Euler considered also the question of enumerating the solutions of the indeterminate simultaneous equation in positive integers ax + by + cz + .
In general, partition problems present themselves which depend upon the solution of a number of simultaneous relations in integers of the form [lambda]1.
The succession of integers thus has a twofold signification: (i.
He showed that the number of ways of composing n with p integersdrawn from the series a, b, c, .
The theorem of reciprocity just established shows that the number of partitions of n into; parts or fewer, is the same as the number of ways of composing n with the integers 1, 2, 3, .
Thus, for instance, the smallest of the infinite integers is the limit of the finite integers, though all finite integers are at an infinite distance from it.
For these reasons among others our interest in Dickens's novels as integershas diminished and is diminishing.
Thus, all the integers form an unlimited or infinite group, while the integers between ten and one hundred (or the two-digit numbers) form a limited or finite group.
As we shall soon see, the series of integers is the purest type of a series so arranged.
Stoney, “may be obtained in one of its graphical forms by plotting down a series of integers as abscissas, and the successive atomic weights as ordinates.
Addition of integers (F) will, however, correlate less closely with any of the above, being apparently dependent on simpler and more isolated abilities.
How many times will he have exercised each of the various bonds involved in the four operations with integers shown below?
The ability to operate with integers plus the two habits of prefixing $ and separating dollars from cents in the result will enable him to operate with United States money.
The habits that have been confirmed by every multiplication and division by integers are, in this particular of 'the ratio of result to number operated upon,' directly opposed to the formation of the habits required with fractions.
The measurement of even so simple a matter as the efficiency of a pupil's responses to these tests in adding integers is really rather complex.
It will save more time in the work with integers than was spent in teaching it!
The scores of ten 6 B classes in a 12-minute test in computation with integers (the Courtis Test 7).
Why disturb the learning of the four operations with integers by adding at each step a second 'procedure with United States money'?
The aggregate of positive integers is an infinite aggregate.
It is therefore impossible to demonstrate the assumptions for certain integers without proving them for all; we must give up proof by example.
Addition, multiplication and involution are direct processes; and, if we start with positive integers, we continue with positive integers throughout.
Hence it follows that there are pairs ofintegers which never have been and never will be thought of by human beings, and that all of them deal with integers the product of which is over 100.
But we also know that the number of integers is infinite, and that only a finite number of pairs of integers ever have been or ever will be thought of by human beings.
We know that all pairs of integers the product of which is less than 100 have been actually multiplied together, and the value of the product recorded in the multiplication table.
Thus there poured into the West from many different directions, but chiefly from two right-angling directions which intersected on the Plains, a diverse population whose integers were later with phenomenal swiftness to merge and blend.
Here they lay, these integers of the line, huddled, jumbled.
The solution in integers of the indeterminate equation ax + by = c may be effected by means of continued fractions.
Each of these integers would be absolutely self-governing and independent, being only united for purposes of mutual good.
Such loyalty was a power and a help in the land, for it knew danger in every form; and anything which aided the cohesion of its integers was a natural asset.
To its own integers it would offer a lasting settlement of interests which at present conflicted, and a share in a new world-power.
The above list will hopefully give you a few useful examples demonstrating the appropriate usage of "integers" in a variety of sentences. We hope that you will now be able to make sentences using this word.
