42 Scientific Notation

Scientific Notation is a standardized method for writing very large or very small numbers compactly, expressing any nonzero real number as the product of a value between 1 and 10 and an integer power of ten, using the exponent rules established for integer exponents to support conversion, comparison, and arithmetic without writing out long strings of digits.

The Scope of Scientific Notation

Scientific notation exists to make extremely large numbers, such as the distance to a star, and extremely small numbers, such as the mass of an atomic particle, manageable to write, read, and compute with, replacing a long sequence of digits and zeros with a compact expression consisting of a small decimal factor and a power of ten. Every nonzero real number has exactly one correct scientific notation representation, making the form both compact and unambiguous.

The Normalized Scientific Form

A number is written in normalized scientific notation as a × 10ⁿ, where a is a value satisfying 1 ≤ |a| < 10 (called the coefficient or mantissa), and n is an integer (called the exponent). The restriction that |a| must fall between 1 and 10 is what makes the representation unique: 45 × 10³ and 4.5 × 10⁴ represent the same number, but only the second form satisfies the normalization requirement and is considered correctly written scientific notation.

Converting a Decimal to Scientific Notation

Converting a standard decimal number into scientific notation requires moving the decimal point until exactly one nonzero digit remains to its left, counting the number of places the decimal point was moved, and using that count as the exponent on 10 — positive if the original number was 10 or greater (decimal moved left), negative if the original number was between 0 and 1 (decimal moved right).

Converting Scientific Notation to a Decimal

Converting a number from scientific notation back to standard decimal form requires moving the decimal point in the opposite direction by the number of places indicated by the exponent — to the right for a positive exponent, adding zeros as placeholders if necessary, or to the left for a negative exponent, again adding zeros as placeholders if necessary.

Comparing Numbers in Scientific Notation

Two numbers written in normalized scientific notation are compared primarily by their exponents: the number with the larger exponent is greater, regardless of the coefficients, since a larger power of ten always dominates a coefficient restricted to the range 1 to 10. Only when the two exponents are equal does the comparison fall to the coefficients themselves, compared as ordinary decimals. This exponent-first comparison rule makes scientific notation especially convenient for quickly ordering numbers that differ by many orders of magnitude.

Multiplying and Dividing Numbers in Scientific Notation

Multiplying two numbers in scientific notation is performed by multiplying their coefficients together and adding their exponents together, using the product rule for exponents, then renormalizing the result if the resulting coefficient falls outside the required 1-to-10 range.

Dividing two numbers in scientific notation is performed by dividing their coefficients and subtracting their exponents, using the quotient rule for exponents, again renormalizing if necessary.

Adding and Subtracting Numbers in Scientific Notation

Adding or subtracting numbers in scientific notation requires the two numbers to share the same exponent before their coefficients can be combined directly; if the exponents differ, one number's coefficient and exponent must first be rewritten to match the other's exponent, adjusting the coefficient accordingly, after which the coefficients are added or subtracted and the shared exponent is kept.

Raising Scientific Notation to an Integer Power

Raising a number in scientific notation to an integer power requires raising the coefficient to that power and multiplying the exponent by that power, using the power of a power and power of a product rules for exponents, then renormalizing the resulting coefficient if it falls outside the required range.

Verifying Scientific Notation and Diagnosing Errors

A scientific notation conversion is verified by confirming the coefficient satisfies 1 ≤ |a| < 10 and by mentally reconstructing the original decimal from the normalized form to check that the exponent's sign and magnitude are correct. Common errors in this area include leaving a coefficient outside the required 1-to-10 range without renormalizing, miscounting the number of places the decimal point was moved, applying the wrong sign to the exponent (confusing a large number's positive exponent with a small number's negative exponent), and attempting to add or subtract two scientific notation values without first matching their exponents.