"Complex numbers are just two real numbers added together with I."
"How many real numbers are there between 0 and 1? An infinite number, right?"
"Surrounding every real number is an island, sometimes referred to as a monad, that contains an entire universe of infinitesimals."
"Conway's real genius is he was able to come up with an ordering and arithmetic operations that agree with the real number subfield."
"Conway introduced arithmetic operations that agree with the real number subfield."
"So my domain in this case is the set of all real numbers and there's a lot of ways we can express that this is one way that we can do it but that's basically it."
"The function taking lambda to the exponential of lambda A is a homomorphism of groups from the reals to the group."
"Every number x on the real number line has one of these star-friends, the number x plus star, which is 'confused' with x, but not equal to it."
"In fact, you get more than just all the fractions; you get all the other real numbers!"
"If I have an imaginary number in this hand and I have an imaginary number in this hand, oftentimes if I combine them together, I don't have an imaginary number anymore; I have a real number back."
"The important thing here is that the real part of \( p \) is bigger than zero. It doesn't matter what the imaginary part of \( p \) is."
"Our goal is to find all functions from the real numbers to the real numbers."
"We talk about what it really means to be a real number, what it really means to be a derivative, to be an integral."
"Real numbers are cleaner, more elegant than ones and zeros."
"Complex numbers have a real component and an imaginary component."
"The domain of F plus G would be all the X's in the real numbers such that X is in the domain of F and simultaneously X is in the domain of G."
"All these numbers together make up what's called the real numbers."
"You get a real part and an imaginary part; it's very important to be comfortable with imaginary numbers."
"If what is under the radical is a positive number, then you'll be able to take the square root of it and you'll get real answers."
"Set interval notation... deals with the study of real numbers."
"For N equals one, it's the usual R. For N equals two, you will get the complex numbers."
"Floating point numbers are an approximation to the set of real numbers."
"The domain of F is all the X's in the real numbers such that X is not equal to minus 1."
"We have a set of real numbers... all the fractions, all the decimals, all the integers we could have."
"Random variables are functions from the sample space to the real numbers."
"Squares of real numbers are always positive or zero."
"Real numbers... are used for positive or negative numbers that can or may have a decimal fraction."
"The real numbers are not countable."
"An inner product space over reals... produces a real number and inner products have these properties: they're symmetric, they are linear, and importantly they have this positive definite property."
"The domain is \( x \) can be any real number."
"The real numbers... will also have the property that they'll solve equations like x squared equals two but they will in some sense fill in the holes in the line, the number line that we began to get a picture of."
"The real numbers is a massive, massive great set, containing all integers, all of the rationals, loads of irrational numbers like the square root of 2, the transcendental numbers such as pi."
"Y can be any real number because this relation continues forever this way and this way."
"Y could be any real number but Y cannot equal 0."
"...we've been spending the first couple of weeks constructing the real numbers and now we're going to turn to a particular proof technique that's going to come up a lot as we move forward and that's the principle of induction."
"The domain, the x's which are included in this white graph, is going to be all real numbers."
"A real number is any number that can be located on a number line."
"The set of complex numbers is all numbers of the form a plus bi, where a and b are both real numbers, and i squared is negative 1."
"In regression, our input spaces are \( \mathbb{R}^d \) and what we're producing, our action, is a real number."
"A random variable can take values randomly from either the entire set of real numbers or a subset of real numbers."
"The numbers between 0 and 1 are uncountable."
"All the real numbers can be represented on the number line."
"Every non-empty set of real numbers that is bounded above has a least upper bound."
"For any two real numbers a and B where a is less than B, there is a rational number between them."
"Our range for logarithms are going to be all real numbers or from negative Infinity to positive Infinity."
"The real numbers often turn out to be the perfect number system in which to do math."
"In the real numbers, every Cauchy sequence is convergent."
"The most common metric for the real numbers is the Euclidean metric."
"Every Cauchy sequence of real numbers is guaranteed to be convergent to a real number."
"The real numbers are the completion of the rational numbers."
"The real numbers complete the number line; if you can point to it on the number line, it's a real number."
"Real numbers are really just a combination of all the rational numbers and all the irrational numbers."