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Basic Math

Basic Middle School Math

Most of the basic concepts in Physics can be explained with the basic math taught in Middle School. The five most important functions are the well known add, subtract, multiply, divide and exponentiation. In addition are the functions of slope and area. The math shown on this page will be in Basic Format. Basic is a computer program developed at Dartmouth College about 40 years ago. There are many versions of Basic Interpreters available on the Web that can be downloaded for free for personal use.

To get a free copy off the web, go to http://macinsearch.com/users/MacBasix/MacBasixLinks1.html and look at the free Basic Interpreters available. There are also versions there that will run under a Windows platform. These recognizes the math operators plus (+), minus (-), multiply (*), divide (/), exponent (^), and many other functions. For example typing the following computer input in brown: (computer results in green and comments in red)

Exponentiaton:

the fifth most important mathematical function.

x = 3
y = 4
print 5*(x+y)
35
print x/y
0.75
print y^2
16 (y squared is 16)
print 12^0
1 (twelve to the zero power)
print .25^-1
4 (reciprocal of .25 is 4)
print 16^0.5
4 (square root of 16)

Exponentiation is the 5th most important arithmetic function, right behind add, subtract, multiply and divide. All powers and roots can be written as exponents. Powers are usually written as an integer exponent such as: 23 or 104. Negative integer exponents such as 10-6 are used to describe small numbers. Roots are usually written as fractional exponents such as 161/2 that is the same as the square root of 16 which is equal to 4. It could also be written as 160.5. In mathematics, an exponent can be any number, integer, negative, fractional, or even imaginary or complex. The California State Standard for high school math covers integer, negative and fractional exponents.

The first lesson of exponentiation is the definition that the exponent is the number of times the base is multiplied by itself, as:
Xn = X times itself 'n' times.

The next lesson of exponentiation is that exponents with the same base can be added together. Thus:
21 x 23 x 25 = 29.

Also, negative exponents are number of times the variable 'X' is divided into one.
X-1 is identical to 1/X.
As in:
10-3 is identical to 1/1000 or 0.001.

And as a special case is a number divided by itself which is one for any number 'X' except 0.
X0 is 1 for any X except 0,
as for example:
120 = 121 x 12-1 = 12/12 = 1.

Fractional exponents follow the same rules as integers.
21 x 21 = 22.
If all the exponents are divided by 2, we get:
21/2 x 21/2 = 22/2 = 21 = 2.

A calculator with the function X,Y can prove this, or any of the examples that follow. Of course BASIC can also be used to prove these examples. Note that the most common root is the square root and that:
X1/2 is identically equal to the square root of X.

Examples:

a. 120 = 1.
b. 161/2 = 4.
c. 272/3 = 9.
d. (0.2)-1= 5.
e. (-0.125)-1/3= -2.
f. 21/12 = 1.05946309435929526456182529494634...

Note the example 'f.' above is an irrational number. It was developed by Johann Sebastian Bach when he divided the musical octave (twice the frequency) into 12 geometric steps and called it the tempered scale. The twelveth root of two therefore discribes the spacing of frets on a guitar, or the length of organ pipes in a church organ. It is the basis of all Western Music.

Another example would be Carbon Dating. Carbon-14 has a half life of 5700 years. Therefor an ancient artifact containing carbon would have less carbon-14 then a new sample of carbon of the same weight. The ratio 'R' of radioactivity from carbon-14 in the ancient artifact to the radioactivity from a new sample of carbon of the same weight would be equal to two to the minus power of age divided by 5700.
2-(age/5700 yr) = R

The last example is the decibel. Since the gain of an amplifier in Bels is the log of the output power divided by the input power, an amplifier with a gain of 23 db [=2.3 b] and an input of 0.1 watt would output of 0.1 x 102.3 watts or about 19.95 W. [almost 20 watts]

Exponentiation is a commonly used function that needs to be understood by any student of math.



Basic Slope and Area Calculations

Physics often uses the term "rate", to mean rate of change. If a function is plotted as a function of time, at any instant, the rate of change is the slope of that function. Taking the slope of a time dependent function adds inverse time to the unit. For example if a person runs 40 meters in 20 seconds, his speed would be 2 meters per second [2 m/s] . The mathmatical term for rate is differentiation.

In Figure 1 is a graph showing a vehicle accellerating from a stop to 40 m/s in 20 seconds, crusing at 40 m/s for 15 more seconds, and then decellerating to a stop in another 15 seconds. The accelleration would be 2 m/s2, about 0.2 g. The decelleration would be -40 m/s in 15 seconds or about -2.76 m/s2, less than -0.27 g.

The opposite of differentiation is integration. Integration is useful for finding the area under a curve. For a plot of velocity as a function of time, the area under the curve is proportional to distance. In Figure 1, the area under the curve is about 1300 meters, which is the distance the vehicle traveled from 0 to 50 seconds. Taking the area of a time dependent function adds time to the unit. Sometimes this is done by removing time from the demonimator as in dropping a "per second" in the calculated area below.


Figure 1

Differentiation and Integration are part of The Calculas. Differentiation is instantaneous slope, the slope calculated at a single point on the function. Similiarally, integration is the sum of almost an infinite number of areas under the curve, as the width (time) of each area approaches zero.

For school math, it is sufficient to approximate a complex function with a series of straight lines as is is shown in figure 1. In figure 1, total area, representing distance traveled, can be calculated by summing the areas under the three lines that represent accelleration, steady speed (cruse), and decelleration. Accuracy improves as more time samples are taken with the corresponding increase in straight line segments used to approximate the function.

Another example is the relationship between power and energy. Power is the rate energy is used. In a graph showing power as a function of time, the area under the power curve would represent energy used. If the power is in watts and the time in seconds, the area representing energy would be in joules. If the power is in kilowatts and the time in hours, the energy would be in kilowatt-hours. [1 kw-hr = 3.6 Mj]

Exercise Problem

To solve the following example problem, you must know that one horsepower is equal to 375 lb-mph, and that something dropped on Earth will accellerate at about 22 mph per second. [1 g]

A 3000 pound vehicle coasting on a level road takes 20 seconds to slow from 66 mph to 55 mph. With the engine engaged at full throttle, it takes 4 seconds to accellerate from 55 to 66 mph. What is the approximate horsepower at the wheels at 60 mph?

Before going to the next page for the answer, do your best to solve this problem.
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This Web Site was designed by Russ Lemon 2007