Scientific Notation Calculator
Convert decimal numbers to scientific ($a \times 10^b$) and engineering notation, or execute arithmetic operations ($+$, $-$, $\times$, $\div$) with step-by-step exponent normalization rules.
Calculation Results
2. Add exponents: 5 + (-3) = 2
3. Final normalized form: 7.0 × 102
Scientific & Engineering Notation Rules
Scientific notation is an standard convention in mathematics, astronomy, physics, and chemistry. It allows scientists and engineers to comfortably write and compute with numbers that are either unimaginably large (like the mass of the Sun: $1.989 \times 10^{30} \text{ kg}$) or infinitesimally small (like the mass of an electron: $9.109 \times 10^{-31} \text{ kg}$).
Core Exponent Laws
- Multiplication: $(a \times 10^b) \times (c \times 10^d) = (a \times c) \times 10^{b+d}$. If $(a \times c) \ge 10$, shift the decimal point left and add 1 to the exponent.
- Division: $(a \times 10^b) \div (c \times 10^d) = (a / c) \times 10^{b-d}$. If $(a / c) < 1$, shift the decimal right and subtract 1 from the exponent.
- Addition & Subtraction: To add or subtract, both numbers must first be rewritten so that they share the exact same exponent: $10^{\max(b, d)}$. Then the coefficients are simply added or subtracted.
Scientific & Engineering Notation: Rules of Normalized Powers of 10
Scientific notation is the standardized mathematical format used by scientists, astronomers, and engineers to express astronomical and subatomic numbers concisely without cumbersome strings of trailing or leading zeros. It decomposes numbers into a normalized mantissa (coefficient) multiplied by an integer power of ten.
For deeper analysis and related planning, you can also explore our Area Calculator and Average Calculator.
Notation Rules and Arithmetic Operations
Standard mathematical conventions govern scientific notation:
- Normalized Scientific Form: a × 10b, where 1 ≤ |a| < 10 and b is an integer.
- Engineering Notation: Similar to scientific notation, but exponent 'b' must be an integer multiple of 3 (103, 106, 10-3, etc.) matching SI metric unit prefixes (kilo, mega, micro, nano).
- Multiplication: (a × 10b) × (c × 10d) = (a × c) × 10b + d
- Division: (a × 10b) ÷ (c × 10d) = (a ÷ c) × 10b - d
Frequently Asked Questions
What is scientific notation format?
Scientific notation represents numbers as a × 10^b, where the coefficient (mantissa) 'a' is a real number such that 1 ≤ |a| < 10, and 'b' is an integer exponent.
How does engineering notation differ from scientific notation?
In engineering notation, the exponent 'b' must be a multiple of 3 (e.g. 10^3, 10^6, 10^-3), and the coefficient 'a' is between 1 and 1000. This corresponds directly to metric SI prefixes such as kilo, mega, micro, and nano.
How do you multiply two numbers in scientific notation?
To multiply (a × 10^b) × (c × 10^d), multiply the coefficients (a × c) and add the exponents (b + d). Then normalize the result so the coefficient is between 1 and 10.