Direct answer
Scientific notation writes a nonzero value as a coefficient with magnitude from 1 to less than 10 multiplied by an integer power of ten; the exponent records how the decimal point is scaled.
What this calculation tells you
Scientific notation separates significant digits from decimal position, making extreme values easier to read and compare. It also makes ambiguous trailing zeros explicit when the intended significant figures are retained.
Moving the decimal left creates a positive exponent; moving it right creates a negative exponent. Zero is a special case because it has no unique normalised nonzero coefficient.
Where it is used
Science and laboratories
Report atomic, astronomical, chemical, and measurement quantities across wide scales.
Engineering
Communicate component values, tolerances, loads, and unit conversions compactly.
Computing and data
Understand exponential display formats and recognise overflow, underflow, or scale mismatches.
Education and public communication
Compare orders of magnitude without long strings of zeros obscuring the relationship.
Common situations
- Normalising a long decimal into coefficient-and-exponent form.
- Expanding scientific notation into an ordinary decimal.
- Comparing values with different exponents.
- Keeping trailing-zero precision unambiguous.
Normalise the coefficient
Shift the decimal until exactly one nonzero digit remains to its left. Count the places moved to determine the exponent and preserve the sign of the original number.
For values between −1 and 1, the exponent is negative because multiplication by a negative power of ten restores the small scale.
Calculate by separating roles
For multiplication, multiply coefficients and add exponents; for division, divide coefficients and subtract exponents, then normalise again. Addition and subtraction first require compatible powers of ten.
Carry extra precision until the reporting step, especially when coefficients nearly cancel.
Distinguish scale from certainty
The exponent indicates decimal scale. The written digits in the coefficient can communicate significant figures, but they do not establish accuracy or uncertainty by themselves.
Engineering notation uses exponents in multiples of three and often pairs naturally with SI prefixes.
Common mistakes
Typical errors include reversing the exponent sign, leaving a coefficient outside the normalised range, and adding exponents during addition instead of aligning them.
- Check by expanding the result.
- Keep units attached.
- Use explicit notation for meaningful trailing zeros.
Practical questions
Frequently asked questions
Why must the coefficient be between 1 and 10?
That convention creates one normalised representation for every nonzero finite value, making comparison straightforward.
Is 45 × 10³ scientific notation?
It is valid power-of-ten notation but not normalised scientific notation; the normalised form is 4.5 × 10⁴.
What does a negative exponent mean?
It represents repeated division by ten, so the magnitude is smaller than the coefficient.