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The set of rational numbers form an Abelian group under addition. That is, if x, y and z are any rational numbers, then
- x + y is a rational number (closure)
- (x + y) + z = x + (y + z) (associative property), and this means that either can be written as x + y + z
- there is a rational number, 0, such that x + 0 = x = 0 + x (existence of identity)
- for any x, there is a rational number x' such that x + x' = 0 = x' + x (existence of additive inverse). x' is denoted by -x.
- x + y = y + x (commutative or Abelian property).
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How are the rules for adding integers applied to operations with rational numbers?
You need the rules of multiplication as well as of addition. But multiplication of integers can be viewed as repeated addition. Thus, if p/q and r/s are two rational numbers then their sum is(p*s + q*r)/(q*s)
Are rational numbers closed under addition?
Yes.
Are rational numbers under addition a group?
Yes.
Can the sum of two rational numbers always be written as a fraction?
Yes, the set of rational numbers is closed under addition.
What are the identity elements for the addition and multiplication of rational numbers?
For addition, 0 and for multiplication, 1.
How are the rules for adding integers applied to operations with rational numbers?
You need the rules of multiplication as well as of addition. But multiplication of integers can be viewed as repeated addition. Thus, if p/q and r/s are two rational numbers then their sum is(p*s + q*r)/(q*s)
Is a rational number closed for addition and for multiplication?
A rational number is not. But the set of ALL rational numbers is.
What are the rules for dividing rational numbers are the same as the rules for dividing integers?
The rules are the same.
How do you know that the sum of (-2 34) and 59 rational?
Because both of those numbers are rational. The sum of any two rational numbers is rational.
Can you add two rational numbers and get an irrational number?
No. The set of rational numbers is closed under addition (and multiplication).
Are the rational numbers closed under addition?
Yes.
Can rational numbers be closed under addition?
Yes, they can.
Are rational numbers closed under addition?
Yes.
Are rational numbers under addition a group?
Yes.
Can the sum of two rational numbers always be written as a fraction?
Yes, the set of rational numbers is closed under addition.
Is the sum of rational numbers always rational?
Yes. In general, the set of rational numbers is closed under addition, subtraction, and multiplication; and the set of rational numbers without zero is closed under division.
What are the identity elements for the addition and multiplication of rational numbers?
For addition, 0 and for multiplication, 1.
