3. Field properties of real numbers

Question 1

What are FIELD PROPERTIES of real numbers?

Answer 1

Basic assumptions about how real numbers behave.

Question 2

What is a BINARY OPERATION?

Answer 2

An operation (such as addition) that works on exactly two numbers of a set at a time.

Question 3

What is the definition of a CLOSED SET under an operation?

Answer 3

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.

Question 4

Is the set of natural numbers closed under addition?

Answer 4

Yes. Adding any two numbers in the set always produces another member of the set. 1 + 1 = 2 for example.

Question 5

Is the natural set of numbers closed under subtraction?

Answer 5

No. Because 2 - 3 = -1. Negative one is not an element of the set of natural numbers.

Question 6

How many FIELD PROPERTIES does the set of REAL NUMBERS have?

Answer 6

Six.

Question 7

What is F1:Closure for the Set of Real Numbers? (Addition)

Answer 7

a + b is a real number

Question 8

What is F1: Closure for the Set of Real Numbers? (Multiplication)

Answer 8

(a)(b) is a real number

Question 9

What is F2: Commutative for the set of real numbers? (Addition)

Answer 9

a + b = b + a

Question 10

What is F2: Commutative for the set of real numbers? (Multiplication)

Answer 10

ab = ba

Question 11

What is F3: Associative for the set of real numbers? (Addition)

Answer 11

(a + b) + c = a + (b + c)

Question 12

What is F3: Associative for the set of real numbers? (Multiplication)_

Answer 12

(ab)c = a(bc)

Question 13

**What is F4: Identity for the set of real numbers? (Addition)

Answer 13

0 is the real number such that a + 0 = 0 +a = 0

Question 14

** What is F4: Identity for the set of real numbers? (Multiplication)

Answer 14

1 is the real number such that (a)(1) = (1)(a) = 0

Question 15

What is F5: Inverse for the set of real numbers? (Addition)

Answer 15

For each real number a, -a exists such that a + (-a) = (-a) + a = 0

Question 16

What is F5: Inverse for the set of real numbers? (Multipliication)

Answer 16

For each real number a except 0, 1/a exists such that a(1/a) = (1/a)a = 1

Question 17

What is F6: Distributive for the set of real numbers? (Addition/Multiplication)

Answer 17

a (b + c) = ab + ac and (b + c) a = ba + ca

Question 18

How is a/b defined in terms of multiplication?

Answer 18

a (1/b)

Question 19

How i the difference a - b defined in terms of addition?

Answer 19

a + (-b)

Question 20

In a (1/b) or a/b, which variable is the dividend?

Answer 20

a

Question 21

In a (1/b) or a/b, which variable is the divisor?

Answer 21

b

Question 22

Is division by zero define?

Answer 22

No it is not.

Question 23

Whenever a variable appears in the denominator of a fraction we assume…

Answer 23

that it cannot have a value that makes the denominator zero.

Question 24

A negative sign that precedes a parenthesized expression may be interpreted as…

Answer 24

take the opposite of whatever is in the parentheses. (Sometimes it is helpful to replace the negative sign by -1 and to multiply.