What is Scientific Notation?
Scientific notation is a compact way of writing very large or very small numbers. Any number is expressed as a coefficient multiplied by a power of ten: a × 10n. The coefficient a has exactly one non-zero digit before the decimal point (so 1 is less than or equal to |a| which is less than 10). The exponent n is an integer that can be positive, negative, or zero.
Scientific notation is standard in science, engineering, astronomy, chemistry, and computing because it makes very large and very small numbers much easier to read and compare. For example, the distance from the Earth to the Sun (approximately 150,000,000,000 metres) is written as 1.5 × 1011 m, and the diameter of a hydrogen atom (approximately 0.0000000001 metres) is written as 1 × 10-10 m.
How to Convert Scientific Notation to Decimal
To convert a × 10n to a decimal number:
- Write out the coefficient a.
- If n is positive, move the decimal point n places to the right (towards larger numbers), filling in zeros where needed.
- If n is negative, move the decimal point |n| places to the left (towards smaller numbers), filling in leading zeros where needed.
- If n is zero, the number equals the coefficient exactly (since 100 = 1).
How to Convert Decimal to Scientific Notation
To convert a decimal number to scientific notation:
- Move the decimal point until there is exactly one non-zero digit to its left.
- Count the number of places you moved the decimal point. This is the exponent n.
- If you moved the decimal left, n is positive. If you moved it right, n is negative.
- Write the result as a × 10n.
Examples
| Scientific Notation | Decimal Notation | Description |
|---|
| 2.5 × 104 | 25,000 | Move decimal 4 places right |
| 1.0 × 100 | 1 | Any number to the power zero is 1 |
| 6.022 × 1023 | 602,200,000,000,000,000,000,000 | Avogadro's number |
| 3.0 × 108 | 300,000,000 | Speed of light (m/s, approx) |
| 1.6 × 10-19 | 0.00000000000000000016 | Charge of an electron (C) |
| 9.8 × 10-3 | 0.0098 | Move decimal 3 places left |
| 4.8 × 104 | 48,000 | Move decimal 4 places right |
Common Uses
Scientific notation is used whenever numbers span many orders of magnitude:
- Chemistry: Avogadro's number (6.022 × 1023), molar masses, and concentrations expressed in scientific notation.
- Physics: Distances in astronomy (light years, parsecs), sizes of atomic particles, and physical constants.
- Computing: Floating-point numbers in computers use a form of binary scientific notation. Very small or very large values in programming are often expressed as 1.5e9 (meaning 1.5 × 109).
- Finance: National GDPs, global debt, and market capitalisation figures are sometimes expressed in scientific notation in academic work.
- Engineering: Tolerances, electrical resistances, and frequencies often involve very small or very large values.
Normalised vs Non-Normalised Scientific Notation
Standard (normalised) scientific notation requires the coefficient to satisfy 1 ≤ |a| < 10. For example, 25,000 in normalised scientific notation is 2.5 × 104, not 25 × 103. This converter uses normalised form when converting decimal to scientific notation. When converting scientific to decimal, any coefficient value is accepted (including non-standard forms such as 25 × 103).
Related Calculators
Method: Conversion formula: decimal = a × 10n. For scientific to decimal, the result is computed as coefficient multiplied by Math.pow(10, exponent). For decimal to scientific, the exponent is floor(log10(|x|)) and the coefficient is x divided by 10^exponent. Large integer results are formatted without floating-point rounding errors by using string manipulation for whole-number exponents.
This converter handles numbers within JavaScript's safe integer range accurately. For very large exponents (above 20) the decimal result is displayed in exponential form, as writing out hundreds of digits is not practical. For scientific work, always verify results using your institution's preferred tool or a scientific calculator.
