**Table of Contents**

In this guide, you will learn about Standard International (SI) Fundamental Units and how to convert between orders of magnitude in the metric system. These are an internationally recognized standard for measuring different physical quantities. We will primarily focus on the basic, most frequently used units.

## 1. Basic SI Units & Their Origins

Some of the most common measurements you’ll encounter in any scientific field include time, length, mass, temperature, and current.

### Time

The SI base measurement for time is in **seconds**. Originally, the time in a second was determined based on the length of a full day, but as the world is rotating ever so slightly slower each year, it is not a fully accurate way to base time measurements off of. Currently, time in seconds is measured by the time it takes 9,192,631,770 caesium atoms to vibrate in an atomic clock. This was selected because the vibrations of caesium are predictable and measurable, unlike the time the Earth takes to rotate, and will not be different over time.

### Length

Another essential unit is length, whose base unit is **meters**. Length is used to measure the distance from one point to another. The amount of length represented by a meter used to be calculated as a small fraction (one ten millionth) of the distance from the equator to the north pole. A standard meter bar made of platinum and iridium was kept in Paris and used as the basis of meter measurement. For a long time, people derived all other meter measurements from it. Now, it is defined as the distance light travels in one 299,792,458th of a second in a vacuum. As the speed of light does not change, this is more accurate than a piece of metal sitting in a museum, which may slowly decay.

### Temperature

Temperature is measured in **degrees celsius.** 100 degrees celsius is the temperature at which water boils, and 0 degrees is the temperature at which it freezes. For calculations where it’s not practical to use negative numbers, degrees **kelvin** is used. Kelvin is based on the same system as celsius. The only difference is Kelvin can never be negative, as the scale for Kelvin starts at absolute zero, which is -273.15 degrees celsius. Therefore, to convert between the two, you only need to add or subtract 273.15.

### Current

A current refers to the flow of a medium, such as the flow of electricity. The base unit for current is an **ampere**, which is commonly shortened to amp. Its name comes from the famous French Physicist André-Marie Ampère, one of the founders of electromagnetism. You’ve likely encountered labels on electronics showing how much current they use. For example, most home circuit breakers in the US are 15 to 20 amps. A typical hair dryer uses 10 to 15 amps on high power. An ampere is defined as the current of one coulomb of electrical charge moving in a second.

### Speed

Speed is a unit of distance over time. Since the base SI unit for distance is the meter, and the base SI unit for time is the second, you can use meters per second for speed.

One important measure of speed is the speed of light, which is calculated at approximately 299,792,458 meters per second.

You’re likely familiar with a speedometer in a vehicle, which measures the current speed the vehicle traveling in km/h or miles/hr. Note that speedometers are often off by a few units. A more accurate way to measure speed is with GPS positioning.

## 2. The Metric System & Orders of Magnitude

If you’re in United States of America you’ve likely dealt with the metric system before in limited capacity, but the rest of the world grew up using the metric system. The USA uses the Imperial Measurement System, which consists of things such as pounds, ounces, feet, miles, etc. The inherent problem with the Imperial system is that it’s difficult to convert between different size measurements. For example, there are twelve inches in a foot, three feet in a yard, and 1760 yards in a mile. This is not a very intuitive system for math or science, as we use a base-10 numbering system.

The metric system uses base-10, so it’s much easier to convert between units at scale. For example, in the metric system, one kilometer is one thousand meters, and one centimeter is one one-hundredth of a meter. The order of magnitude refers to the classification of a unit by scale size. The order-of-magnitude prefixes are below. If you’re in any science classes, familiarize yourself with these prefixes, at least pico and up.

### Orders of Magnitude Table

Prefix | Symbol | Value |
---|---|---|

exa | E | 10^{18} |

peta | P | 10^{15} |

tera | T | 10^{12} |

giga | G | 10^{9} |

mega | M | 10^{6} |

kilo | k | 10^{3} |

hecto | h | 10^{2} |

deca | da | 10^{1} |

unit | unit | 1 |

deci | d | 10^{-1} |

centi | c | 10^{-2} |

milli | m | 10^{-3} |

micro | µ | 10^{-6} |

nano | n | 10^{-9} |

pico | p | 10^{-12} |

femto | f | 10^{-15} |

atto | a | 10^{-18} |

If you’re familiar with computers or phones, you’ve probably heard terms like kilobyte, megabyte, gigabyte, and terabyte before. They’re used to calculate file sizes and storage capacities. Each step up in size is 1000x (10^{3}) more than the last. So if you already know that a megabyte is bigger than a kilobyte, a gigabyte is bigger than a megabyte, and so on, then you already know 4 of the prefixes!

