Numbers

Reading Activity

Memorize the following word combinations

1. zero is neither positive nor negative-ноль не является ни положительным, ни отрицательным

2. to label a point on the line- отметить точку на прямой

3. to each point on the line we assign a number-каждой точке на кривой мы ставим в соответствие число

The beginning of our number system dates back to antiquity where symbols, which we call positive integers, were used as an aid in counting, and only in the nineteenth century the system, which we know today, was completed. As an aid in studying this number system, let’s use the diagram.

The first numbers we use are the positive integers, and the fundamental fact that there is a first integer, unity, but not a last is soon established. Later positive fractions, or numbers, which can be expressed as the ratio of two of these integers, are used and understood. Then it is seen that these integers and fractions can be negative as well as positive. The division point between the positive and negative numbers which is the position from which we start to count in either direction, is occupied by the number zero. This number is different from all others in that we are not allowed to use it as a divisor.

The positive integers are often written without the plus sign, thus we may write 789 instead of + 789. Since zero is neither positive nor negative, it has no sign.

If we take a straight line and label a point on the line 0 and another point +1, we impose a scale on the line in terms of which we can mark off the line with the positive numbers to the right of 0 and the negative numbers to the left.

To each point on the line we assign a number whose length is the distance of the point from zero and whose sign + or - is determined whether the point is to the left or right of zero. The numbers in this uncountable set are known as the real numbers. The integers correspond to a small subset of the reals.

The positive and negative integers and fractions, together with zero, are called rational numbers.

Besides rational numbers we find irrationals, which are defined as numbers that cannot be expressed as the quotient of two integers. The √2; -√3 and π are examples of such numbers.

The two classes of numbers, rational and irrational, form the real number system, which we shall use in the first part of our course. Later we shall study such numbers as √-2, -√-1, etc., which are called imaginaries; and finally it will be seen that the basic system of all numbers is the complex, in which the reals and imaginaries are included as special cases. 2+ √-3 is such a number and we see, that it is composed of a real and an imaginary parts.

To denote the part of a complex number, we use the notation R (a + bi) = a for the imaginary part.

Arithmetic is performed on complex numbers in the same way as on real numbers, except that i² is replaced by - 1 whenever it occurs.

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