41.9 Scientific Notation Definitions

Scientific Notation Definitions establishes the terminology and structural requirements used to express very large or very small numbers compactly as a product of a coefficient and a power of ten, defining the standard form, its components, and the concept of magnitude that this notation relies on.

Elementary Algebra Scientific Notation Definition

Definition

Scientific notation is a way of writing a number as the product of two parts: a coefficient and a power of ten, expressed in the general form:

where $c$ represents the coefficient and $n$ represents an integer exponent.

Purpose

This form allows numbers with many digits, whether very large or very small, to be written and compared using far fewer symbols than their full decimal representation would require.

Elementary Algebra Scientific Coefficient Definition

Definition

The coefficient in scientific notation is the numerical factor $c$ that is multiplied by the power of ten, and it is required to have an absolute value that is at least one but strictly less than ten:

Example

In $4.5·{10}^7$, the coefficient is $4.5$, a value satisfying this required range.

Elementary Algebra Normalized Scientific Form Definition

Definition

A number is said to be in normalized scientific form when its coefficient satisfies the required range of at least one and less than ten, and the power of ten carries the appropriate integer exponent needed to restore the original value when multiplied together.

Distinguishing Normalized from Non-Normalized Forms

The expression $45·{10}^6$ represents the same value as the earlier example but is not in normalized form, since its coefficient of $45$ exceeds the allowed range; only $4.5·{10}^7$ satisfies the normalized definition.

Elementary Algebra Decimal Order of Magnitude Definition

Definition

The order of magnitude of a number refers to the integer exponent $n$ attached to the power of ten in its normalized scientific notation form, indicating the general scale of the number in terms of powers of ten.

Example

The number $4.5·{10}^7$ has an order of magnitude of $7$, reflecting that it is on the scale of tens of millions.