A worked example: 250,000 in scientific notation
Feed 250000 into the scientific notation calculator with the "Decimal to Scientific notation" direction, and the exact result is 2.5 × 10^5. The mantissa, 2.5, sits between 1 and 10 as required, and the exponent, 5, records how many places the decimal point moved to get there — from 250000. down to 2.5.
The formula: mantissa times a power of ten
Any nonzero number can be written as a × 10^b, where a is the mantissa (1 ≤ |a| < 10) and b is an integer exponent. For 250000, that means finding the largest power of ten it divides into within that range: 250000 / 10^5 = 2.5, so a = 2.5 and b = 5, giving 2.5 × 10^5 — the same value the calculator returned above, computed with arbitrary-precision decimal arithmetic so there's no rounding drift even at extreme magnitudes.
A worked example: 0.000047 in scientific notation
Small numbers work the same way, just with a negative exponent. Entering 0.000047 with the same "Decimal to Scientific notation" direction returns 4.7 × 10^-5 — the decimal point moved 5 places to the right to turn 0.000047 into 4.7, so the exponent is -5 rather than +5. The sign of the exponent alone tells you whether the original number was larger or smaller than 1.
Why the direction toggle matters
The calculator also runs the conversion in reverse. Entering 6.022e23 (Avogadro's number, in the standard "e" notation) with the "Scientific notation to Decimal" direction expands it to the full integer 602200000000000000000000 — every one of the 24 digits, computed exactly rather than approximated. That reversibility matters whenever a number arrives in one form but needs to be read, compared, or plugged into another calculation in the other form, such as a huge exact integer that a big-number calculation just produced.