Why use powers of ten
Engineering numbers span huge ranges. A wavelength might be a few hundred billionths of a metre, and a clock frequency several billion cycles per second. Writing them as 0.0000005 or 3,000,000,000 invites miscounted zeros. Scientific notation puts the size of the number in an exponent so that it can be read and compared at a glance.
The format
A number in scientific notation is written as a × 10^n, where a (the coefficient or mantissa) is at least 1 and less than 10, and n is a whole number (the exponent).
- 4,500,000 = 4.5 × 10^6 (the decimal point moves six places left)
- 0.00032 = 3.2 × 10^-4 (the decimal point moves four places right)
- 7 = 7 × 10^0, since any number to the power zero is one
A quick rule: moving the decimal point left to make the coefficient smaller makes the exponent larger by the same number of places, and the reverse for moving right. A large number has a positive exponent; a number less than one has a negative one.
Multiplying
Multiply the coefficients and add the exponents.
(3 × 10^4) × (2 × 10^3) = 6 × 10^7.
If the coefficient product reaches 10 or more, adjust:
(4 × 10^5) × (5 × 10^3) = 20 × 10^8 = 2 × 10^9.
Dividing
Divide the coefficients and subtract the exponents.
(6 × 10^8) ÷ (2 × 10^-3) = 3 × 10^(8 - (-3)) = 3 × 10^11.
Subtracting a negative exponent is a classic slip. Writing the subtraction explicitly, as above, helps.
Adding and subtracting
You can only add the coefficients directly when the exponents match, so rewrite one number first.
3.2 × 10^5 + 4 × 10^4 = 3.2 × 10^5 + 0.4 × 10^5 = 3.6 × 10^5.
A sanity check: 320,000 + 40,000 = 360,000. Correct.
Notice that adding does not add the exponents. That rule belongs to multiplication.
Powers and roots
Raising to a power multiplies the exponent.
(2 × 10^3)² = 4 × 10^6, because 2² = 4 and 3 × 2 = 6.
Square roots halve the exponent, which is easiest when the exponent is even.
√(9 × 10^6) = 3 × 10^3, since √9 = 3 and 6 ÷ 2 = 3.
If the exponent is odd, rewrite first: √(1.6 × 10^5) = √(16 × 10^4) = 4 × 10^2 = 400. Check: 400² = 160,000. Correct.
Engineering notation
Engineering notation is the same idea, but the exponent is restricted to multiples of three: 10^3, 10^6, 10^-3, 10^-9 and so on. The coefficient may then be anywhere from 1 up to (but not including) 1000. For instance, 0.000047 farad is 47 × 10^-6 F, which is 47 µF. This links directly to the standard prefixes: kilo is 10^3, mega is 10^6, milli is 10^-3, micro is 10^-6. The article on SI prefixes covers these in more detail.
How calculators and software display it
Calculators and spreadsheets often show 3.2E5 or 3.2e5. This means 3.2 × 10^5, and has nothing to do with the mathematical constant e. A common mistake is typing 3.2 × 10^5 into a calculator as “3.2 × 10 ^ 5” without brackets in a longer expression, which can change the order of operations. Use the dedicated exponent key or E notation when entering data. In engineering spreadsheets, a cell shown as 1.5E-06 is a number, not text, and it can be formatted differently without changing the value.
Orders of magnitude and the logarithmic view
The exponent tells you the order of magnitude. Two numbers whose exponents differ by 3 differ by roughly a factor of a thousand. This is the basis of order-of-magnitude estimates, where you work only with the exponent and one digit. The same idea underlies the decibel scale, which turns ratios into additions through logarithms.
Rounding and digits
The number of digits in the coefficient indicates how precisely a value is known. Writing 3.0 × 10^8 suggests two significant figures, while 3 × 10^8 suggests one, and 2.998 × 10^8 suggests four. This is why scientific notation is helpful for removing ambiguity: the number 1,500 could have two, three or four meaningful digits, while 1.5 × 10^3 and 1.500 × 10^3 are explicit. See significant figures for guidance on how many to keep.
Common mistakes
- Wrong sign on the exponent. Small numbers have negative exponents.
- Adding exponents when adding numbers. Only multiplication adds exponents.
- Coefficient outside 1 to 10. Not always wrong in intermediate steps, but tidy up at the end.
- Dropping the unit prefix. 47 × 10^-6 F needs its unit; “47 × 10^-6” alone does not say what it is.
- Moving the decimal the wrong way. Check with a rough comparison: 3.2 × 10^-4 must be smaller than 1.
Practice
Work these by hand, then confirm with a calculator:
- (2.5 × 10^3) × (4 × 10^-5) = 10 × 10^-2 = 1 × 10^-1 = 0.1
- (9 × 10^6) ÷ (3 × 10^2) = 3 × 10^4
- 5 × 10^3 + 2 × 10^2 = 5.2 × 10^3
- (3 × 10^-2)² = 9 × 10^-4
Each answer above has been checked, so treat them as a way to test your method rather than only a result to memorise.