Analysis
The set ℝ of all real numbers can be characterized by axioms. (The real numbers can also be constructed by successively extending the domains of the natural numbers ℕ, the integers ℤ, and the rational numbers ℚ. We cannot go into this approach.)
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Analysis
3.1
Differential and integral calculus of functions of one and several variables
3.1.1
Real numbers
3.1.1.1
System of axioms for the real numbers
The set R of all real numbers can be characterized by axioms. (The real numbers can also be constructed by succes~ively extending the domains of the natural numbers N, the integers 71., and the rational numbers Q. We cannot go into this approach.) Axioms of addition
1. For any two real numbers a and b there is a real number a + b, called the sum of a and b; here a and b are called the summands of a + b. 2. a + b = b + a for all a, bE R (commutative law of addition). The symbol = of equality denotes the identity of two real numbers; if a and b are distinct real numbers, one writes a oF b. 3. a + (b + c) = (a + b) + c for all a, b, c E IR (associative law of addition). 4. There exists a real number 0 such that a + 0 = a for all a E JR. This number 0 E JR is called zero. 5. For every a E R there is a number bE JR such that a + b = O. Thus, the set JR is a commutative group under addition. Consequence: For two real numbers a and b there exists exactly one real number x such that a + x = b. This number x is called the difference of b and a and is denoted by b - a. One says: a is subtracted from b. For 0 - a one writes -a. Hence, the number b in 5. is uniquely determined. Further, from the axioms 1.-5. one obtains the following consequences: Consequences:
a) For all aE JR: a
= -(-a);
= O. = d - c=- b + c = a + d; (b + d) - (a + c) = (b - a) + (d (b + c) - (a + d) = (b - a) - (d -0
b) For all a, b, c, dE IR: b - a
c); c).
Axioms of multiplication
I. here 2. 3. 4. 5. 6.
For any two real numbers a and b there is a real number a . b, called the product of a and b; a and b are called the factors of a . b. a . b = b . a for all a, bE IR (commutative law of multiplication). a . (b . c) = (a' b) . c for all a, b, c E IR (associative law of multiplication). There exists a real number I such that I . a a for all a E JR. The number I E JR is called one. For every real number a oF 0 there is a real number b such that a • b = I. a . (b + c) = a . b + a . c for all a, b, c E JR (distributive law).
=
Thus, the set of all non-zero real numbers is a commutative group under multiplication. Consequences of the axioms of addition and multiplication:
1. From a . b = 0 it follows that a
= 0 or b =
O.
I. N. Bronshtein et al., Handbook of Mathematics © Springer Fachmedien Wiesbaden 1979
216
3.1.1.1
3.1 Differential and integral calculus
2. For two real numbers a and b with a 'I 0 there exists eaxctly one real number x such that a'x = b. This number x is called the quotient of b and a and is denoted by b/a. One says: b is divided by a. The number b is called the numerator and a the denominator. I
3. For all aE R \ {O}:-I/a
= a.
b d 4. For all a, b, c, dE R \ {O}: - = - b· c = a' d; a c d b · db/a b· c b -; . ~ = -;;:-;:' dr; = -;;-:-J
a 5. For all a, bE R, for all c E R \ {O}: c 6. ForallaER: -a=(-I)·a.
7. For alla,b E R: -(a·b)
=
+ -cb
a +b = --. c
(-a)·b.
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