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Ta có\(\frac{-543}{546}=\frac{3-546}{546}=\frac{3}{546}-1\)
\(\frac{-789}{792}=\frac{3-792}{792}=\frac{3}{792}-1\)
Vì \(546< 792\Rightarrow\frac{1}{546}>\frac{1}{792}\Rightarrow\frac{3}{546}>\frac{3}{792}\Rightarrow\frac{3}{546}-1>\frac{3}{792}-1\)
Do đó \(\frac{-543}{546}>\frac{-789}{792}\)
vì B<1 => \(B=\frac{10^{2013}+1}{10^{2014}+1}< \frac{10^{2013}+1+9}{10^{2014}+1+9}=\)\(\frac{10^{2013}+10}{10^{2014}+10}=\frac{10\left(10^{2012}+1\right)}{10\left(10^{2013}+1\right)}\)\(=\frac{10^{2012}+1}{10^{2013}+1}=A\)
\(\Rightarrow A>B\)
\(\frac{10^{2012}+1}{10^{2013}+1}=\frac{\left(10^{2012}+1\right)\cdot10}{\left(10^{2013}+1\right)\cdot10}=\frac{10^{2013}+10}{\left(10^{2013}+1\right)\cdot10}=\frac{10^{2013}+1+9}{\left(10^{2013}+1\right)\cdot10}=\frac{10^{2013}+1}{\left(10^{2013}+1\right)\cdot10}+\frac{9}{\left(10^{2013}+1\right)\cdot10}=\frac{1}{10}+\frac{9}{\left(10^{2013}+1\right)\cdot10}\left(1\right)\)
\(\frac{10^{2013}+1}{10^{2014}+1}=\frac{\left(10^{2013}+1\right)\cdot10}{\left(10^{2014}+1\right)\cdot10}=\frac{10^{2014}+10}{\left(10^{2014}+1\right)\cdot10}=\frac{10^{2014}+1+9}{\left(10^{2014}+1\right)\cdot10}=\frac{10^{2014}+1}{\left(10^{2014}+1\right)\cdot10}+\frac{9}{\left(10^{2014}+1\right)\cdot10}=\frac{1}{10}+\frac{9}{\left(10^{2014}+1\right)\cdot10}\left(2\right)\)Từ (1)(2) => A > B
Xét : \(B=\frac{196+197}{197+198}=\frac{196}{197+198}+\frac{197}{197+198}\)
Ta có : \(\frac{196}{197}>\frac{196}{197+198}\) và \(\frac{197}{198}>\frac{197}{197+198}\)
Hay A>B
Suy ra : \(\frac{196}{197}+\frac{197}{198}>\frac{196+197}{197+198}\)
Ta có \(\frac{196+197}{197+198}\)= \(\frac{196}{197+198}\)+ \(\frac{197}{197+198}\)
Vì \(\frac{196}{197}\)> \(\frac{196}{197+198}\)và \(\frac{197}{198}\)>\(\frac{197}{197+198}\)
\(\Rightarrow\)\(\frac{196}{197}\)\(+\)\(\frac{197}{198}\)> \(\frac{196+197}{197+198}\)
Vậy A > B
Có 196/197+198<196/197
197/197+198<197/198
Mà B=196+197/197+198=196/197+198 +197/197+198
Suy ra A>B
B<A
Ta có:
196/197+198 < 196/197
197/197+198 < 197/198
\(\Rightarrow\)196/197+198 + 197/197+198 <196/197 + 197/198 (1)
Mà B=196+197/197+198 = 196/197+198 + 197/197+198 (2)
Từ (1) và (2):
\(\Rightarrow\) B < 196/197 + 197/198
\(\Rightarrow\) B < A
Vậy trong hai số A và B, số A lớn hơn
A=-543/546=-181/182
B=-789/792=-263/264
Xét trên trục số thì:
Do A,B đều là phân số có giá trị<0
=>A>B