Convert any decimal number into scientific notation (a × 10ⁿ), or back.
A number is expressed as a × 10n where 1 ≤ |a| < 10. We find n by counting decimal places moved.
Scientific notation writes any number as a value between 1 and 10, multiplied by a power of 10 — 6.022 × 10²³ instead of writing out Avogadro's number as 602,200,000,000,000,000,000,000. The format exists for a genuinely practical reason: once numbers get astronomically large (distances between stars) or vanishingly small (the mass of an atom), writing out every digit and zero becomes both impractical to read and dangerously easy to miscount — losing or adding a single zero in a string of twenty changes the value by a factor of 10, and that kind of error is far easier to make (and far harder to catch) in expanded form than in scientific notation, where the exponent makes the magnitude explicit at a glance.
Scientific notation also interacts directly with significant figures, which is why the two concepts are usually taught together: writing a measured value as 6.02 × 10²³ versus 6.022 × 10²³ makes an explicit, unambiguous statement about measurement precision (three significant figures versus four) in a way that's genuinely difficult to convey clearly in standard decimal notation, especially for numbers containing trailing zeros where it's often unclear whether those zeros are meaningful measured digits or just placeholders. This is exactly why scientific and engineering fields default to scientific notation for very large or very small quantities — it's not a stylistic preference, it's a format that makes both the magnitude and the precision of a number explicit and unambiguous at the same time.
Scientific notation expresses a number as a value between 1 and 10, multiplied by a power of 10. For example, Avogadro's number (602,200,000,000,000,000,000,000) is written as 6.022 × 10²³ — far more compact and easier to read than the fully expanded form.
Because very large or very small numbers become impractical to read and dangerously easy to miscount when fully written out — losing or adding a single zero in a long string changes the value by a factor of 10, an error that's much easier to make in expanded decimal form. Scientific notation makes the magnitude explicit through the exponent, reducing that risk.
Scientific notation makes the number of significant figures in a measurement explicit and unambiguous — 6.02 × 10²³ clearly shows three significant figures, while 6.022 × 10²³ shows four. This is harder to convey clearly in standard decimal notation, especially with trailing zeros where it's often unclear whether they're meaningful measured digits or just placeholders.
Move the decimal point until only one non-zero digit remains before it, then count how many places you moved it — that count becomes the exponent on 10 (positive if you moved left for a large number, negative if you moved right for a small number). For example, 45,000 becomes 4.5 × 10⁴, and 0.00032 becomes 3.2 × 10⁻⁴.