By Chris Deziel
Updated August 30, 2022
Farknot_Architect/iStock/GettyImages
NASA reports that the distance from Earth to the nearest star is 40,208,000,000,000 km. Such colossal figures are unwieldy for manual calculation, which is why scientists express them in scientific notation— a decimal multiplied by a power of ten. For example, the star‑distance becomes 4.02 × 1013 km.
To convert any number to scientific notation: 1) Move the decimal so that only the first non‑zero digit remains to its left; 2) Count how many places you moved the decimal to obtain the exponent; 3) If the original number is less than 1, the exponent is negative. Round the decimal to two significant figures unless more precision is required.
When writing large integers, it is conventional to separate digits into groups of three with commas (e.g., 10,835,921). The first three digits of the integer always form the coefficient in scientific notation, regardless of whether the first group contains one or two digits.
Very large numbers receive a positive exponent equal to the number of digits after the decimal point once it is positioned after the first digit. Very small numbers (less than 1) receive a negative exponent equal to the number of leading zeros plus one. For instance:
Numbers can be added or subtracted directly only when they share the same exponent. If exponents differ, adjust one number to match the other’s exponent before operating.
Example 1: 3.45 × 1010 + 2.75 × 108
Example 2: 4.00 × 1012 + 7.55 × 1012 = 11.55 × 1012 = 1.16 × 1013.
When multiplying, multiply the coefficients and add the exponents. When dividing, divide the coefficients and subtract the exponents.
Example 1: 3.25 × 108 × 1.42 × 104 = 4.62 × 1012.
Example 2: 3.25 × 108 ÷ 1.42 × 104 = 2.29 × 104.
Mastering scientific notation simplifies complex calculations and enhances precision across scientific disciplines.
