Perform arithmetic operations on scientific notation and convert numbers into standard scientific, engineering, and E-notations.
Coefficients are multiplied; exponents of 10 are added.
How to express, multiply, divide, and convert extremely large and small real numbers using powers of 10
Multiply coefficients and add exponents: (a₁ × 10^{b₁}) × (a₂ × 10^{b₂}) = (a₁ · a₂) × 10^{b₁ + b₂}. If the product exceeds 10, renormalize by shifting the decimal left.
Divide coefficients and subtract exponents: (a₁ × 10^{b₁}) ÷ (a₂ × 10^{b₂}) = (a₁ / a₂) × 10^{b₁ - b₂}. If quotient is below 1, shift the decimal right.
Exponents must be converted to match the same power of 10 before adding or subtracting mantissas: (a₁ × 10^b) + (a₂ × 10^b) = (a₁ + a₂) × 10^b.
Moving decimal LEFT increases the exponent by 1 per step (e.g., 45,000 = 4.5 × 10⁴). Moving decimal RIGHT decreases the exponent by 1 per step (e.g., 0.0072 = 7.2 × 10⁻³).
Multiply (3.0 × 10⁸) by (2.0 × 10⁴)
Scientific notation restricts the coefficient so there is exactly one non-zero digit before the decimal point (1 <= |a| < 10) with any integer exponent. Engineering notation allows coefficients up to 1000 and restricts exponents to multiples of 3, corresponding to SI metric units (kilo, mega, milli, micro).
It eliminates cumbersome strings of leading or trailing zeros, clarifies significant figures without ambiguity, and prevents arithmetic overflow or underflow in calculations.
E-notation is shorthand for "times 10 to the power of". For example, 4.5e+8 means 4.5 × 10⁸, and 1.2e-4 means 1.2 × 10⁻⁴.
A negative exponent represents division by powers of 10: 10⁻ⁿ = 1 / 10ⁿ. For example, 3.5 × 10⁻³ = 3.5 / 1000 = 0.0035.
Understanding your calculation
Scientific notation writes a nonzero number as a × 10^n, conventionally with 1 ≤ |a| < 10. Moving the decimal point changes the exponent.
0.00045 = 4.5 × 10^−4, and 6.2 × 10³ = 6200.
The exponent is a power of ten, not a count of significant figures. Displayed digits do not establish measurement accuracy.