42.7 Scientific Notation Addition and Subtraction

Scientific Notation Addition and Subtraction describes how two numbers written in scientific notation are combined under addition or subtraction, requiring both numbers to first share the same power of ten before their coefficients can be added or subtracted directly.

Equal Power-of-Ten Requirement

Requirement

Before two numbers in scientific notation can be added or subtracted, both must share the exact same exponent on their power of ten; unlike multiplication and division, addition and subtraction cannot be performed directly when the exponents differ.

Reasoning

Adding or subtracting coefficients attached to different powers of ten would combine quantities of very different scale incorrectly, similar to adding a number of ones to a number of thousands without accounting for the difference in place value.

Common Scientific Exponent Selection

Procedure

When the two numbers have different exponents, one exponent is selected as the common exponent to which both expressions will be converted, typically the larger of the two exponents.

Example

For $3.2·{10}^5$ and $4.5·{10}^4$, the larger exponent, $5$, is selected as the common exponent.

Coefficient Rescaling

Procedure

The coefficient of the number whose exponent was not selected is rescaled by shifting its decimal point, adjusting its value so that its new exponent matches the common exponent chosen in the previous step.

Example

Rewriting $4.5·{10}^4$ with an exponent of $5$ requires shifting its decimal point one place to the left:

Scientific Coefficient Addition

Procedure

Once both numbers share the same exponent, their coefficients are added together directly, using ordinary decimal addition.

Example

Adding the rescaled coefficients from the running example:

Scientific Coefficient Subtraction

Procedure

Once both numbers share the same exponent, their coefficients are subtracted directly, using ordinary decimal subtraction, following the same rescaling requirement as addition.

Example

Subtracting the rescaled coefficients from the running example:

Shared Power-of-Ten Retention

Procedure

The power of ten, using the common exponent selected earlier, is carried forward unchanged into the result of the addition or subtraction, since only the coefficients change during this operation.

Sum or Difference Normalization

Procedure

If the resulting coefficient from the addition or subtraction falls outside the required range for normalized scientific form, either at or above ten or below one, the result is renormalized by shifting the decimal point and adjusting the shared exponent accordingly.

Example

If adding two coefficients produced $10.8$ alongside an exponent of $5$, the result $10.8·{10}^5$ is renormalized to $1.08·{10}^6$.