Future Beacon



        
Chapter F

Rational Numbers





Introduction


Definition - Number r is a rational number if and only if r is a counting number or r is an integer.

The word ratio is the root word of the word rational. Characterizing these numbers as rational was originally intended to emphasize their ability to quantify ratios, whether of whole things or of parts of things.

Counting numbers are will now also be called rational numbers.



Addition


Definition - Rational number z is the sum of all of the rational numbers in Q if and only if the following statements are true:
E is an event set.
E = (A, B, [C :: B, D], D)
A = Q is a set of rational numbers, and z is a rational number equal to zero.
B = If x is in Q, then z becomes the sum of x and the previous value of z, and x is removed from Q.
C = If Q is not empty, repeat B; otherwise continue to D.
D = Stop.
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Definition - A subscript is a label, name, numeral or number, or combination of same, that is an element in an ordered set, and is assigned to an element.

Notation - If V is a set of numbers, such that V = (3, 5, 7), and N stands for any element in V, then V may have its elements subscripted this way: Na, Nb, Nc. The subscripts used here are members of the alphabet. If the set V were a larger set, This notation could be used: Na, Nb, Nc, . . . Nt. The set of letters from a to t is well defined. The three dots could be understood as meaning "and so on" (until the last element is written at the end). If an element is to be written multiple times in a calculation or list, the only difference between one subscripted element and another may be the subscript itself.

Notation - The sum of all of the rational numbers in Q may be written this way: K1 + K2 + K3 + . . . + Kn, where K1 through Kn are all of the elements in set K.

Saying that "z is a rational number" implies, from the definition of rational number that z is an element of some counting set. The explicit mention of such sets is not necessary if the meaning of the number designation is clearly known. Similarly, "x is an integer" will not require recounting parts of the definition of integer.



Multiplication


Definition - The product of rational number x and rational number y equals rational number z if and only if the following statements are true:
C is a counting set containing x, y and z.
Q is a counting set containing x and y.
Each element in Q is in C.
The unit string size of C is d, and the unit string size of Q is e, and d = e2.
If a is associated with b in Q, then e elements in C are between a and b in C.
Zero in C is associated with w in C.
Element p in C is associated with 0 in C.
S is a set.
Each element in S is a set.
The intersection of all of the sets in S is empty.
If v is a set in S, then the string from w to the absolute value of x in C has a one-to-one correspondence with v.
S has a one-to-one correspondence with the string from w to the absolute value of y in Q.
If x and y are of the same sign, the string from w to z in C has a one-to-one correspondence with the union of all the sets in S.
If x and y are not of the same sign, the string from z to p in C has a one-to-one correspondence with the union of all the sets in S.
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Q is a subset of C. The numbers to be multiplied must be chosen from the subset Q of C in order to insure that the product is in C. If the numbers to be multiplied are chosen from a counting set Q having a unit string size of ten, the product may not be in Q. For example, 0.2 ⋅ 0.3 = 0.06. The product is an element in a counting set having a unit string size of 100. The product of two rational numbers chosen from any counting set is always an element in a counting set of some unit string size. In fact, it is always an element of a counting set that has a unit string size that is the square of the original. The product may not be in the original subset, but it is no less a rational number because it is an element in a correctly defined counting set.

While they form a rigorous foundation, these technicalities will not add to the difficulty of calculating. The length of the string of numerals to the right of the decimal point is routinely modified to fit the precision of the calculation. Now, it is hoped, that the practice has a development in terms of sets.

Notation - The product of a pair of integers maybe written this way: x ⋅ y

Definition - Rational number x multiplied by rational number y equals rational number z if and only if z is the product of x and y.
Definition - Rational number z is the product of all of the rational numbers in Q if and only if the following statements are true:
E is an event set.
E = (A, B, [C :: B, D], D)
A = Rational number z = 0, and Q is a set of rational numbers;
B = If x is in Q, then z becomes the product of x and the previous value of z, and x is removed from Q.
C = If Q is not empty, repeat B; otherwise continue to D.
D = Stop.
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Notation - The product of all of the rational numbers in Q may be written this way: Q1 ⋅ Q2 ⋅ Q3 ⋅ . . . ⋅ Qn, where Q1 through Qn are all of the elements in set Q.

Definition - The quotient of integer x and integer y equals integer z if and only if z ⋅ y = x.
Definition - Integer n divided by integer m equals integer x if and only if x is the quotient of n and m.
Definition - Integer a divides integer b if and only if integer c = b divided by a.
Definition - F is a factor in G if and only F divides G.

Notation - Number p divided by number q may be written this way: p/q

Definition - R is a ratio of p and q if and only if R = p/q.
Definition - Integer j is divisible by integer k if and only if j/k is an integer.
Definition - P is a prime number if and only if P is an integer greater than 1, and no positive integer greater than 1 other than itself divides P.
Definition - S is the set of prime factors of N if and only if the following statements are true:
Each element in S is an integer equal to Pk where P is a prime number and k is a positive integer.
The product of all of the integers in S equals N.
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Definition - The quotient of rational number x and rational number y equals rational number z if and only if z ⋅ y = x.
Definition - Rational number x divided by rational number y equals rational number z if and only if z is the quotient of x and y.

Notice that any quotient p/q would be wholly ambiguous if q were allowed to be zero. Such would-be ratios are considered undefined.

Definition - The square root of y is x if and only if x ⋅ x = y.
Definition - The nth root of y is x if and only if x1 ⋅ x2 ⋅ x3 ⋅ . . . ⋅ xn = y, where n is greater than 1.

Notation - The square root of x may be written this way: √x
Notation - The nth root of x may be written this way: n√x

Definition - N factorial = 1 ⋅ 2 ⋅ . . . ⋅ N.

Notation - N factorial may be written this way: N!

Example - If N = 3, then N! = 1 ⋅ 2 ⋅ 3 = 6.



Exponents for Rational Numbers


Definition - Rational number x raised to the zero equals 1.
Definition - Rational number z equals rational number y raised to the positive integer x if and only if the following statements are true:
E is an event set.
E = (A, B, [C :: B, D], D)
A = Q is a counting set containing positive integer x and rational number y; and z in Q equals 0, and K is a non-empty set containing x elements.
B = If any element q is in K, then z becomes the product of y and the previous value of z, and q is removed from K.
C = If K is not empty, repeat B; otherwise continue to D.
D = Stop.
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Notation - Integer z equals integer y raised to the positive integer x may be written this way: z = yx.

Definition - Rational number Z equals rational number y raised to the negative integer x if and only if the following statements are true:
R = the absolute value of x.
T = yR.
Z = 1/T.
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Definition - Rational number y raised to the 1/n = n√y if and only if n is an integer and n is not zero.
Definition - Rational number y raised to the m/n = n√ym if and only if m and n are integers and n is not zero.

All ratios p/q where p and q are integers and q ≠ 0 can be named by means of some counting set. Notice that these definitions define yx where x and y are rational numbers.

Examples

         4-0.5 = 1/40.5 = 1/41/2 = 1/√4 = 1/2 = 0.5

         4-1.5 = 1/41.5 = 1/43/2 = 1/√(43) = 1/√64 = 1/8 = 1.25

         2-1.5 = 1/21.5 = 1/23/2 = 1/√(23) = 1/√8

The square root of 8 in the last example can only be approximated by rational numbers. It can, however, be approximated to any desired precision in any base. This is also true of π (the circumference of a circle divided by its diameter). This issue will be addressed later in this development.