Real Numbers in the Neighborhood of Infinity by Jonathan W. Tooker Abstract We demonstrate the existence of a broad class of real numbers which are not elements of any number field: those in the neighborhood of infinity. After considering the reals and the affinely extended reals we prove that numbers in the neighborhood of infinity are ordinary real numbers. As an application in complex analysis, we show that the Riemann zeta function has infinitely many non-trivial zeros off the critical line in the neighborhood of infinity. 2010 Mathematics Subject Classification: Primary 58-02. Keywords: Real analysis, Extended real analysis, Complex analysis, Riemann hypothesis. x1. Real Numbers Definition 1.1. A real number x 2 R is a cut in the real number line. Definition 1.2. A cut in a line separates one line into two pieces. Remark. A number is a cut in a line. A line is defined a priori. All lines can be cut so all lines are number lines. A given line is the real line by definition. A real number separates the real number line into a set of \larger" real numbers and a set of \smaller" real numbers. Definition 1.3. The real numbers are defined in interval notation as R ≡ (−∞; 1) : Definition 1.4. Call real numbers in the neighborhood of the origin x 2 R0. Define them such that x 2 R0 () x 2 R ; and − n < x < n ; for some n 2 N. Tooker: [email protected]; 2 Jonathan W. Tooker Definition 1.5. Call real numbers in the neighborhood of infinity x 2 R1. Define them as all real numbers except for real numbers in the neighborhood of the origin: R1 ≡ R=R0 : Definition 1.6. For x 2 R and n 2 N, we have the property n 1 X lim = diverges ; and lim k = diverges : x!0± x n!1 k=1 x2. Affinely Extended Real Numbers Definition 2.1. Define two affinely extended real numbers ±∞ such that for x 2 R and n 2 N n 1 X lim = ±∞ ; and lim k = 1 : x!0± x n!1 k=1 Definition 2.2. The set of all affinely extended real numbers is R ≡ R [ {±∞} : Definition 2.3. In R, ±∞ are such that the limit any sequence of real numbers which diverges in R is equal to 1 or −∞. Theorem 2.4. If x 2 R and x 6= ±∞ then x 2 R. Proof. Proof follows from Definition 2.2. Definition 2.5. The affinely extended real numbers are defined in interval nota- tion as R ≡ [−∞; 1] : Definition 2.6. Infinity is such that 1 1 − 1 = undefined ; and = undefined : 1 Definition 2.7. Infinity does not have the distributive property of multiplication. For two non-zero real numbers a and b we have 1 · b + a = 1 : To the contrary, we write for two orthogonal unit vectorse ^1 ande ^2 1 · e^1 +e ^2 = undefined : Real Numbers in the Neighborhood of Infinity 3 Remark. In Definition 2.7 the former case gives the appearance of a distributive property because we can sum b + a and then use the multiplicative absorptive property of 1 to obtain a simplified result but we cannot do so in the latter case which is only mentioned in anticipation of the expression 1·x+iy which appears in Section 5. x3. Modified Infinity Definition 3.1. When the ±∞ symbols appear as ±c1, let the hat be an instruc- tion to delay the additive absorption of ±∞ indefinitely or until such delay causes a contradiction. At that time the hat must be removed. The hat is inserted and removed by choice except in the case where it invokes a contradiction and must be removed by definition. Example 3.2. An example of a statement in which the hat does not invoke a contradiction and may be left in place is x = c1 − b : Example 3.3. An example of a statement in which the hat invokes a contradic- tion and may not be left in place is given by two sequences n n X X xn = k ; and yn = c0 + k ; k=1 k=1 where n 2 N and c0 is some non-zero real number. Since 1 and c1 are the same number we can use Definitions 2.1 and 3.1 to write lim xn = 1 = 1 ; and lim yn = 1 = 1 : n!1 c n!1 c We may also write, however, lim yn = lim c0 + lim xn = c0 + 1 : n!1 n!1 n!1 c This delivers an equality c1 = c0 + c1 ; which contradicts the delayed additive absorption of c1. At this point, we must cease to delay additive absorption by removing the hat. Then 1 = c0 + 1 ; demonstrates the usual additive absorptive property of infinity and there is no contradiction. 