Number Systems
Scientific Notation Explained
Scientific notation writes a nonzero number as a x 10^n, where 1 <= |a| < 10 and n is an integer. It separates significant digits from the number's scale.
Reviewed
Normalized scientific notation
The coefficient contains the meaningful digits and has exactly one nonzero digit before the decimal point. The exponent tells how many places the decimal point moves.
A positive exponent moves the decimal point right when converting to ordinary notation. A negative exponent moves it left.
N = a x 10^n, with 1 <= |a| < 10Converting in both directions
To convert an ordinary number, move its decimal point until the coefficient is normalized. Count the moves: moving left gives a positive exponent, while moving right gives a negative exponent.
Zero is a special case and is commonly written 0 rather than with a unique scientific exponent.
Arithmetic with powers of ten
For multiplication, multiply coefficients and add exponents; for division, divide coefficients and subtract exponents. Normalize the coefficient afterward.
Addition and subtraction require matching exponents first because the coefficients must represent the same place-value scale.
(a x 10^m)(b x 10^n) = ab x 10^(m+n)Worked examples
Convert 0.0000725
Move the decimal five places right to make 7.25.
coefficient = 7.25moving right gives exponent -5
Result: 0.0000725 = 7.25 x 10^-5
Multiply (3.2 x 10^5)(4 x 10^-3)
Combine coefficients and powers separately.
3.2 x 4 = 12.810^5 x 10^-3 = 10^212.8 x 10^2 = 1.28 x 10^3
Result: 1.28 x 10^3
Common mistakes
- Leaving a coefficient such as 12.8 without normalizing it.
- Adding exponents during addition instead of first matching them.
- Reversing the exponent sign for small decimal numbers.
Where this idea is used
- Recording astronomical distances and microscopic measurements.
- Keeping significant figures visible during calculations.
- Comparing orders of magnitude quickly.
Questions about scientific notation
Is E notation the same thing?
Yes. 3.2e5 is a plain-text form of 3.2 x 10^5.
Can the coefficient be negative?
Yes. Its magnitude must be at least 1 and less than 10.
How do I add two scientific-notation values?
Rewrite one so both powers of ten match, add the coefficients, and normalize the result.