# Tagged Questions

53 views

### Definition of Rational/ Irrational Numbers reguarding denominators

The definition of a Irrational number is "Irrational numbers don't include integers OR fractions. However, irrational numbers can have a decimal value that continues forever WITHOUT a pattern." So ...
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### Prove that if $a$ is a rational number and $a^2$ is an integer then $a$ is an integer.

Question on a proof's review: Proof by contradiction: Suppose $a$ is not an integer. Then $a=p/q$ where $p$ and $q$ are coprime, $q$ is not 0, and $q$ is not 1. Then $a^2 = p^2/q^2$. This is ...
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### show that if $|x-\frac{m}{n}| \leq \epsilon$ then $n$ is very large

I am working on my calculus homework currently, and in order to solve a question, I need to prove this more simple statement: if $|x-\frac{m}{n}| \leq \epsilon$ for all $\epsilon>0$ then $n$ has ...
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### number between 17 and 18, and has a rational square root

"number between 17 and 18, and has a rational square root" Is there even one? They all keep coming up irrational for me
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### Let $a,b \in R$ where $a < b$. Prove that there exist a rational number $c$ and an irrational number $d$ such that $a <c<b$ and $a<d<b$.

Question : Let $a,b \in R$ where $a < b$. Prove that there exist a rational number $c$ and an irrational number $d$ such that $a <c<b$ and $a<d<b$. Hint: consider decimal expansions ...
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### Prove that if $y,z \in Q$ then $y^z \in A$

Question : Prove that if $y,z \in Q$ then $y^z \in A$ My attempt: Definition 2.7.8 states that a number s is an algebraic number when there exists some $p \in Z[x]$ such that $p(s) =0$. Let us ...
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### Show that if S=a+b√2 : a,b are rational numbers and T=r+s√3 :r,s are rational numbers, then$S \cap T$ = rational

Someone please correct a formatting error in the problem [still a newbie] ; "S&T" (And = upside down U) Here's a bonus question that was on a test we received that I couldn't figure out. I'd ...
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### Finding non-zero rational numbers to fit $a^2+b^2=C$

This is a question on my homework. Specifically, find non-zero rationals $a,b$ such that $a^2+b^2=9$. I think that this is related to work that Diophantus did, but I'm not really sure and I just don't ...
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### Does the set of rational numbers between 0 and 2 have the least upper bound property?

Let $A = \{ a \in Q : 0 < a < 2\}$ Does A have the least upper bound property? Definition: $A$ has the least upper bound property if $\forall$ nonempty $B \subseteq A$, if $B$ has an upper ...
261 views

### Proving supremum for non-empty, bounded subsets of Q iff supremum in R is rational

Let E be a nonempty bounded subset of ℚ. Prove that E has a supremum in ℚ if and only if its supremum in ℝ is rational and that in this case, the two are equal. This seems intuitive enough, and I ...
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### Show that a rational number has no good rational approximations

This is homework question. The teacher proved that if $a$ is irrational, there are infinitely many rational numbers $\frac{x}{y}$ such that $|\frac{x}{y} - a|<\frac{1}{y^2}$. What we need to ...
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### $n ^ 2 +5$, $n ^ 2 +10$ are rational number square

Seeking a nonzero rational number $n$, such that $n ^ 2 +5$, $n ^ 2 +10$ are rational number square。 This is a high school students asked the question, answer $n=\frac{31}{12}$, but no answer ...
### Proof that there is an infinite amount of rationals $t$, $0<t<1$ using the definitions of $=$ and $\leq$
This is a question to which the answer is of course intuitively obvious. We are asked to prove that there is an infinite amount of rationals $t$ with $0<t<1$ using the definitions of $=$ and ...