3. Field properties of real numbers Flashcards
What are FIELD PROPERTIES of real numbers?
Basic assumptions about how real numbers behave.
What is a BINARY OPERATION?
An operation (such as addition) that works on exactly two numbers of a set at a time.
What is the definition of a CLOSED SET under an operation?
If the outcome of a binary operation is also a member of the same set, the set is said to be closed under the operation.
Is the set of natural numbers closed under addition?
Yes. Adding any two numbers in the set always produces another member of the set. 1 + 1 = 2 for example.
Is the natural set of numbers closed under subtraction?
No. Because 2 - 3 = -1. Negative one is not an element of the set of natural numbers.
How many FIELD PROPERTIES does the set of REAL NUMBERS have?
Six.
What is F1:Closure for the Set of Real Numbers? (Addition)
a + b is a real number
What is F1: Closure for the Set of Real Numbers? (Multiplication)
(a)(b) is a real number
What is F2: Commutative for the set of real numbers? (Addition)
a + b = b + a
What is F2: Commutative for the set of real numbers? (Multiplication)
ab = ba
What is F3: Associative for the set of real numbers? (Addition)
(a + b) + c = a + (b + c)
What is F3: Associative for the set of real numbers? (Multiplication)_
(ab)c = a(bc)
**What is F4: Identity for the set of real numbers? (Addition)
0 is the real number such that a + 0 = 0 +a = 0
** What is F4: Identity for the set of real numbers? (Multiplication)
1 is the real number such that (a)(1) = (1)(a) = 0
What is F5: Inverse for the set of real numbers? (Addition)
For each real number a, -a exists such that a + (-a) = (-a) + a = 0
What is F5: Inverse for the set of real numbers? (Multipliication)
For each real number a except 0, 1/a exists such that a(1/a) = (1/a)a = 1
What is F6: Distributive for the set of real numbers? (Addition/Multiplication)
a (b + c) = ab + ac and (b + c) a = ba + ca
How is a/b defined in terms of multiplication?
a (1/b)
How i the difference a - b defined in terms of addition?
a + (-b)
In a (1/b) or a/b, which variable is the dividend?
a
In a (1/b) or a/b, which variable is the divisor?
b
Is division by zero define?
No it is not.
Whenever a variable appears in the denominator of a fraction we assume…
that it cannot have a value that makes the denominator zero.
A negative sign that precedes a parenthesized expression may be interpreted as…
take the opposite of whatever is in the parentheses. (Sometimes it is helpful to replace the negative sign by -1 and to multiply.
