What the calculator is trying to do

Scientific notation is a neat way to write very big or very small numbers without drowning in zeros. It uses the format a × 10^b, where:

• a (the mantissa) is a number with one non-zero digit before the decimal point, usually 1 ≤ |a| < 10.
• b (the exponent) says how many times to move the decimal point.

The calculator’s job is to take whatever number style you give it—plain decimal like 123000, scientific like 1.23 × 10^5, or E-notation like 1.23e5—and turn it into clean, normalized forms. It also offers engineering notation (exponents that are multiples of 3), optional rounding, and a standard decimal “expanded” view.

What you can enter

You can paste or type:

• A normal number: 0.0000123 or 123456789
• Scientific with a multiplication sign: 3 × 10^-4 or 6.02 × 10^23
• E-notation: 4.56e7 or -9.81E-3
• Even 10^b forms like 10^6

The calculator cleans up spaces, commas, and even the unicode “×” and “−” so you don’t have to be picky.

How it “normalizes” your number

1. Under the hood, it finds the mantissa and exponent.
   It figures out where the first non-zero digit sits, then counts how far the decimal point needs to shift to land the mantissa between 1 and 10 (unless the value is 0). That shift becomes the exponent b.
2. Scientific vs. engineering notation.
   
   • Scientific: mantissa has one digit before the decimal (1 to 9), exponent b can be any integer.
   • Engineering: the exponent is forced to a multiple of 3 (… −9, −6, −3, 0, 3, 6, 9 …). To make that work, the mantissa may show up as something like 470 × 10^3 instead of 4.7 × 10^5—it’s the same value, but grouped by thousands, millions, etc. This pairs nicely with SI prefixes (kilo, mega, micro…).
3. Rounding the mantissa (optional).
   You get two choices:
   • Significant figures (sig figs): “Keep this many meaningful digits overall.” Great for scientific work where precision counts, e.g., 3 sig figs turns 1.23456 into 1.23.
   • Decimal places: “Keep this many digits after the decimal point.” Useful when you want a tidy mantissa like 1.234 (3 decimal places).
     After rounding, if the mantissa accidentally becomes 10.000… (that can happen!), the calculator nudges it back to 1.0 and increases the exponent by 1. This preserves the same value.
4. Formatting choices.
   You can choose to see the main result as a × 10^b or E-notation (aEb), pick uppercase/lowercase E, and turn thousands separators on/off for readability.

What the outputs mean

• Scientific notation: the normalized a × 10^b form.
• E-notation: the same number as aEb (e.g., 1.23E+5).
• Engineering notation: exponent is a multiple of 3; handy for SI prefixes and “human scale” groupings.
• Mantissa (a) & Exponent (b): split out so you can copy them separately.
• Standard decimal (expanded): shows the full number (or switches to E-notation if it would be ridiculously long).
• Some versions also show order of magnitude (basically the exponent in scientific form) and SI group name/prefix (e.g., thousand/kilo, million/mega).

Friendly examples

Example 1: A big number
Input: 123456789

• Scientific: 1.2346 × 10^8 (rounded to 5 sig figs, say)
• E-notation: 1.2346E+8
• Engineering: 123.46 × 10^6 (or 123.46 × 10^6), which lines up with “millions/mega”
• Mantissa: 1.2346, Exponent: 8
• Expanded: 123,456,789

Example 2: A small number
Input: 0.0000123

• Scientific: 1.23 × 10^-5
• E-notation: 1.23E-5
• Engineering: 12.3 × 10^-6 (micro range)
• Expanded: 0.0000123

Example 3: Using E-notation directly
Input: -9.81E-3

• Scientific: -9.81 × 10^-3 (already normalized since 9.81 is between 1 and 10)
• Engineering: -9.81 × 10^-3 (still fine; −3 is a multiple of 3)
• Expanded: -0.00981

Example 4: Rounding choices
Suppose the unrounded scientific form is 1.234567 × 10^5.

• 3 sig figs → 1.23 × 10^5
• 4 decimal places on the mantissa → 1.2346 × 10^5
  If rounding produced 9.9999 → 10.000, the tool renormalizes to 1.0000 × 10^6.

Edge cases it handles

• Zero: always shows as 0 with exponent 0 (there’s no unique scientific form for zero).
• Very large or tiny values: the “expanded” decimal may switch to E-notation so your page doesn’t explode with thousands of characters.

Negative numbers: the sign simply sits on the mantissa (e.g., −1.2 × 10^3).