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If the and are real numbers, a more elegant proof is available: the identity expresses the fact that the absolute value of the product of two quaternions is equal to the product of their absolute values, in the same way that the Brahmagupta – Fibonacci two-square identity does for complex numbers.
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If and are
If it were not for an old professor who made me read the classics I would have been stymied on what to do, and now I understand why they are classics ; ;
If I even hint at it do you think it will matter that you are his nephew -- and not even a blood nephew ''??
If the circumstances are faced frankly it is not reasonable to expect this to be true.
If his dancers are sometimes made to look as if they might be creatures from Mars, this is consistent with his intention of placing them in the orbit of another world, a world in which they are freed of their pedestrian identities.
If love reflects the nature of man, as Ortega Y Gasset believes, if the person in love betrays decisively what he is by his behavior in love, then the writers of the beat generation are creating a new literary genre.
If to be innocent is to be helpless, then I had been -- as are we all -- helpless at the start.
Defoe then commented, `` If they Could Draw that young Gentleman into Their Measures They would show themselves quickly, for they are not asham'd to Say They want only a head to Make a beginning ''.
If Jews are identified as a religious body in a controversy that comes before a national or international tribunal, it is obviously compatible with the goal of human dignity to protect freedom of worship.
If they are right, they will prevail of and by themselves.
Without preliminaries, Esther asked him, `` If you are a world citizen, will you take Garry Davis' place in his tent while he goes to the hospital ''??
If we are to believe the list of titles printed in Malraux's latest book, La Metamorphose Des Dieux, Vol. 1 ( ( 1957 ), he is still engaged in writing a large novel under his original title.
If the would-be joiner asks these questions he is not likely to be duped by extremists who are seeking to capitalize on the confusions and the patriotic apprehensions of Americans in a troubled time.
If we break the minister to our bit, we are buying back our own sins.
If the record buyer's tastes are somewhat eclectic or even the slightest bit esoteric, he will find them satisfied on educational records.
If Daddy's books are out of bounds his own picture books are not.
If it will simply delay the debates until the qualifications are closed next spring, and then carry all the candidates on a tour of debates, it can provide a service to the state.
If they are to be commended for foresight in their planning, what then is the judgment of a town council that compounds this problem during the planning stage??
If the Communists are sincere in wanting a united, neutral and disarmed Germany, it might well be advantageous for the German people in this nuclear age.
If only this could be done more often -- with such heartening results -- many of the earth's `` big problems '' would shrink to the insignificances they really are.
If any are left, presently, we may expect to see signs specifically prohibiting the feeding of them too.
If the raw population figures are crucially relevant, then it is idle to think of liberation, as idle as to suppose that Poland might liberate Russia.
If it is not enough that all of our internationalist One Worlders are advocating that we join this market, I refer you to an article in the New York Times' magazine section ( Nov. 12, 1961 ), by Mr. Eric Johnston, entitled `` We Must Join The Common Market ''.
If the UN troops are not mercenaries then the Hessians were not mercenaries either.
If and real
If competition beats you to it, this exciting new product era can have real headaches in store.
If the photographically realistic continuity of dreams, however bizarre their combinations, denies that it is purely a composition of the brain, it must be compounded from views of diverse realities, although some of them may never be encountered in what we are pleased to call the real life.
I kept saying, `` If I could just build up a reputation for myself, make some real money, get to be well known as an illustrator -- like Peter Askington, for instance -- then I could take some time off and paint ''.
If one quantizes a real scalar field, then one finds that there is only one kind of annihilation operator ; therefore, real scalar fields describe neutral bosons.
Existence is the principle that gives reality to an essence not the same in any way as the existence: " If things having essences are real, and it is not of their essence to be, then the reality of these things must be found in some principle other than ( really distinct from ) their essence.
If X is a Banach space and K is the underlying field ( either the real or the complex numbers ), then K is itself a Banach space ( using the absolute value as norm ) and we can define the continuous dual space as X ′ = B ( X, K ), the space of continuous linear maps into K.
If X is a real Banach space, then the polarization identity is
If A is a self-adjoint operator, then is always a real number ( not complex ).
If the Czech ethnic origin was to be stressed, combinations like " Bohemian of Bohemian language "( Čech českého jazyka ), " a real Bohemian " ( pravý Čech ) etc.
If the exponent r is even, then the inequality is valid for all real numbers x.
If the wave function is regarded as ontologically real, and collapse is entirely rejected, a many worlds theory results.
If wave function collapse is regarded as ontologically real as well, an objective collapse theory is obtained.
If the eigenvalues are all positive, then the frequencies are all real and the stationary point is a local minimum.
If zeal for orthodoxy caused him to overstep the limits of discretion, his real attitude towards Rome is sufficiently clear.
If a simply stable system response neither decays nor grows over time, and has no oscillations, it is marginally stable: in this case the system transfer function has non-repeated poles at complex plane origin ( i. e. their real and complex component is zero in the continuous time case ).
If this limit exists, then it may be computed by taking the limit as h → 0 along the real axis or imaginary axis ; in either case it should give the same result.
If x and y are real numbers, and if the graph of y is plotted against x, the derivative measures the slope of this graph at each point.
If a DBMS system responses users ' request in a given time period, it can be regarded as a real time database.
If the matrix entries are real numbers, the matrix can be used to represent two linear mappings: one that maps the standard basis vectors to the rows of, and one that maps them to the columns of.
If this operation has to be done in real time video games there is an easy trick to boost performance.
* If we think of as the set of real numbers, then the direct product is precisely just the cartesian product,.
* If we think of as the group of real numbers under addition, then the direct product still consists of.
* If we think of as the ring of real numbers, then the direct product again consists of.
The best known example is the ring of quaternions H. If we allow only rational instead of real coefficients in the constructions of the quaternions, we obtain another division ring.
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