What is this in scientific notation?

How it works

A number in scientific notation is a × 10ⁿ, where the coefficient a has exactly one non-zero digit before the decimal point (1 ≤ |a| is less than 10) and the exponent n is a whole number.

To convert a number:

1. Find its significant digits (the rules are below).
2. Put the decimal point after the first of them. That is a.
3. n is how many places the decimal point moved: positive when it moved left (a large number), negative when it moved right (a small number).

The calculator also gives:

• E notation: a e n, the same number as calculators and spreadsheets write it (4.5e-4).
• Engineering notation: the power is the multiple of 3 at or below n, so 1 to 3 digits come before the point (123.4 × 10³, 450 × 10⁻⁶).
• Decimal notation: the number written out, for n from −30 to 30.

Significant figures

• Every non-zero digit counts.
• Zeros between non-zero digits count (1,005 has 4).
• Leading zeros never count (0.00045 has 2).
• Trailing zeros count after a decimal point (0.0010 has 2, 1.200 × 10³ has 4).
• Trailing zeros of a whole number written without a decimal point do not count (123,400 has 4, 1200 has 2). Type a point at the end, 1200., to count them (4).

Rounding to significant figures

Keep the first N significant digits and look at the digits dropped:

• more than half of a unit in the last kept place: round up (2.451 → 2.5)
• less than half: round down (2.449 → 2.4)
• exactly half: round to the even digit (2.45 → 2.4 and 2.35 → 2.4)

A carry can add a digit (9.96 to 2 figures is 10, written 1.0 × 10¹). Asking for more figures than the number has adds zeros (1.2 to 4 figures is 1.200).

Assumptions

• Digits are kept exactly as typed. Nothing passes through binary floating point, so 0.1 stays 0.1 and very long numbers keep every digit.
• Commas, underscores, and white space (spaces, tabs, no-break spaces) are removed before reading, wherever they are: 123,400, 1_000, and 1 000 are all read. A minus sign may be a hyphen, the minus sign −, or an en dash –, and a plus sign is allowed.
• Accepted forms are decimals (0.00045, 123,400, −5.2, .5, 5.), E notation with e or E and an optional sign (4.5e-4, 1.2E+5), and times-ten notation: ×, x, X, *, or · then 10, then ^ or ** and the power (4.5 × 10^-4, 4.5x10^-4, 4.5*10**-4, 4.5·10^3), or 10 with a superscript power (4.5 × 10⁻⁴). The power is a whole number of 1 to 9 digits. "4.5 × 104" is not read as a power of ten.
• Anything else has no answer, including text with no digits: "," alone, "abc", or "4.5e". An empty box (or only spaces) counts as 0.
• Zero is written as 0 in every form.

How the results are written

• Scientific and engineering notation: the coefficient, then " × 10" and the exponent in superscript digits (1.234 × 10⁵, 4.5 × 10⁻⁴). The coefficient keeps exactly the significant figures, so trailing zeros stay (3.00 × 10⁸).
• E notation: the same coefficient, a lower-case e, and the exponent with no plus sign (1.234e5, 4.5e-4). A negative number starts with a plain hyphen (-1.0e-3), so it pastes into spreadsheets and code.
• Scientific, engineering, and decimal notation start a negative number with the minus sign − (−1.0 × 10⁻³).
• Decimal notation groups the whole-number part in threes with commas (123,400) and keeps the significant trailing zeros after the point (0.0010).
• Zero gives 0 × 10⁰, 0e0, 0 × 10⁰, and 0, with exponent 0 and no count of significant figures.

Worked examples by hand

123,400. The significant digits are 1234 (the two trailing zeros of a whole number do not count), so 4 figures. The point moves 5 places left: 1.234 × 10⁵, or 1.234e5. The multiple of 3 at or below 5 is 3, so the engineering form is 123.4 × 10³.

37,000 (OpenStax Prealgebra 2e, section 10.5). The point moves 4 places left: 3.7 × 10⁴.

0.00045. The leading zeros do not count, so the digits are 45 (2 figures). The point moves 4 places right: 4.5 × 10⁻⁴, or 4.5e-4. The multiple of 3 at or below −4 is −6, so the engineering form is 450 × 10⁻⁶.

299,792,458 to 3 figures (the speed of light in m/s). Keep 299 and drop 792,458. That is more than half, so round up: 300, or 3.00 × 10⁸.

6.02214076e23 (the Avogadro constant). This is 6.02214076 × 10²³, with 9 figures. Written out: 602,214,076,000,000,000,000,000. Engineering: 602.214076 × 10²¹.

9.1093837139 × 10^-31 to 4 figures (the electron mass in kg). Keep 9109 and drop 3837139, which is less than half: 9.109 × 10⁻³¹.

2.45 to 2 figures. The dropped digit is exactly 5 with nothing after it, so round to the even digit: 2.4, or 2.4 × 10⁰.

−0.0010. The trailing zero after the point counts, so the digits are 10 (2 figures): −1.0 × 10⁻³.