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This is always a non-negative real number, and it is the same as the Euclidean norm on considered as the vector space . Multiplying a quaternion by a real number scales its norm by the absolute value of the number. That is, if α is real, then
It follows that p 2 is a real number if and only p is either a real number (q = 0 and p = w) or a purely nonreal split quaternion (w = 0 and p = q). The structure of the subalgebra R [ p ] {\displaystyle \mathbb {R} [p]} generated by p follows straightforwardly.
A valid number sentence that is true: 83 + 19 = 102. A valid number sentence that is false: 1 + 1 = 3. A valid number sentence using a 'less than' symbol: 3 + 6 < 10. A valid number sentence using a 'more than' symbol: 3 + 9 > 11. An example from a lesson plan: [6] Some students will use a direct computational approach.
Quine's paradox is a paradox concerning truth values, stated by Willard Van Orman Quine. [1] It is related to the liar paradox as a problem, and it purports to show that a sentence can be paradoxical even if it is not self-referring and does not use demonstratives or indexicals (i.e. it does not explicitly refer to itself).
For real numbers, this formula is true if we substitute (arbitrarily) =, but is false if = It is the presence of a free variable, rather than the inconstant truth value, that is important; for example, even for complex numbers, where the formula is always true, it is still not considered a sentence.
State of the art embeddings are based on the learned hidden layer representation of dedicated sentence transformer models. BERT pioneered an approach involving the use of a dedicated [CLS] token prepended to the beginning of each sentence inputted into the model; the final hidden state vector of this token encodes information about the sentence ...
Even the trusty manual transmission that automakers have been phasing out for years now is increasing in desirability. “Manual transmissions were definitely at kind of a premium,” Hagerty said.
The long real line pastes together ℵ 1 * + ℵ 1 copies of the real line plus a single point (here ℵ 1 * denotes the reversed ordering of ℵ 1) to create an ordered set that is "locally" identical to the real numbers, but somehow longer; for instance, there is an order-preserving embedding of ℵ 1 in the long real line but not in the real ...