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nietzsche
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My prof gave us twelve basic properties of numbers, and I think I'm supposed to use those in my proof, but I'm not sure how to incorporate them.
The properties are:
P1 Associative law for addition
P2 Additive identity
P3 Additive inverse
P4 Commutative law for addition
P5 Associative law for multiplication
P6 Multiplicative identity
P7 Multiplicative inverse
P8 Commutative law for multiplication
P9 Distributive law
P10 Trichotomy law
P11 Closure under addition
P12 Closure under multiplication
Prove that if 0 < a < b, then
[tex]a < \sqrt{ab} < \frac{a + b}{2} < b[/tex]
N/A
Part I:
[tex]
\begin{align*}
a&<b\\
a^2&<ab\\
\sqrt{a^2}&<\sqrt{ab}\\
a&<\sqrt{ab}
\end{align*}
[/tex]
Part II:
[tex]
Suppose:
\begin{align*}
\sqrt{ab}&\geq\frac{a+b}{2}\\
ab&\geq\left(\frac{a+b}{2}\right)^2\\
ab&\geq\frac{a^2+2ab+b^2}{4}\\
4ab&\geq a^2+2ab+b^2\\
0&\geq a^2-2ab+b^2\\
0&\geq (a-b)^2\\
\end{align*}
[/tex]
[tex]
but:
\begin{align*}
0&<(a-b)^2\\
\therefore \sqrt{ab}&<\frac{a+b}{2}
\end{align*}
[/tex]
Part III:
[tex]
\begin{align*}
a&<b\\
a+b &< b+b\\
a+b &< 2b\\
\frac{a+b}{2} &< b
\end{align*}
[/tex]
So I'm able to prove them, but I don't know if I used the properties correctly (if at all). Any opinions or suggestions?
The properties are:
P1 Associative law for addition
P2 Additive identity
P3 Additive inverse
P4 Commutative law for addition
P5 Associative law for multiplication
P6 Multiplicative identity
P7 Multiplicative inverse
P8 Commutative law for multiplication
P9 Distributive law
P10 Trichotomy law
P11 Closure under addition
P12 Closure under multiplication
Homework Statement
Prove that if 0 < a < b, then
[tex]a < \sqrt{ab} < \frac{a + b}{2} < b[/tex]
Homework Equations
N/A
The Attempt at a Solution
Part I:
[tex]
\begin{align*}
a&<b\\
a^2&<ab\\
\sqrt{a^2}&<\sqrt{ab}\\
a&<\sqrt{ab}
\end{align*}
[/tex]
Part II:
[tex]
Suppose:
\begin{align*}
\sqrt{ab}&\geq\frac{a+b}{2}\\
ab&\geq\left(\frac{a+b}{2}\right)^2\\
ab&\geq\frac{a^2+2ab+b^2}{4}\\
4ab&\geq a^2+2ab+b^2\\
0&\geq a^2-2ab+b^2\\
0&\geq (a-b)^2\\
\end{align*}
[/tex]
[tex]
but:
\begin{align*}
0&<(a-b)^2\\
\therefore \sqrt{ab}&<\frac{a+b}{2}
\end{align*}
[/tex]
Part III:
[tex]
\begin{align*}
a&<b\\
a+b &< b+b\\
a+b &< 2b\\
\frac{a+b}{2} &< b
\end{align*}
[/tex]
So I'm able to prove them, but I don't know if I used the properties correctly (if at all). Any opinions or suggestions?