important quotes of the fault in our stars
Some people don't understand the promises they're making when they make them."
"Right, of course. But you keep that promise anyway. That's what love is. Love is keeping the promise anyway."
When Hazel and Isaac have this conversation, it leads up to a point between Augustus and Hazel's relationship later in the story. When Hazel commits to a relationship with Augustus, she promises to love Augustus always ("okay.") and when she first committed to a relationship with Augustus, she didn't know that she would be the one to deal with the other dying young, yet she does because she loves Augustus so much that she will keep her promise.
That's the thing about pain. It demands to be felt."
Augustus tells Isaac this after Isaac;s long time girlfriend breaks up with him, but this advice goes much farther than just over heartbreak. It's a life lesson not only for in this story, but also in real life. A person cannot ignore pain, because it will come back stronger the more you ignore it. Hazel (despite not wanting to feel it) is in pain a lot, whether it be because the love of her life (Augustus) is dead or because her cancer is making it so that she can't breathe.
There are infinite numbers between 0 and 1. There's .1 and .12 and .112 and an infinite collection of others. Of course, there is a bigger infinite set of numbers between 0 and 2, or between 0 and a million. Some infinities are bigger than other infinities. A writer we used to like taught us that. There are days, many of them, when I resent the size of my unbounded set. I want more numbers than I'm likely to get, and God, I want more numbers for Augustus Waters than he got. But, Gus, my love, I cannot tell you how thankful I am for our little infinity. I wouldn't trade it for the world. You gave me a forever within the numbered days, and I'm grateful.”
This quote is important because it is Hazel's way of saying that despite wanting more days for her to live, and more days for Augustus to live, she is extremely happy and glad for the days they had had together. It is a lesson to be learned that everyone has numbered days and so you should enjoy your "infinities", the moments with loved ones, that you have to the maximum.