The difference between 认为 and 以为
The verbs 认为 and 以为 are translated as “to think”, “to believe”, but they are used in different situations.
The verb 认为
认为 
is used to express an opinion in which the speaker is sure of himself.
is used to express an opinion in which the speaker is sure of himself.Examples
他们认为他行为反常
They think he acts unusual
tāmen rènwéi tā xíngwéi fǎncháng
他们认为他行为反常
What do you think about this job?
nǐ rènwéi zhè piān lùnwén zěnyàng
The verb 以为
以为 is also used to express an opinion. However, 以为 indicates that an opinion is wrong.
is also used to express an opinion. However, 以为 indicates that an opinion is wrong.Examples
我以为你还在北京呢
I thought you were still in Beijing (but it turns out you weren’t)
wǒ yǐwéi nǐ hái zài běijīng ne
起初他们都以为我是个疯子
At first they all thought I was crazy (actually, I wasn’t)
qǐchū tāmen dōu yǐwéi wǒ shì gè fēngzi