1) The mathematical system has real numbers as elements and defines the operation @ as a@b=(a-b)+a/b.
2) The operation @ is not closed, commutative, or associative for real numbers since division by zero is undefined and order of operations affects results.
3) There is no identity element for @ since no single element i satisfies a@i=i@a=a for all real numbers a.
Presentacion de matematica
Definición de Conjuntos.
Operaciones con conjuntos.
Números Reales
Desigualdades.
Definición de Valor
Absoluto
Desigualdades con
Valor Absoluto
Presentacion de matematica
Definición de Conjuntos.
Operaciones con conjuntos.
Números Reales
Desigualdades.
Definición de Valor
Absoluto
Desigualdades con
Valor Absoluto
Hi ....these slides have been taken from resources shared by Fiona Clark, Group 4 curriculum manager of IBO, in the DP workshop in Mumbai, November 2011
Rajashree Basu
Math 7 lesson 11 properties of real numbersAriel Gilbuena
At the end of the lesson, the learner should be able to:
recall the different properties of real numbers
write equivalent statements involving variables using the properties of real numbers
Lecture Notes on Number Theory : Introduction to Numbers, Natural Numbers, Integers, Rational and Irrational Numbers, Real Number
Mathematical Induction.
1. Examples of Mathematical Systems(Blitzer, Thinking Mathematically, 4e, Section 13.1) By Gideon Weinstein
2. Example 1 Mathematical System = Elements and Operation(s) Elements = “Things” are the integers Operation # = “Action on two” is defined by a#b=ab+(a+b) Now we check each of the properties and see which ones the integers have under # Closure, Commutative, Associative, Identity, Inverse
3. a#b=ab+(a+b) Closure: is a#b always an integer? YES, because ab and (a+b) are integers because we know integers are closed under addition and multiplication
4. a#b=ab+(a+b) Commutative: is a#b = b#a all the time? Better check by working it out…. Compute a#b and get ab+(a+b). Compute b#a and get (ba)+(a+b). Is ab+(a+b) = ba+(b+a) always true? YES, because ba=ab and b+a=a+b because the integers are commutative for addition and multiplication
5. a#b=ab+(a+b) Associative: is (a#b)#c=a#(b#c)? YES, left as an exercise for the listener. HINT: use the same trick as before to reduce it to a question about whether the integers are associative under multiplication and addition.
6. a#b=ab+(a+b) Identity: Is there a special element i so that a#i=a? Try to figure it out. We know a#i=ai+(a+i) by the definition of #. And we are trying to solve a#i = a. So we need to solve a=ai+(a+i) for i a=ai+(a+i) a=ai+a+i by simplifying parentheses 0=ai+i by subtracting a from both sides 0=i(a+1) by factoring out the i 0/(a+1)=i by dividing by (a+1) 0=i because zero divided by anything is zero So this algebra tells us that 0 acts as the identity Let’s double-check: a#0=a0+(a+0)=0+a=a YES, 0 is the identity for #
7. a#b=ab+(a+b) Inverse: Since 0 is the identity, this becomes the question that given an a, is it possible to find an integer x so that a#x=0? As before, we try to figure it out by solving the equation a#x=0 for x. Since a#x=ax+(a+x), you need to try to solve ax+(a+x)=0 for x ax+(a+x)=0 ax+a+x =0 by simplifying parentheses ax+x+a =0 because addition of integers is commutative x(a+1)+a =0 by factoring x(a+1)=-a by subtracting a from both sides x=-a/(a+1) by dividing by a+1 on both sides. It is NOT always an integer, so the inverse does NOT always exist. NO, # does not have the Inverse Property
8. Example 2 Mathematical System = Elements and Operation(s) Elements = “Things” are the real numbers Operation @ = “Action on two” is defined by a@b=(a-b)+a/b Now we check each of the properties and see which ones the real numbers have under @ Closure, Commutative , Associative , Identity, Inverse
9. a@b=(a-b)+a/b Closure: is a@b always a real number? NO. It is true that (a-b) is always a real number because real numbers are closed under subtraction. But a/b is not always a real number because a/0 is undefined and therefore not a real number. we know integers are closed under addition and multiplication.
10. a@b=(a-b)+a/b Commutative: is a@b = b@a all the time? Check by working it out…. Compute a@b and get (a-b)+a/b. Compute b@a and get (b-a)+b/a. Is (a-b)+a/b= (b-a)+b/a always true? It really doesn’t feel like it could be true because division and subtraction give different results when done in the opposite order. Let’s try a couple of numbers like 5 and 10. 5@10=(5-10)+5/10=-5+0.5=-4.5 10@5=(10-5)+10/5=5+2=7 -4.5 and 7 and NOT equal, so NO, @ is not commutative.
11. a@b=(a-b)+a/b Associative: is (a@b)@c=a@(b@c)? NO, left as an exercise for the listener. HINT: finding a counterexample.
12. a@b=(a-b)+a/b Identity: TWO questions, because @ is not commutative Is there a special element i so that a@i=a AND i@a=a for any value of a? It turns out the algebra is quite horrific, so it would not be a textbook or exam question. The answer is NO, by the way. It might be legitimate to ask something like “Could the number 2 be the identity element for the operation @?” You would check by calculating if a@2=a=2@a, and you’d very quickly see it is not true.
13. a@b=(a-b)+a/b Inverse There can’t be an inverse for every element, because the identity doesn’t exist. The same horrific algebra done before does show that 1@1=1 (and you might stumble on it just by exploring @ for some numbers), so in some sense 1 is it’s own inverse and its own identity, but that is not the standard way to use those terms