We can get a better idea of the meaning of imaginary numbers by examining some specific cases. This then implied a fourth dimension to the complex number space, a dimension given by the product of i and j, to which Hamilton then assigned the variable k:  ij = k. Even better, this is another imaginary number. We can even factor 2, as (1 + i)(1 - i), even though 2 is not generated by 4n + 1. Pro, Vedantu The reciprocal of i is equal to -i:  1/i = -i. However, we have already seen, in effect, the move of understanding imaginaries as pairs of real numbers; for the use of the "complex number plane" is no less than to substitute something real, the plane, for the meaning of √-1, something which, as it happens, is governed by an ordered pair of coordinates. There seem to me to be several problems with such a principle. This leaves me wondering if Martínez appreciates the meaning of the issue. Thus, although division is not symmetrical, and we cannot commute the divisor and divident as we could the multiplier and multiplicand, we actually can commute the signs. This does mean, however, that the notion, as found in Asimov above, that the complex number plane is somehow the real meaning of complex numbers, is put in an odd position. If I want to calculate the square roots of -4, I can say that -4 = 4 × -1. But since we can always transpose a negation from a divisor to a dividend, we can avoid the contradiction -- just as we avoid division by zero -- so perhaps this is just the reductio ad absurdum of division by negatives. The diagram illustrates the Bose-Einstein Statistics in the way they were explained by Albert Einstein himself in a 1925 letter to Irwin Schrödinger [cf. It now becomes evident what happens when both multiplier and multiplicand are negative. If we think that all reality is absolutely limited by non-contradiction, then there will be no square circles. They assumed [?] This is "simpler"? (0, 3). The most troubling thing about the approaches I see are the implications of either formalism or conventionalism. The laws of Geometry and numbers seemed like the laws of God, and therefore mathematics was valued as a preparation to discipline the mind for studies of metaphysics and theology. It will work, indeed, if a negative multiplier means that we substract rather than add the numbers. Ordinarily, if we toss two coins in the air, half the time (2/4) we will get one coin heads and the other tails. Bombelli’s Breakthrough 5. The exception to algebraic commutation creates some problems for determining values involving i, j, & k. Thus, if we begin with ij = k and multiply both sides by i, we get i2j = ik = -j.