GRAPHICAL REPRESENTATION

165. We have seen that positive integers and fractions can be represented by lines.

Thus, the line AB represents 3, and

the line BC represents 31.

A

B

៖

B

Similarly, we have seen that negative integers and fractions, which for a long time were considered to be meaningless, can be represented by lines.

Irrational numbers can also be represented a by lines.

Thus, in the right-angled triangle abc, the line ab represents the √2.

OA and

A

0

-2

B

y'

the opposite direction. 3√- 1 or 3 i would then be represented by

- 3 i by OB, as in the figure, and others similarly.

The reason for placing √−1 or i on a line at right angles to the line on which real numbers are plotted may be seen in the fact that multiplying 1 by √-1 twice changes +1 into

-

1. On the graph +1 can be changed into 1 by turning it through 180°. If multiplying 1 by V-1 twice turns the line 1 through 180°, multiplying 1 by √-1 once should turn +1 through 90°.

For example:

1. Represent graphically V-4:

√ − 4 = √ 4 i = 2 i; this is represented by a line 2 spaces long drawn upward on the y-axis.

2. Represent graphically-√3:

-V

3 = − √3 i = − 1.7 i (approximately); this is represented by a line 1.7 spaces long drawn downward on the y-axis.

(yy'). As in other graphical work this locates the point P1 which is taken to represent the complex number, 3 + i.

The number Va2 + b2 is called the modulus of the complex number a + bi. As appears from the figure OP1 = √32 + 12, and hence OPı represents the modulus of 3 + i.

2. Represent.graphically 3-i.

The point P2 is the graph of the complex number 3 i, and OP represents its modulus.

The point P3 is the graph of the complex number represents its modulus.

We have thus interpreted by means of diagrams positive and negative integers, positive and negative fractions, positive and negative irrational numbers, and positive and negative complex numbers; in fact, all of the numbers used in elementary algebra.

SUMMARY

The following questions summarize the definitions and processes treated in this chapter:

1. What is an imaginary number?

Sec. 154.

2. What term is used to designate numbers not imaginary?

Sec. 156.

3. Define and illustrate a complex number.

Sec. 158.

4. What is the meaning of the symbol i in complex numbers?

5. How may any integral power of i be expressed?

Sec. 158.

Secs. 161, 162.

6. How do imaginary numbers often occur in practice? What do imaginary roots indicate in solving problems?

Secs. 163, 164.

HISTORICAL NOTE

Negative numbers were so long a stumbling block to mathematicians that their square roots were naturally regarded as impossible until recent times. The Greeks understood irrationals and could express many of these numbers concretely; for example, the √2 was shown by them to be the length of the hypotenuse of a right triangle whose sides are unity, as in the figure, but the √2 had no meaning to them. The Hindoo, Bhaskara, said:

V2

1

"The square of a positive, as also of a negative number, is positive, but there is no square root of a negative number, for it is not a square." Even the great scholars of the sixteenth and seventeenth centuries did little more than to accept imaginaries as numbers, and it remained for Caspar Wessel (1797) to make the first concrete representation of complex numbers; but his discovery made little impression until Gauss emphasized its importance.

Karl Friedrich Gauss was born at Brunswick, Germany, in 1777. His father was a mason and took little interest in his son's education, but

KARL FRIEDRICH GAUSS

the boy's wonderful genius for numbers attracted the attention of his teachers, who induced the Duke of Brunswick to send young Gauss to a preparatory school. He entered the University of Göttingen in 1795 and soon made discoveries in the properties of numbers that won for him high rank among mathematicians. On the appearance of his great work, Disquisitiones Arithmetica, published in 1801 when Gauss was only twentyfour years old, his contemporary, Laplace, declared Gauss to be the greatest mathematician of all Europe. Gauss died in 1855 after a life devoted to mathematics. He enriched all its branches, including its applica

tions in astronomy and physics, with the lasting products of his wonderful genius. He has been worthily called: Princeps Mathematicorum, the Prince of Mathematicians.

CHAPTER VI

GRAPHS OF QUADRATIC EQUATIONS

166. PREPARATORY.

1. By counting the spaces read the length of EF in the figure.

2. Is it the square of the length of OE?

3. Answer similar questions for GH and он.

Every point of the curve is so located that the length of its ordinate is the square of its abscissa.

167. Quadratic expressions may be represented graphically.

For example:

The curve in the figure is the graph of y = x2. That is, the length of CD is the square of that of OC; the length of AB is the square of that of OA; etc.

ORAL EXERCISES

1. In y = x2 what is y when x = 2?

point having these values of x and y.

Locate in the figure the

2. Answer the same question when x = 1; also 3; 0; 4.

x=