The metric system uses several symbols to represent different powers of ten. The most common include:

The MKS (meter-kilogram-second) system of units is part of the SI (Systeme Internationale) that is the standard in much of the world outside of the United States, and in science.

Length

• The meter was originally defined as 1/10⁷ of the distance from the equator to the North pole measured along a longitude passing through Paris.
• In 1899 it was defined as the length between two marks on a particular platinum-iridium rod kept in Paris.
• In 1960 the meter was redefined as 1,650,763.73 wavelengths of a particular orange light emitted by krypton-86.
• One meter is about the length of your stride.

Mass

• One kilogram is about 2.2 pounds.
• The standard kilogram is a particular platinum-iridium cylinder at the International Bureau of Weights and Measures near Paris.
• One litre of water has a mass of 1 kg.

Time

• The second is defined as the time required for 9,192,631,770 periods of the radiation emitted when an electron in a cesium atom makes a particular transition.
• One year = π x 10⁷ s
• Average human lifespan = 2 x 10⁹ s
• Age of the Universe = 10¹⁸ s

Exercise

Determine your height in meters and your mass in kilograms. Figure out how many seconds it took you to figure this out.

Example of a unit conversion:
Michael Johnson ran 200 m in 19.32 seconds when he set the world record for that distance. Express his average speed in km per hour.

Average speed = distance/time = 200/19.32 = 10.35 m/s.

10.35 m/s x (1 km / 1000 m) x (3600 s / hr)

= 10.35 x (3.6) = 37.27 km/hr

For the non-metric types, this is over 23 mph!

Squaring, cubing, etc.

If 100 cm = 1 m, how many cm² = 1 m²?

How many cubic cm are in 1 cubic meter?

If you filled 1 cubic meter with water, what mass of water would you have?

Example of a unit conversion:
Michael Johnson ran 200 m in 19.32 seconds when he set the world record for that distance. Express his average speed in km per hour.

Average speed = distance/time = 200/19.32 = 10.35 m/s.

10.35 m/s x (1 km / 1000 m) x (3600 s / hr)

= 10.35 x (3.6) = 37.27 km/hr

For the non-metric types, this is over 23 mph!

Squaring, cubing, etc.

If 100 cm = 1 m, how many cm² = 1 m²?

How many cubic cm are in 1 cubic meter?

If you filled 1 cubic meter with water, what mass of water would you have?

Basic geometry

First, let's start with the right-angled triangle. You should know the Pythagorean theorem:
Pythagorean theorem: a² + b² = c²

You should also have no trouble remembering SOHCAHTOA:
sin(θ) = opp./hyp.
cos(θ) = adj./hyp.
tan(θ) = opp./adj.

Although we often deal with right-angled triangles, many triangles do not have a right angle. The Sine Law and Cosine Law are often useful for those triangles.

The Sine Law:
sin(θ_a) / a = sin(θ_b) / b = sin(θ_c) / c

The Cosine Law:
c² = a² + b² - 2ab cos(θ_c)

You should have no problem rearranging equations to solve for particular variables.
For instance, solve for v in the equation:
4v² - 7 = 3 - v²

The quadratic equation

If you have an equation that looks like ax² + bx + c = 0, the quadratic equation can be used to solve for x. Generally there are two solutions:

Integrals

∫ 3t dt

∫ (1/x) dx

∫ sin(4t) dt

∫ 3t dt = (3/2)t² + c

∫ (1/x) dx = ln(x) + c

∫ sin(4t) dt = (-1/4)cos(4t) + c

Derivatives

d(12t³)/dt

d(e^kx)/dx

d(sin[4t])/dt

d(12t³)/dt = 36t²

d(e^kx)/dx = k e^kx

d(sin[4t])/dt = 4 cos(4t)

Other important concepts from calculus:

• the integral is the area under the curve
• the derivative is the slope
• to maximize or minimize a function take its derivative and set it equal to zero