Convert numbers to and from scientific notation.
N = a × 10ⁿ
To express any non-zero number N in scientific notation, move the decimal point so that exactly one non-zero digit sits to the left of it. The number of places you moved the decimal becomes the exponent n on the base 10. Moving left gives a positive exponent; moving right gives a negative exponent.
Standard form: N = a × 10ⁿ
Where 1 ≤ |a| < 10 and n is an integer.
Move the decimal point 4 places to the right to get a coefficient of 6.02 (since 1 ≤ 6.02 < 10). Because we moved the decimal RIGHT, the exponent is negative: n = −4. Therefore: 0.000602 = 6.02 × 10⁻⁴
Result: 6.02 × 10⁻⁴
The result 6.02 × 10⁻⁴ means that 0.000602 equals 6.02 divided by 10,000. The coefficient 6.02 is between 1 and 10 (satisfying the scientific notation rule), and the exponent −4 tells you the original number is a small decimal. To verify: 6.02 × 10⁻⁴ = 6.02 ÷ 10,000 = 0.000602 ✓
Scientific notation is a standardized way of writing very large or very small numbers using powers of ten. Instead of writing 93,000,000 miles (the average Earth-Sun distance), scientists write 9.3 × 10⁷ miles. Instead of 0.00000000167 kg (the mass of a proton in atomic mass context), physicists write 1.67 × 10⁻²⁷ kg.
Multiplication: (a × 10ᵐ) × (b × 10ⁿ) = (a × b) × 10^(m+n)
Example: (3 × 10⁴) × (2 × 10³) = 6 × 10⁷
Division: (a × 10ᵐ) ÷ (b × 10ⁿ) = (a ÷ b) × 10^(m−n)
Example: (8 × 10⁶) ÷ (4 × 10²) = 2 × 10⁴
In calculators and programming languages, scientific notation is often written as 6.02E-4 or 6.02e-4, where 'E' or 'e' stands for '× 10^'. This calculator accepts and outputs both formats.
1,000,000 = 1 × 10⁶. Move the decimal point 6 places to the left to obtain the coefficient 1, so the exponent is +6.
0.00045 = 4.5 × 10⁻⁴. Move the decimal 4 places to the right to get coefficient 4.5, giving exponent −4.
To add numbers in scientific notation, first rewrite both numbers so they share the same exponent, then add the coefficients. Example: (3.0 × 10⁴) + (2.0 × 10³) = (3.0 × 10⁴) + (0.2 × 10⁴) = 3.2 × 10⁴.
Avogadro's number is approximately 6.022 × 10²³ mol⁻¹. This is a classic example of scientific notation used for an extremely large quantity.
In most countries outside the US, 'standard form' IS the term used for scientific notation (a × 10ⁿ with 1 ≤ |a| < 10). In the US, 'standard form' typically refers to the plain decimal representation. This calculator handles both directions of conversion.
The coefficient (a) must satisfy 1 ≤ |a| < 10. This means the absolute value of a is at least 1 but strictly less than 10 — exactly one non-zero digit appears before the decimal point.
Multiply the coefficient by the power of ten. For a positive exponent n, move the decimal point n places to the right. For a negative exponent, move it |n| places to the left, filling with zeros as needed. Example: 3.7 × 10⁵ = 370,000.
Yes. Negative numbers follow the same rule. For example, −4,500 in scientific notation is −4.5 × 10³. The negative sign applies to the coefficient, not the exponent.
'E notation' (e.g., 1.23E6) is shorthand for scientific notation used by calculators and computers because they cannot display superscripts. '1.23E6' means 1.23 × 10⁶ = 1,230,000.
Use as many significant figures as your original measurement or data has. Scientific notation does not change precision — it only changes the way the number is written. If your number has 4 significant figures, your coefficient should too.
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