Everyday Usage and Use in Mathematics

The term "order of magnitude" is used to describe a big difference in size, scale, or quantity between two things. This phrase is often used in areas like economics, science, and technology to show important differences. For example, if a company says its revenue increased by an order of magnitude, it means the revenue is now ten times more than it was before.

In mathematics and science, "order of magnitude" means the size of a number when expressed as a power of 10. For example, if one item is ten times larger than another, it is said to be "an order of magnitude larger." This helps us quickly compare the sizes of different numbers by looking at their exponent when the number is written in scientific notation.

While "order of magnitude" deals with comparing the scale of quantities, the magnitude of a vector has a different meaning that refers specifically to the length of a vector.

Importance in Mathematics

Understanding orders of magnitude is useful for quickly estimating and comparing quantities that are vastly different in size. For example, a city with a population of \(10^6\) (one million) is much larger than a town with \(10^4\) (ten thousand).

Orders of magnitude also simplify complex calculations, especially in fields like physics and astronomy, where numbers often differ by many orders of magnitude. For instance, the distance from Earth to the Sun (~\(1.496 \times 10^8\) km) and to the nearest star, Proxima Centauri (~\(4.24 \times 10^{13}\) km), differ by several orders of magnitude becuase of the vastness of space.

Examples

The order of magnitude of a number is shown by the exponent of the power of ten when the number is written in scientific notation. For example:

• The number 1,000 can be written as \(10^3\), so it is said to be of the order of magnitude \(10^3\).
• A number like 25,000 can be rounded to \(2.5 \times 10^4\), so it is considered to be of the order of magnitude \(10^4\).

In these examples, \(10^3\) and \(10^4\) represent the orders of magnitude. Although 25,000 is not exactly a power of ten, it is closest to \(10^4\) and can be approximated as \(2.5 \times 10^4\), placing it within the same order of magnitude as \(10^4\).

1. Comparing the mass of the Earth (~\(5.97 \times 10^{24}\) kg) to the mass of a human (~\(7 \times 10^1\) kg) shows a difference of more than 20 orders of magnitude.

2. The wavelength of visible light ranges from about 400 nm (\(4 \times 10^{-7}\) meters) to 700 nm (\(7 \times 10^{-7}\) meters), while the wavelength of X-rays is on the order of \(10^{-10}\) meters. This difference in wavelength is a difference in orders of magnitude, which is important for understanding the behavior of different types of light.

Exercises

1. Express the number 5,200,000 in scientific notation and determine its order of magnitude.
2. Compare the order of magnitude of the population of Earth (~\(7.8 \times 10^9\)) with that of an ant colony (~\(5 \times 10^6\)).
3. If the mass of a proton is about \(1.67 \times 10^{-27}\) kg and the mass of the Earth is \(5.97 \times 10^{24}\) kg, by how many orders of magnitude do these two masses differ?