真正有意义的是旅途中不期而至的快乐。

It is the unexpected joy on the journey that really has a meaning.

youdao

然而，假如多重打击同时不期而至呢？

But what if several blows land at the same time?

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这一切在日益暗淡的金融背景下不期而至。

This comes against an increasingly bleak backdrop.

youdao

So always amazing to us how we go into the problem, our eye or mind can see one class of solutions, but the math will tell you sometimes there are new solutions and you've got to respect it and understand and interpret the unwanted solutions, and this is a simple example where you can follow what the meaning of the second solution is.

解决问题的方式总能让人惊喜不已,我们的眼睛或思想可以看见一类解,但数学有时会告诉你还有新的解,你不能忽视它,而是解释这些不期而至的解,这只是一个简单的例子,由此你可以理解第二个解的意义

耶鲁公开课 - 基础物理课程节选