## Metadata - Author: [[Timothy Gowers]] - Full Title: Mathematics - Category: #books ## Highlights - The real number system consists of all numbers that can be represented by infinite decimals. This concept is more sophisticated than it seems, for reasons that will be explained in Chapter 4. For now, let me say that the reason for extending our number system from rational to real numbers is similar to the reason for introducing negative numbers and fractions: they allow us to solve equations that we could not otherwise solve. ([Location 573](https://readwise.io/to_kindle?action=open&asin=B000SEP2T2&location=573)) - If we introduced i in order to have a solution to the equation x2 = −1, then what about other, similar equations such as x4 = −3, or 2x6 + 3x + 17 = 0? Remarkably, it turns out that every such equation can be solved within the complex number system. In other words, we make the small investment of accepting the number i, and are then repaid many times over. This fact, which has a complicated history but is usually attributed to Gauss, is known as the fundamental theorem of algebra and it provides very convincing evidence that there is something natural about i. ([Location 611](https://readwise.io/to_kindle?action=open&asin=B000SEP2T2&location=611)) - One of the greatest virtues of the abstract method is that it allows us to make sense of familiar concepts in unfamiliar situations. The phrase ‘make sense’ is quite appropriate, since that is just what we do, rather than discovering some pre-existing sense. ([Location 632](https://readwise.io/to_kindle?action=open&asin=B000SEP2T2&location=632)) - What I have just said in the last paragraph is far from obvious: in fact it was one of the great discoveries of the early 20th century, largely due to Frege, Russell, and Whitehead (see Further reading). This discovery has had a profound impact on mathematics, because it means that any dispute about the validity of a mathematical proof can always be resolved. ([Location 748](https://readwise.io/to_kindle?action=open&asin=B000SEP2T2&location=748))