Introduction
7. Central portion of the number linenear 0
However, one of the great achievements of the 19th century was the full realization that the true domain of number is not one-, but rather is two-dimensional. The plane of the complex numbers is the natural arena of discourse for much of mathematics. This has been brought home to mathematicians and scientists through problem solving–to be able to carry out the investigations required to solve real-world problems, many of which seem to be only about ordinary counting numbers, it becomes necessary to expand your number horizon. The explanation as to how this extra dimension emerges will come towards the end of this chapter and be explored further in Chapter 8.
Pluses and minuses
The integers is the name applied to the set of all whole numbers,positive negative, and zero. This set, often symbolized by the letter Z, is therefore infinite in both directions:
{···-4,-3,-2,-1,0,1,2,3,4,···}.
The integers are often pictured as lying at equally spaced points along a horizontal number line, in the order indicated. The additional rules that we need to know in order to do arithmetic with the integers can be summarized as follows:
(a)to add or subtract a negative integer,-m,wemovemspaces to the left in the case of addition, and m spaces to the right for subtraction;
(b)to multiply an integer by-m, we multiply the integer by m,and then change sign.
In other words, the direction of addition and subtraction of negative numbers is the opposite to that of positive numbers, while multiplying a number by-1 swaps its sign for the alternative. For example,8+(-11)=-3,3×(-8)=-24, and(-1)×(-1)=1.
You should not be troubled by this last sum. First, it is reasonable that multiplying a negative number by a positive one yields a negative answer:when a debt(a negative amount)is subject to interest(a positive multiplier greater than 1)the outcome is greater debt, that is to say a larger negative number. We are all well aware of this. That multiplication of a negative number by another negative number should have the opposite outcome, that is a positive result, would then appear consistent. The fact that the product of two negative numbers is positive can readily be given formal proof. The proof is based on the assumptions that we want our expanded number system of the integers to subsume the original one of the natural numbers, and that the augmented system should continue to obey all the normal rules of algebra.Indeed, the result on the product of two negatives follows from the fact that any number multiplied by zero equals zero.(This too is not an assumption but rather is also a consequence of the laws of algebra.)For we now have: