it lines up with the metric prefixes people already use: 10^3 is kilo, 10^6 is mega, 10^9 is giga. That is why 45,800,000 reads naturally as 45.8 x 10^6, or 45.8 mega, instead of 4.58 x 10^7. The convert-to-scientific-notation mode above shows this line automatically.
Frequently asked questions
What is scientific notation?
Scientific notation writes a number as a coefficient between 1 and 10 multiplied by a power of ten, like 4.58 x 10^7. It exists so people working with very large or very small numbers, astronomers and chemists especially, do not have to write or count long strings of zeros.
How do you convert a number to scientific notation?
Move the decimal point until exactly one nonzero digit sits to its left, then multiply by 10 raised to the number of places you moved. Moving the point left gives a positive exponent, moving it right gives a negative exponent. For 45,800,000 the point moves left 7 places, so it becomes 4.58 x 10^7.
How do you convert scientific notation back to a normal number?
Move the decimal point the number of places shown by the exponent: right for a positive exponent, left for a negative one, filling in zeros as needed. 3.2 x 10^-6 moves the point left 6 places to give 0.0000032.
Why is 310 written as 3.10 x 10^2 sometimes and 3.1 x 10^2 other times?
Scientific notation makes significant figures unambiguous, which plain decimal notation cannot. Writing 3.10 x 10^2 states three significant figures on purpose, while 3.1 x 10^2 states two. The trailing zero in 310 by itself is ambiguous: you cannot tell if it was measured precisely or just rounded.
How do you multiply or divide numbers in scientific notation?
Multiply or divide the coefficients as normal numbers, then add the exponents for multiplication or subtract them for division. If the resulting coefficient is not between 1 and 10, shift the decimal point one place and adjust the exponent by 1 to renormalize, for example 20 x 10^7 becomes 2 x 10^8.
What is engineering notation?
Engineering notation is scientific notation with the exponent restricted to a multiple of 3, so the coefficient runs from 1 to 999 instead of 1 to 10. Engineers use it because it lines up with metric prefixes: 10^3 is kilo, 10^6 is mega, 10^9 is giga, so 45,800,000 reads naturally as 45.8 x 10^6, or 45.8 mega.
Why is Avogadro's number written in scientific notation?
Avogadro's number is about 602,000,000,000,000,000,000,000, and an electron's mass is about 0.00000000000000000000000000091 kilograms. Writing either one in full decimal form invites miscounted zeros. Scientific notation, 6.022 x 10^23 and 9.1 x 10^-31, keeps both numbers exact and easy to compare at a glance.
Does a negative number have a negative exponent in scientific notation?
Not necessarily; the sign of the number and the sign of the exponent are independent. Negative numbers 45,800,000 and -0.0000032 look like -4.58 x 10^7 and -3.2 x 10^-6: the minus sign stays on the coefficient, while the exponent's sign depends only on whether the original number is larger or smaller than 1.