42.5 Scientific Notation Comparison

Scientific Notation Comparison is the process of determining which of two or more numbers written in scientific notation is greater or lesser, by first comparing their signs, then their exponents, and finally their coefficients whenever the exponents match.

Scientific Number Sign Comparison

Procedure

The first comparison made between two numbers in scientific notation is their overall sign: any positive number is automatically greater than any negative number, regardless of the specific coefficients or exponents involved.

Example

Positive Exponent Magnitude Comparison

Procedure

Among positive numbers, once the exponents are examined, the number with the larger exponent represents the greater value, since a larger exponent indicates a greater number of factors of ten.

Example

Negative Exponent Magnitude Comparison

Procedure

Among positive numbers with negative exponents, the number whose exponent is closer to zero, meaning less negative, represents the greater value, since a less negative exponent indicates fewer divisions by ten.

Example

Equal Exponent Coefficient Comparison

Procedure

When two numbers share the exact same exponent, their coefficients are compared directly using ordinary decimal comparison, since the shared power of ten does not affect which coefficient represents the greater value.

Example

Common Exponent Conversion for Comparison

Procedure

When comparing numbers that are not already in normalized scientific form, or when a more direct comparison is desired, each number can be rewritten so that all involved expressions share the same exponent, allowing their coefficients to be compared directly.

Example

Rewriting $34·{10}^3$ as $3.4·{10}^4$ allows a direct coefficient comparison against another number already expressed with an exponent of $4$.

Ordered Scientific Number Sequence

Procedure

A group of numbers expressed in scientific notation is arranged from least to greatest, or greatest to least, by applying sign comparison first, exponent comparison second, and coefficient comparison last, in that specific order, to every pair of numbers in the group.

Example

Given $-3.1·{10}^5$, $2.4·{10}^2$, and $2.4·{10}^6$, applying sign comparison first places the negative number as least, and exponent comparison then orders the remaining two, producing the sequence from least to greatest: $-3.1·{10}^5$, $2.4·{10}^2$, $2.4·{10}^6$.