Units (in Math & Science)

This page will give an overview of the units used in both mathematics and science, and the different categories that each falls into.

We will go into the metric system vs. the system the United States uses, as well as the NIST denotation for SI units. Dimensional analysis and unit conversion will be discussed as well.

Distance

Distance refers to the measurement of the length from point A to point B. Usually, we use a ruler, tape measure, etc. to calculate this value.

In the metric system (which is the unit system majority of the world uses), distance is measured in terms of the meter (m) and its derivatives; this includes centimeters (cm), millimeters (mm), kilometers (km), micrometers (μm), etc.

Now in the U.S. system specifically to the United States, the unit of distance is inches, feet, yards, miles, and whatever follows. Usually, it doesn't get larger than a mile when describing distance. These units can be converted to each other as:

• 12 inches are in 1 foot
• 3 feet are in 1 yard
• 1760 yards are in 1 mile

And so on...more about unit conversions in the dimensional analysis page.

Weight

Weight refers to how heavy something is as determined by the force of gravity acting upon its mass. Simply put, weight is how much you weight. Usually, a scale or balance is used to measure this.

In the metric system, weight is measured in terms of kilograms (kg) and grams (g).

In the U.S. system, weight is measured using a more complicated system of ounces (oz), pounds (lb), and eventually tons (t). Ounces can also be used to measure volume (which we'll get into later as well).

• 16 ounces are in 1 pound
• 2000 pounds are in 1 ton

If you couldn't already tell, the U.S. system for measurement is a bit...confusing and inconvenient, hence why most science classes and facilities use the metric system.

Volume

Volume refers to how much space something takes up in a container, specifically how much space a liquid takes up. Usually, volume can be measured in any scientific glassware such as a graduated cylinder, beaker, flask, etc. But also, we can measure volume in a container that has a pre-determined, known volume amount.

In the metric system, volume is measured in milliliters (mL) and liters (L).

In the U.S. system, volume is measured in a "chain system" where we go from fluid ounces (fl oz) to cups to pints to quarts and eventually, gallons.

• 8 fluid ounces (fl oz) are in 1 cup
• 2 cups are in 1 pint
• 2 pints are in 1 quart
• 4 quarts are in 1 gallon.

And many more conversions can be done from cups to quarts, or pints to gallons, or fluid ounces to quarts, etc. Usually, this measurement system is used for baking recipes—measurements of flour, butter, sugar are usually made in these units.

SI Units

In science, the National Institute of Science and Technology (NIST) defines SI units (International System of Units) as the universally agreed upon units for scientific measurements. And so, for common science concepts, we will use the agreed upon SI units as measurement.

Mass

Mass refers to the amount of matter present in an object.

For mass, the SI units are kilograms (kg), similarly to the metric system. Balances and scales should be set under the metric system for easy conversion.

Length

For length (distance), the SI units are meters (m), but more commonly on the smaller scale as centimeters (cm) if applicable.

Time

Time is a measure of the events that occur in correlation to the past, present, and eventually future.

For time, the SI units are seconds (s).

Energy

Energy is a measure of the ability to do work. It's a key concept that many science classes will touch upon. Commonly, it can refer to how things move or heat up.

For energy, the SI units are Joules (J). A Joule can be defined as the energy necessary to accelerate an object with a mass of 1 kilogram (kg) using a force of 1 Newton (N) over a distance of 1 meter (m).

Force

Force is a measure of how much push or pull there is on object. Force on an object can cause it to move and change speed and/or direction.

For force, the SI units are Newtons (N), named after Sir Isaac Newton himself who penned the Laws of Motion that force is very much so based on.

Substance

The amount of substance within an object refers to a pre-determined elementary entity that is present in an amount of matter. It's an arbitrary value that helps us better relate units to each other—it's more of a concept if anything.

For the amount of substance, the SI units are moles (mols). Avogadro's number specifies that there are 6.022 x 10^23 elementary entities in 1 mole.

Pressure

Pressure is a measure of the amount of force exerted perpendicularly per area (P = F x A).

For pressure, the SI units are Pascals (Pa).

Temperature

Temperature is basically measure of how hot or cold something is, but in reality, it is a measure of the average amount of kinetic energy of the particles present and how fast/slow the particles move. Depending on the speed of the particles, it will indicate the "hotness" or "coldness" of the object/compound.

For temperature, the SI units are Kelvin (K), often used in scientific problems and calculations. However, under the metric system, Celsius (°C) are most commonly used. And if in the United States, Fahrenheit (°F) is used instead.

Electric Current

Electric current is a measure of the amount of electricity flowing through a circuit or conductor.

For current, the SI units are Amperes (A).

Charge

Charge is a measure of the amount of electrons present or not present in an atom. More electrons will creative a more negative charge whereas less electrons will create more positive charge.

For electric charge, the SI units are Coulombs (C)—not to be confused with the unit symbol for Celsius, which also uses °C, but has the degree symbol.

Power

Power is a measure of the amount of work done in a given unit of time. It is Joules/second essentially.

For power, the SI units are Watts (W), which is the same thing as a J/s (the definition of power).

Resistance

Resistance is a measure of the opposition to current flow in an electrical circuit.

For resistance, the SI units are Ohms (Ω)—the Greek letter for Omega is used as the symbol.

Unit Conversion (Dimensional Analysis)

With all of these units, it's best to know how to convert between each one in order to solve mathematic or scientific problems, or gain a better understanding of what is occurring and interacting with each other.

Firstly for distance conversions:

In the metric system where we use the meter (m) as our primary unit, its derivatives can easily be found using scientific notation and the power of 10.

Using the mnemonic, King Henry Died By Drinking Chocolate Milk where the "By" (B) stands for the base unit—in this case, meters—, we can easily convert from meters to any other metric meter. "King" (K) stands for kilo(meter), "Henry" (H) stands for hecto(meter), "Died" (D) stands for deca(meter), Drinking (D) stands for deci(meter), "Chocolate" (C) stands for centi(meter), and "Milk" (M) stands for milli(meter). Most of the times, the common units will be kilometer, centimeter, and millimeter.

Since the B (base) is in the middle, if move right one word, we multiply the base by 10. For example, if we have 10 meters and we want to convert to decimeters, since decimeter is one space to the right of the base, we will multiply 10 meters by 10 to gain 100 decimeters. If we want centimeters, we will multiply by an additional 10, or 100 from the base—therefore, 10 meters = 100 centimeters.

Conversely, if we move to the left of the base, we divide by 10. For example, if we have 10 meters and we want to decameters, we will divide 10 by 10. 10 meters = 1 decameter. Or, if we want to convert to kilometers, we will divide 10 meters by a total of 1000, equaling 0.01 meters.

Every time we want to convert to a unit that's to the left of our current unit, we will divide by 10 until we get to that unit. Every time we want to convert to a unit that's to the right of our current unit, we will multiply by 10 until we get to that unit.

This method is usable for any kind of unit type with these metric prefixes.