18/08/2026
Mathematical view of Real Numbers.....
Walter Rudin ( Principles of Mathematical Real Analysis) clearly describes and explains the concept of Real Numbers, essential in Real Analysis of Mathematical Concepts in Calculus, Statistics, Algebra and other branches. for further guidance check it out.
Real Numbers are a complete ordered field. This is the definition of Real Numbers.
Do you remember the types of numbers? Natural Numbers,Intergers, Rational Numbers,Prime and others...
Are natural Numbers Real Numbers? Are intergers Real Numbers? Are Rational Numbers Real Numbers? What qualifies a number set to be a real Number?
Field Axioms. Mathematical statements. There exist 11 axioms that must be satisfied for a number set to be a field. The Operations of Addition and Multiplication on sets must be satisfied.
1. Closure under Addition. Let a and b belong to set R.(a+b) e R.
2.Closure under multiplication. a,b are members of R. (a × b) e R.
3. commutative Law on Addition. a,b are in R, if a+b=b+a e R.
4. Commutative Law on *. a×b=b×a are members of R.
5. Assocatiatve laws on Addition. (a+b)+c=a+(b+c) in R
6.Associative law on Multiplication. ab=ba
7. There exist an identity 0. a+0=a
8. There exist an identity 1. a1=a
9. Additive inverse. a+ (-a)= 0
10.Multiplicative inverse holds. a.a-=1
11. Distributive law holds. a(b+c)= ab+ac
If all of these are satisfied that number set is qualified to be called a Field.
Order operations of must be satisfied, there are rules here also. Do a research.
HINT: REAL NUMBERS SATISFY THE 11 FIELD AXIOMS AND ORDER PROPERTIES.
NATURAL NUMBERS ARE NOT REAL NUMBERS BUT SUBSETS OF REAL NUMBERS.
Task: Logically Prove why Intergers are not Real Numbers and give sufficient reasons as a mathematical analysts at an Engineering Firm working on building an application software that solves real life problems in Architectural Designs.