If a work is divided into several large segments, a last-minute drawing of random numbers may determine the order of the segments for any particular performance.

If he can't play with Mommy's magazines, he should have some old numbers of his own.

If we are discussing differentiable complex-valued functions, then Af and V are complex vector spaces, and Af may be any complex numbers.

If he does, it's still better than an even chance he won't notice the transposition of the numbers, and if he should notice it, the thing can be passed off as an honest mistake.

* If it is required to use a single number X as an estimate for the value of numbers, then the arithmetic mean does this best, in the sense of minimizing the sum of squares ( x < sub > i </ sub > − X )< sup > 2 </ sup > of the residuals.

** If the set A is infinite, then there exists an injection from the natural numbers N to A ( see Dedekind infinite ).

If, however, the numbers are referring to allies it could possibly include forces from Tyre, Judah, Edom, and Moab.

If a is algebraic over K, then K, the set of all polynomials in a with coefficients in K, is not only a ring but a field: an algebraic extension of K which has finite degree over K. In the special case where K = Q is the field of rational numbers, Q is an example of an algebraic number field.

Let ( m, n ) be a pair of amicable numbers with m < n, and write m = gM and n = gN where g is the greatest common divisor of m and n. If M and N are both coprime to g and square free then the pair ( m, n ) is said to be regular, otherwise it is called irregular or exotic.

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 the exponent r is even, then the inequality is valid for all real numbers x.

If the convention B < sub > 1 </ sub >=− is used, this sequence is also known as the first Bernoulli numbers ( / in OEIS ); with the convention B < sub > 1 </ sub >=+ is known as the second Bernoulli numbers ( / in OEIS ).

If the shooter rolls one of these numbers on the come-out roll, this establishes the " point "-to " pass " or " win ", the point number must be rolled again before a seven.

If is a convex set, for any in, and any nonnegative numbers such that, then the vector

If n ≥ 1 and is an integer, the numbers coprime to n, taken modulo n, form a group with multiplication as operation ; it is written as ( Z / nZ )< sup >×</ sup > or Z < sub > n </ sub >< sup >*</ sup >.

If the numbers describing the position of the pyramid were changed and this process repeated, the pyramid would appear to move.

From the criterion it also follows that Carmichael numbers are cyclic .< ref > Proof sketch: If is square-free but not cyclic, for two prime factors and of.

This is a Cauchy sequence of rational numbers, but it does not converge towards any rational limit: If the sequence did have a limit x, then necessarily x < sup > 2 </ sup > = 2, yet no rational number has this property.

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 we look for solutions that have the form, where C is a constant, we discover the relationship, and thus must be one of the complex numbers or.

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 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 he had married her, he'd have been asking for trouble.

`` If you hadn't I'd have killed you ''.

If we have to we'll take him apart and see what he's made of ''!!

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 it were not that I knew who it was I could have mistaken it for my Aunt so well did her clothes fit him.

`` If you can conveniently let me have twenty dollars '', he wrote one friend in 1791 when he was Secretary of the Treasury.

If these people were denied a voice ( do they have a moral right to a voice??

If we want respect from ourselves or others, we will have to earn it.

If he had been `` liquidated '' in some way, he would have become a martyr, a rallying point for people who shared his ideas.

If our sincerity is granted, and it is granted, the discrepancy can only be explained by the fact that we have come to believe hearsay and legend about ourselves in preference to an understanding gained by earnest self-examination.

If you cut down these horrible buildings you'll have no more traffic jams.

If I am to speak the whole truth about my knowledge of love, I will have to stop trying to emulate the transcendant nightingale.

If you had screamed right there in the street where we stood, I could not have felt more fear.

If we grasp this opportunity to build an age of productive partnership between the less fortunate nations and those that have already achieved a high state of economic advancement, we will make brighter the outlook for a world order based upon security and freedom.

If his scholarship and formal musicianship were not all they might have been, Mercer demonstrated at an early age that he was gifted with a remarkable ear for rhythm and dialect.

If only for this modest masterpiece of military history, Blenheim is likely to be read and reread long after newer interpretations have perhaps altered our picture of the Marlborough wars.

If Gorton wanted peace and quiet for his complicated meditations this is where he should have had it.

If Krutch is correct, tragedy may have quite the opposite effect.

`` If you can firmly make the good knight sure to pleasure our Corporation '', Sturley wrote, `` besides that ordinary allowance for your diet you shall have 20 for recompence ''.

If he borrowed money from Shakespeare or with his help, he would now have been able to repay the loan.

If any one of them has any power to veto the Secretary General's decisions the nature of the organization will have changed.

If he can bounce back with one of those 25 home runs years, the club will have to be better off offensively.

If there were no West Berlin problem, imperialist quarters would have invented an excuse for stepping up the armaments race to try to solve the internal and external problems besetting the United States and its NATO partners.

If they also defeat the school bill, the GOP task force won't have much research to do.