1.6 Scientific Notation

Definition

Scientific notation is a standard way of writing very large or very small numbers using powers of ten.

It simplifies calculations, improves readability, and reduces writing errors.

General Form

Any number written in scientific notation has the form

a×10na \times 10^{n}

where

  • a is the coefficient and satisfies 1≤a<10
  • n is an integer (positive, negative, or zero).

Why Scientific Notation is Used?

Many physical quantities have values that are either extremely large or extremely small.

For example,

  • the distance between stars is enormous,
  • the diameter of an atom is extremely small.

Writing these numbers in ordinary decimal form is inconvenient and increases the chance of mistakes. Scientific notation provides a compact and readable representation.

Positive Powers of Ten

To write a large number in scientific notation, move the decimal point to obtain a coefficient between 1 and 10. The number of places moved becomes the positive exponent of ten.

Ordinary NumberScientific Notation
30003.0 × 10³
580005.8 × 10⁴
72000007.2 × 10⁶
Table 1.6.1 — Positive Powers of Ten. Examples of large numbers expressed in scientific notation.

Negative Powers of Ten

To write a small number in scientific notation, move the decimal point to obtain a coefficient between 1 and 10. The number of places moved becomes the negative exponent of ten.

Ordinary NumberScientific Notation
0.44.0 × 10⁻¹
0.00323.2 × 10⁻³
0.000000818.1 × 10⁻⁷
Table 1.6.2 — Negative Powers of Ten. Examples of small numbers expressed in scientific notation.

Converting to Scientific Notation

Example 1

Write 68000 in scientific notation.

Move the decimal point four places to the left.

68000=6.8×10468000 = 6.8 \times 10^{4}

Example 2

Write 0.00045 in scientific notation.

Move the decimal point four places to the right.

0.00045=4.5×10−40.00045 = 4.5 \times 10^{-4}

Converting Back to Decimal Form

Example 1

2.5×103=25002.5 \times 10^{3} = 2500

Example 2

7.9×10−5=0.0000797.9 \times 10^{-5} = 0.000079

Scientific Notation and Prefixes

Scientific notation is closely related to SI prefixes. For example,

Scientific NotationSI Prefix
10³kilo (k)
10⁻³milli (m)
10⁻⁶micro (µ)
10⁻⁹nano (n)
Table 1.6.3 — Scientific Notation and SI Prefixes. Examples of commonly used powers of ten and their corresponding SI prefixes.

However, scientific notation can represent any power of ten, whereas SI prefixes exist only for selected powers.

Key Points

  • Scientific notation expresses numbers in the form a × 10ⁿ
  • The coefficient always satisfies 1 ≤ a < 10.
  • Positive exponents represent large numbers.
  • Negative exponents represent small numbers.
  • Scientific notation simplifies calculations and improves readability.
  • SI prefixes are names assigned to some commonly used powers of ten.

← 1.5 SI Prefixes

1.7 Significant Figures →