4 Jonathan W. Tooker Definition 3.4. c1 is such that for any non-zero b 2 R0 ±c1 + b = b ± c1 ±c1 − b = −b ± c1 ±c1 + − b = ±c1 − b ±c1 + b = ±c1 − − b − ± c1 = ∓c1 ( ±c1 if b > 0 ±c1 · b = b · ±c1 = ∓c1 if b < 0 ( ±1 ±1 if b > 0 c = c b ∓c1 if b < 0 b = 0 : ± c1 Definition 3.5. c1 is such that ±c1 + 0 = 0 ± c1 = ±c1 − 0 = undefined ±c1 · 0 = 0 · ±c1 = undefined ±1 c = undefined 0 0 = 0 : ± c1 Remark. We will revisit the lack of an additive identity in Example 4.11. Definition 3.6. ±∞ has all the properties assigned to ±c1 in Definitions 3.4 and 3.5 plus the additive absorptive operation for non-zero b 2 R0 ±∞ ± b = ±∞ ∓ b = ±∞ ; such that b is an additive identity of 1. x4. Real Numbers in the Neighborhood of Infinity Definition 4.1. Let Rb be the set of all numbers of the form x = ± c1 − b ; where b 2 R0 ; b > 0 ; Real Numbers in the Neighborhood of Infinity 5 Definition 4.2. The ordering of Rb numbers is such that ± c1 − b = ± c1 − a () a = b c1 − b > c1 − a () a > b − c1 − b > − c1 − a () a < b c1 − b > − c1 − a : Theorem 4.3. All numbers x 2 Rb are cuts in the affinely extended real number line, i.e.: they are affinely extended real numbers. Proof. Definition 1.2 requires that a cut separates one line into two pieces. Observe that R=(c1 − b) ≡ [−∞; c1 − b) [ (c1 − b; 1] R= − c1 + b ≡ [−∞; −c1 + b) [ (−c1 + b; 1] : Main Theorem 4.4. All numbers x 2 Rb are real numbers. Proof. If a number is an affinely extended real number x 2 R and x 6= ±∞ then by Theorem 2.4 we have x 2 R. In the absence of additive absorption, ± c1 − b 6= ±c1 = ±∞ ; because it is the definition of Rb that b 6= 0. Also note that R=(c1 − b) ≡ (−∞; c1 − b) [ (c1 − b; 1) : All numbers x 2 Rb satisfy Definition 1.1. Theorem 4.5. All numbers x 2 Rb are real numbers in the neighborhood of infinity x 2 R1. Proof. We have shown in Main Theorem 4.4 that Rb ⊂ R ; so we will satisfy Definition 1.5 if we show Rb \ R0 ≡ ; : Definition 1.4 requires that elements of R0 satisfy −n < x < n ; 6 Jonathan W. Tooker so assume −n < ± c1 − b < n : Since b 2 R0 we know it has an additive inverse. Add or subtract b to obtain −n + b < c1 < n + b ; and − n − b < −c1 < n − b : We obtain a contradiction as 1 cannot be less than the sum of two finite numbers and −∞ cannot be greater than the difference of two finite numbers. All Rb numbers satisfy the definition of R1. Remark. The remaining definitions in the section define the arithmetic opera- tions for Rb numbers. The purpose in defining these operations is to supplement the canonical operations for R0 and 1 ∼ c1. Every Rb number can be decomposed and its pieces manipulated separately but the main purpose of defining special operations for Rb is to define new operations for expressions which are undefined under the arithmetic operations of R0 and 1 alone or whose structure vanishes under additive absorption. Definition 4.6. The arithmetic operations of Rb numbers with R0 numbers are − c1 − b = −c1 + b − − c1 + b = c1 − b ( ±c1 ∓ b − x if b 6= x ± c1 − b + x = x ± c1 − b = ±c1 if b = x ( ±1 − xb if x 6= 0 ±1 − b · x = x · ±1 − b = c c c undefined if x = 0 8 b ±1 − b <±1 ∓ if x 6= 0 c = c x x :undefined if x = 0 x = 0 : ± c1 − b Theorem 4.7. The quotient of a number x 2 R0 divided by a number y 2 Rb is identically zero. Proof. Let z be any non-zero real number such that x = z : y Real Numbers in the Neighborhood of Infinity 7 Since jjxjj < jjyjj, we have jjzjj < 1 which implies z 2 R0. All R0 numbers have a multiplicative inverse. We find, therefore, that x = 1 () x = zy : zy The hat on c1 only suppresses additive absorption so zy = z · ± c1 − b = ± c1 − zb : This delivers a contradiction because it requires that x is a real number in the neighborhood of infinity while we have already defined it to be a real number in the neighborhood of the origin. Therefore, the only possible numerical value for x=y is 0. Definition 4.8. The arithmetic operations of Rb numbers with Rb numbers are ( ±c1 ∓ b + a if a 6= −b ± c1 − b ± c1 − a = ±c1 if a = −b ± c1 − b ∓ c1 − a = ± a − b c1 − b c1 − a = undefined 1 − b c = undefined : c1 − a Theorem 4.9.