Question

Prove the following statement directly from the definition of rational number. The difference of any two rational numbers is a rational number. -Select- integers complex numbers -Select- b?0 and d? 0 a?0 and c?0 b+d?0 negative numbers r=\frac{a}{b} and s = \frac{c}{d} for some [real numbers a, b, c, and d with a + c ? 0 positive numbers Proof: Suppose r and s are any two rational numbers. By definition of rational, r Write r - s in terms of a, b, c, and d as a quotient of two integers whose numerator and denominator are simplified as much as possible. The result is the following. r - s = -Select- products and differences of integers are rational numbers Both the numerator and the denominator are integers because products and quotients of integers are rational numbers products and differences of integers are integers products and quotients of rational numbers are integers In addition, bd ? 0 by the -Select- zero product property definition of integer definition of rational Hence r - s is a -Select- of two integers with a nonzero denominator, and so by definition of rational, r - s is rational. product sum difference quotient

          Prove the following statement directly from the definition of rational number.
The difference of any two rational numbers is a rational number.
-Select-
integers
complex numbers
-Select-
b?0 and d? 0
a?0 and c?0
b+d?0
negative numbers
r=\frac{a}{b} and s = \frac{c}{d} for some [real numbers
a, b, c, and d with a + c ? 0
positive numbers
Proof: Suppose r and s are any two rational numbers. By definition of rational, r
Write r - s in terms of a, b, c, and d as a quotient of two integers whose numerator and denominator are simplified as much as possible. The result is the following.
r - s =
-Select-
products and differences of integers are rational numbers
Both the numerator and the denominator are integers because products and quotients of integers are rational numbers
products and differences of integers are integers
products and quotients of rational numbers are integers
In addition, bd ? 0 by the -Select-
zero product property
definition of integer
definition of rational
Hence r - s is a -Select- of two integers with a nonzero denominator, and so by definition of rational, r - s is rational.
product
sum
difference
quotient

Prove the following statement directly from the definition of rational number.
The difference of any two rational numbers is a rational number.
-Select-
integers
complex numbers
-Select-
b?0 and d? 0
a?0 and c?0
b+d?0
negative numbers
r=(a)/(b) and s = (c)/(d) for some [real numbers
a, b, c, and d with a + c ? 0
positive numbers
Proof: Suppose r and s are any two rational numbers. By definition of rational, r
Write r - s in terms of a, b, c, and d as a quotient of two integers whose numerator and denominator are simplified as much as possible. The result is the following.
r - s =
-Select-
products and differences of integers are rational numbers
Both the numerator and the denominator are integers because products and quotients of integers are rational numbers
products and differences of integers are integers
products and quotients of rational numbers are integers
In addition, bd ? 0 by the -Select-
zero product property
definition of integer
definition of rational
Hence r - s is a -Select- of two integers with a nonzero denominator, and so by definition of rational, r - s is rational.
product
sum
difference
quotient

Added by Christopher C.

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