## 3. Order of Magnitude Conversions

#### Basic Examples

Now that you’ve seen all the orders of magnitude, let’s try some sample problems in which we convert between them.

**Example 1:** Convert 359 centimeters to hectometers.

When you are first learning to convert, you may find it helpful to draw this out on paper. Since we need to convert from a small scale to a large scale, we know the decimal must be moving towards the left. You can draw a line as shown, write each unit, and then move the decimal place accordingly to find the answer.

**Example 2:** Convert 1.5 centimeters to megameters.

Mathematically you can convert this on paper by using the following steps. For the first step, create a fraction to convert from centimeters to meters, then, create another fraction to convert from meters to megameters. Since there are 100 cm in a meter, and 1,000,000 (or 10^{6}) meters in a megameter, the setup looks like this:

\( 1.5 cm \times \frac{1 m}{100 cm} \times \frac{1 Mm}{1000000m} \)

Next, we can cancel out the units that are divisible by one another.

\( 1.5 \cancel{cm} \times \frac{1 \cancel{m}}{100 \cancel{cm}} \times \frac{1 Mm}{1000000\cancel{m}} = 1.5 \times \frac{1}{100} \times \frac{1 Mm}{1000000}\)

When we multiply this out, our result is:

\( 1.5 \times \frac{1}{100} \times \frac{1 Mm}{1000000} = \frac{1.5 Mm}{100,000,000}\)

Since there are 8 zeros in the denominator, that means in scientific notation it can be written as \(1.5 \times 10^{-8}\) megameters. So that is our final answer.

Another option is to look at the table of orders of magnitude above. The value for centi is 10^{-2} and the value for mega is 10^{6}. Next, we need to find the difference between these two exponents. You can simply subtract 6 from -2 in the exponent, for a result of 10^{-8}. We know the exponent must be negative, since we’re going from a smaller unit to a larger one, so our actual answer will be a much smaller number. Therefore, 1.5 centimeters is equal to \( 1.5 \times 10^{-8}\) megameters.

\(10^{-2}\times10^{-6}= 10^{-2-6} = 10^{-8}\)

**Example 3:** Convert 0.59 kilograms to decagrams. Following the same logic as the above example, we can write this out as:

\(0.59kg \times \frac{1000 g}{1kg} \times \frac{1dag}{10g}\)

All the units cancel out, and 1000 divided by 10 is 100.

\(0.59 \xcancel{kg} \times \frac{1000 \cancel{g}}{1\cancel{kg}} \times \frac{1dag}{10\cancel{g}} = 0.59 \times \frac{100dag}{1} = 59dag\)

You could have figured this out by experimenting with the exponents. We know kilo is 10^{3}, and deca is 10^{1}. the difference between 3 and 1 is 2, therefore the answer is \(0.59 \times 10^2\), which is 59 dag. We know the exponent must be positive since we’re going from a larger unit to a smaller one, and therefore require more zeros for a larger number.

\(10^3 \times 10^{-1} = 10^{3-1} = 10^{2}\)

**Example 4:** A car is driving on a road at 65 miles per hour. Given that there are 1.60934 kilometers in a mile, how fast is the vehicle traveling in meters per second?

First, convert from miles per hour to kilometers per hour.

\(\frac{65\text{miles}}{1\text{hr}} \times \frac{1.60934\text{km}}{1\text{mile}}=\frac{104.6071\text{km}}{\text{hr}}\)

Next, convert from kilometers per hour to meters per hour.

\( \frac{104.6071\text{km}}{\text{hr}} \times \frac{1000\text{m}}{\text{km}} = \frac{104607.1\text{m}}{\text{hr}} \)

Finally, convert from meters per hour to meters per second.

\( \frac{104607.1\text{m}}{\text{hr}} \times \frac{1\text{hr}}{60\text{min}} \times \frac{1\text{min}}{60\text{sec}} = \frac{29.06\text{m}}{\text{sec}} \)

A car going 65mph is moving approximately 29 meters per second.