I have heard that according to Hubble Telescope the universe is expanding like a balloon . A balloon expands in 3 dimensions . The Hubble Telescope sends reports to NASA and NASA said that universe is flat. Is NASA itself confused?
I am not a cosmologist, but I think the answer is roughly as follows...
I believe there may be a little terminology and analogy confusion here.
It's not accurate to imagine the universe expanding out into three dimensions from a point when we think of the "Big Bang."
The key word here is "surface," and the balloon analogy is a lower dimensional analogue. If my understanding is correct, the universe is expanding like the three dimensional hypersurface of a four dimensional balloon. This does not imply that the universe is a three dimensional manifold embedded in some four dimensional space, but rather it serves as a picture.
I suppose NASA intended to say that the universe is roughly flat in that it is broadly uniform, or that it appears similar in all directions.
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No, NASA is not confused. The concept of the universe being flat refers to its overall geometry, not its expansion. The Hubble Telescope's observations support the idea that the universe is expanding, similar to a balloon inflating. The term "flat" in cosmology signifies that on a large scale, parallel lines remain parallel, indicating a Euclidean geometry. This does not contradict the understanding of the universe's expansion.
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When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
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