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| < 10The exponent records the decimal shift
The coefficient is normalized so its absolute value is at least 1 and less than 10.
| Ordinary number | Scientific notation | Direction of shift |
|---|---|---|
| 72,500 | 7.25 x 10^4 | Decimal moved 4 places left |
| 0.00725 | 7.25 x 10^-3 | Decimal moved 3 places right |
| -0.00081 | -8.1 x 10^-4 | Sign stays with the coefficient |
Worked example 1
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
Converting 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.
Normalize after the arithmetic
A correct intermediate product may still need one more power-of-ten adjustment.
1. Multiply (6 x 10^4)(3 x 10^2).
Hint: First get 18 x 10^6, then normalize 18.
Answer: 1.8 x 10^7
2. Add 2.4 x 10^5 and 7 x 10^4.
Hint: Express both terms with exponent 5.
Answer: 2.4 x 10^5 + 0.7 x 10^5 = 3.1 x 10^5
Normalization and exponent checks
- 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.
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)Questions about scientific notation
Is E notation the same thing?
Yes. 3.2e5 is a plain-text form of 3.2 x 10^5.