Question

Verify that 1=0.99999....​

Answer 1

Answer:

guess you are probably confusing two different symbols ‘ := ' and ′= '. The symbol ‘ := ' , read as defined to equal, is used in definitions, whereas the symbol ‘ = ', which is read equals, is a relation linking two objects of the same type.

On writing x:=y , we intend to mean that the object x is defined to equal y , it is the definition for object x . For instance, we write A:=2 to express the fact that A is a natural number and we define it to be the number 2 . However, writing 2:=A makes no sense because it is not a definition of the object 2 . Instead, we can write 2:=S(1) , where S(1) denotes the successor of 1 , because 2:=S(1) is a definition of 2 (see ).

Now coming to the symbol ‘ = ', as stated earlier it is a relation (see and ) linking two mathematical objects of the same type. Not only equality is a relation, it is an equivalence relation. The equality satisfies following four axioms.

(Reflexive axiom). Given any object x , we have x=x .

(Symmetry axiom). Given any two objects x and y of same type, if x=y , then y=x .

(Transitive axiom). Given any three objects x , y , z of the same type, if x=y and y=z , then x=z .

(Substitution axiom). Given any two objects x and y of the same type, if x=y , then f(x)=f(y) for all functions or operations f .

Since equality satisfies the symmetry axiom, it makes perfect sense to write y=x in place of x=y . Thus, we can write 2=A as well as A=2 , both are equally good, but we cannot write 2:=A . The same is true for 0.999⋯:=1 . The notation 0.9999⋯ is a short hand for a more precise statement limn→∞0.99⋯9n . Since the above limit is 1 , we can write limn→∞0.99⋯9n:=1 to mean that the limit is 1 , however we cannot write 1:=limn→∞0.99⋯9n , unless you define 1 by that way. However, we can certainly write 0.999⋯=1 and 1=0.999⋯ since the relation = is symmetric.

Please mark me as brainliest.

Thank you and have a nice day