# Doubt in law of reciprocal

Law of reciprocal: If both sides of inequality have same sign, while taking its reciprocal the sign of inequality gets reversed. Thus,

a >b> 0 → 1/a < 1/b

But if both sides of inequality have opposite sign, then while taking its reciprocal the sign of inequality does not change. Thus,

a < 0 < b → 1/a < 1/b

Q.1 What the rule meant by saying "both sides of inequality have same sign or different sign"? How to know which sign is different and which sign is same? Like the example in the above statements → a > b > 0 ; a < 0 < b

Now, in these two above examples, how to find if "both sides of inequality have same sign or different sign"?

Q.2 Why inequality sign gets reversed if "both sides of inequality have same sign"?

Q.3 Why inequality sign remains same if "both sides of inequality have opposite sign"?

Q.4 What is the reason behind flipping signs? Is there any simple explaination for it? Like some proof?

• These rules can of course be proved, but you would probably find the proofs tedious and not convincing. If you work out some examples and think about what happens on the number line they should be clear. Try $a = \pm 1$ and $b = \pm 2$ (four combinations of signs). – Ethan Bolker Nov 11 '18 at 0:51
• Part of your question suggests to me that you might be unaware that inequalities, like $a<0$ or $b>0$ tell you the signs of numbers. The sign of a real number $x$ is $+$ if $x>0$, and the sign is $-$ if $x<0$. (Apologies if I'm just telling you what you already knew.) – Andreas Blass Nov 11 '18 at 1:05
• @Ethan Bolker Oh ok. Yes sir i tried them. Now, I'm able to understand. – brahamdeep singh Nov 11 '18 at 1:24
• @Andreas Blass No need to apologise sir. Actually, i was very confused because of book. Thanks for your response. I'm able to understand now. – brahamdeep singh Nov 11 '18 at 1:26
• @brahamdeepsingh Good. Now you can answer your own question here, explaining what you understand and why, and accept your own answer (the check mark). Then it won't remain on the unanswered queue and attract unwanted attention. – Ethan Bolker Nov 11 '18 at 1:27