Scientific notation is a writing system for numbers that are too large or too small to be convenient in ordinary form. It writes every number as a mantissa — a number between 1 and 10 — multiplied by a power of 10. The distance to the nearest star, 40,000,000,000,000 kilometres, becomes 4 × 10¹³. The mass of a grain of sand, 0.00000005 kilograms, becomes 5 × 10⁻⁸. The notation does two jobs at once: it keeps the arithmetic manageable, and it makes the precision of a measurement visible. This guide covers converting in both directions, the arithmetic rules, E notation on calculators, and the mistakes that cost marks in exams.

Converting to scientific notation

The conversion is a decimal-point move. For a large number, move the decimal point left until one non-zero digit remains before it, and count the moves: that count is the exponent. 6,200,000 becomes 6.2 when the point moves six places left, so it is 6.2 × 10⁶. 93,000,000 becomes 9.3 × 10⁷. For a number already between 1 and 10, the exponent is 0: 7.4 is 7.4 × 10⁰, though nobody writes it that way outside a textbook exercise.

For a small number, move the decimal point right until one non-zero digit sits before it, and count the moves: the exponent is the negative of that count. 0.00045 becomes 4.5 after four moves right, so it is 4.5 × 10⁻⁴. 0.00000032 is 3.2 × 10⁻⁷. The sign of the exponent is the whole story: positive means the original number was large, negative means it was a fraction of one. A frequent error is to count the zeros rather than the moves — for 0.00045 there are three zeros before the 4, but the exponent is −4, because the point moved four places.

Zero is the special case: 0 has no non-zero digit, so it has no normalised scientific notation. By convention it is written as 0, or as 0 × 10⁰ in contexts that require the form. Negative numbers keep their sign on the mantissa: −52,000 is −5.2 × 10⁴.

Converting back to ordinary form

To convert back, move the decimal point the other way. A positive exponent moves the point right: 3.06 × 10⁵ becomes 306,000 (five moves, filling with zeros). A negative exponent moves it left: 7.8 × 10⁻³ becomes 0.0078 (three moves, with zeros filling the empty places). The mantissa’s digits never change; only their place value does, which is why the method is reliable even for numbers with many digits.

Worked pair, checked both ways: 0.0000912 converts to 9.12 × 10⁻⁵ (five moves right), and 9.12 × 10⁻⁵ converts back to 0.0000912 (five moves left). If a conversion loses or gains a digit, the decimal point moved the wrong number of places — recount the moves, out loud if necessary, because this is the one error that survives checking by eye.

Arithmetic: multiply, divide, add, and subtract

Multiplication is where the notation earns its keep. Multiply the mantissas, add the exponents, then re-normalise: (2 × 10³) × (3 × 10⁴) = 6 × 10⁷. If the mantissa product reaches 10 or more, shift it: (4 × 10⁵) × (5 × 10²) = 20 × 10⁷, which normalises to 2.0 × 10⁸. Division is the mirror image: divide the mantissas, subtract the exponents: (8 × 10⁶) ÷ (2 × 10³) = 4 × 10³. If the mantissa quotient falls below 1, shift it the other way: (3 × 10⁴) ÷ (6 × 10²) = 0.5 × 10² = 5.0 × 10¹.

Addition and subtraction require a shared exponent first. To add 3 × 10⁵ and 2 × 10⁴, rewrite the smaller exponent to match: 2 × 10⁴ = 0.2 × 10⁵, so the sum is 3.2 × 10⁵. The common error is adding numbers with different exponents directly — 3 × 10⁵ + 2 × 10⁴ is not 5 × 10⁵ or 5 × 10⁹; it is 320,000. Powers and roots follow the same logic: (2 × 10³)² = 4 × 10⁶, squaring both parts; the square root of 4 × 10⁶ is 2 × 10³.

E notation and calculators

Calculators and software write scientific notation with E: 1.23E+06 means 1.23 × 10⁶, and 4.5E−06 means 4.5 × 10⁻⁶. The E is a marker, not a value. On a scientific calculator, enter the mantissa, press EE or EXP, then the exponent — on many models the exponent key also handles the sign, so you press it once for a negative exponent rather than typing a minus. On a phone calculator, the EE key may hide behind a function panel; in spreadsheets, typing 1.23E+06 stores 1,230,000, and choosing scientific number format displays any number this way.

Two entry traps catch students regularly. First, typing the letter E from a keyboard into a calculator app usually produces an error or the constant e (2.718…), not exponent entry — use the dedicated key. Second, entering the exponent without its sign: 4.5E6 is 4,500,000, while 4.5E−6 is 0.0000045, a billion times smaller. When an answer looks wildly wrong, the exponent sign is the first thing to check.

Engineering notation, and where the notation is used

Engineering notation is a close cousin: the exponent is always a multiple of 3, so the mantissa ranges from 1 to 1,000 instead of 1 to 10. 12,300 becomes 12.3 × 10³, and 0.000045 becomes 45 × 10⁻⁶. The advantage is that every exponent corresponds to a metric prefix — 10³ is kilo, 10⁶ is mega, 10⁻³ is milli, 10⁻⁶ is micro — so engineers can read a value and its unit prefix together. Scientific notation remains the standard in pure science and in school mathematics; engineering notation dominates in electronics and measurement.

The notation appears wherever magnitudes span many orders: astronomy (distances in kilometres, masses in kilograms), chemistry (Avogadro’s number, 6.022 × 10²³), physics (the charge of an electron, 1.602 × 10⁻¹⁹ coulombs; the speed of light, 2.998 × 10⁸ metres per second), computing (storage and data rates), and finance (national debts and market capitalisations). In each case the notation is doing the same job: keeping the significant digits visible while the zeros are handled by the exponent.

Frequently asked questions

How do I convert 0.00045 to scientific notation?

Move the decimal point four places right to get 4.5, so the number is 4.5 × 10^−4. The exponent is negative because the original number is less than 1, and its size is the number of places moved, not the number of zeros.

How do I convert 6,200,000 to scientific notation?

Move the decimal point six places left to get 6.2, so the number is 6.2 × 10^6. Check: 6.2 × 1,000,000 = 6,200,000.

What is 2 × 10^3 times 3 × 10^4?

6 × 10^7. Multiply the mantissas (2 × 3 = 6) and add the exponents (3 + 4 = 7). Then check the mantissa is between 1 and 10 — here 6 is, so no re-normalising is needed.

What does 1.23E+06 mean?

1.23 × 10^6, or 1,230,000. E notation is how calculators display scientific notation: the number after E is the power of 10. 1.23E−06 would be 0.00000123.

Why is scientific notation used in science?

Because it handles very large and very small numbers compactly, makes multiplication and division easy through exponent arithmetic, and shows significant figures unambiguously — the digits in the mantissa are the significant digits.

Can scientific notation show negative numbers?

Yes. The sign sits on the mantissa: −52,000 is −5.2 × 10^4. The exponent’s sign is separate: a negative exponent means a small positive or negative number, not a negative number. −4.5 × 10^−4 is −0.00045.