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Scientific Notation

Convert Decimals & Large Numbers
#
OR
× 10^
Scientific Notation
0 × 100
Decimal Form 0
E-Notation 0e0
Engineering Notation 0 × 100

Why Do We Need Scientific Notation?

In science and engineering, we often deal with numbers that are impossibly huge or microscopically small. Writing them out in standard decimal form is not just tedious; it is prone to errors.

For example, the distance from Earth to the Sun is about 149,600,000,000 meters. Writing all those zeros takes time and space. The mass of an electron is approximately 0.00000000000000000000000000000091 kg. Try typing that into a standard calculator without missing a zero!

Scientific Notation simplifies these numbers into a compact format that is easy to read, write, and compute.

The Formula Structure

Scientific notation is always written in the form:

a × 10n

This formula has two parts:

  • The Coefficient (a): A decimal number that is greater than or equal to 1, but less than 10. (e.g., 1.5, 4.23, 9.99).
  • The Exponent (n): An integer (whole number) that tells you how many places the decimal point moved.
The Decimal Point Rule:

Positive Exponent: If you move the decimal to the LEFT, the exponent goes up. This is for big numbers.
Example: 5,000 → Move decimal 3 spots left → 5 × 103

Negative Exponent: If you move the decimal to the RIGHT, the exponent goes down (becomes negative). This is for small decimal numbers.
Example: 0.005 → Move decimal 3 spots right → 5 × 10-3

Real World Examples

Measurement Standard Form Scientific Notation
Speed of Light 300,000,000 m/s 3 × 108 m/s
Avogadro's Number 602,200,000,000,000,000,000,000 6.022 × 1023
Width of Human Hair 0.00005 meters 5 × 10-5 m
One Googol 1 followed by 100 zeros 1 × 10100

What is E-Notation?

Computers and older calculators often cannot display the superscript format ($10^n$). Instead, they use E-Notation (Exponential Notation).

In this format, the "× 10" is replaced by the letter "E" (or "e").

  • 5 × 103 becomes 5E3 or 5e3.
  • 1.2 × 10-5 becomes 1.2E-5.

If you see this on your calculator screen, it doesn't mean "Error"—it just means the number is too big or too small to display normally.

Scientific vs. Engineering Notation

Our calculator also provides Engineering Notation. While similar, there is one strict rule for engineering notation: the exponent must be a multiple of 3 (e.g., 3, 6, 9, -3, -6).

Why multiples of 3? Because they align perfectly with SI Metric Prefixes:

  • 103 = Kilo (Thousands)
  • 106 = Mega (Millions)
  • 109 = Giga (Billions)
  • 10-3 = Milli (Thousandths)
  • 10-6 = Micro (Millionths)

Example: Convert 40,000.

  • Scientific: 4 × 104 (Correct coefficient between 1-10)
  • Engineering: 40 × 103 (Matches "40 Kilo")

How to Perform Math with Scientific Notation

You can use these numbers in multiplication and division without converting them back to decimals first. Here are the shortcut rules:

1. Multiplication

Multiply the coefficients, and add the exponents.

(a × 10n) · (b × 10m) = (a · b) × 10n+m

Example: (2 × 103) · (3 × 105) = 6 × 108

2. Division

Divide the coefficients, and subtract the exponents.

(a × 10n) / (b × 10m) = (a / b) × 10n-m

Example: (6 × 108) / (2 × 105) = 3 × 103

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