Unknown numbers are usually represented by letters of the alphabet. The letter is the all-out favorite, with x used when a second unknown is needed and y for a third. By the way, the name used for the unknown number by early mathematicians (like Al-Khorezmi) was "zthe thing"--"shai" in Arabic, "res" in the Latin which scholars used in Europe.
However, different letters are also used, often hinting at the quantity they represent-- for an unknown time,T or m for unknown masses and M for an unknown force. (These will be used later in the calculation of Lagrangian points). F
"Unknown" may be broadened to include " An example of case (#1) is the ratio between the length of a circle and its diameter, a universal constant appearing in many equations (for example, Kepler's equation in section (12a)), whose value to 11 decimals is 3.14159265359... Although the number is known, it is universally represented in all equations by the letter p (Greek p, pronounced "pi"). It is only replaced with the actual number when the unknown quantity is derived. As an example of case (#2), consider the distance s which a dropped ball covers in a time of t seconds, starting from rest. It is
/2)--get multiplied, remember?).Here is the number giving the strength of the Earth's gravity pull: if g is in meters, s = 9.81, if in feet, g = 32.16 (9.81 meters = 32.16 feet). It gis known, but (as in example #1 above), you don't bother with its actual value until the moment you actually use it. But the time is not yet chosen! Whatever you choose for t, the formula will give you the appropriate distances.t
.sStill, the "digging out" skills you learn with equations are also useful here. Suppose you seek the inverse relationship--given s, what is t? One now views t as the unknown and proceeds to isolate it. Multiply both sides by 2 2 and divide by 2s To go from t SQRT(2s Now, whatever the distance ## Substitution of FormulasContinuing the discussion of substitutions from section (M-1), here is a situation which happens quite frequently. In one of the problems in "Stargazers", for instance, one arrives at two equations:
VT = 2 p R (1)_{1} = 2 p R_{1} (2)
where T
(VT_{1})/(VT)> = 2 p R_{1}/(VT)
However, because of the = sign in (1), we could replace the denominator on the left by (VT_{1})/(VT)> = 2 p R_{1}/(2 p R)Canceling equal multipliers ("factors") on top and on bottom leaves
T_{1}/T = R_{1}/R
which turns out to be useful in the rest of the calculation. |

**Next Stop: #M-4 Identities**

*Author and curator: David P. Stern *

Last updated 25 February 1999