Complex numbers

With the development of mathematics the concept of number has widened to meet the requirements for the solution of increasingly sophisticated problems. The process is illustrated by means of a Venn diagram (Fig. 4.1). We can use quadratic equations as models of the problems to be solved:Integers (positive or negative) solve equations of the form <math display='block'> <mrow> <mrow><mo>(</mo> <mrow> <mi>x</mi><mo>+</mo><mn>1</mn> </mrow> <mo>)</mo></mrow><mrow><mo>(</mo> <mrow> <mi>x</mi><mo>&#x2212;</mo><mn>3</mn> </mrow> <mo>)</mo></mrow><mo>=</mo><mn>0.</mn> </mrow> </math> $$\left( {x + 1} \right)\left( {x - 3} \right) = 0.$$Rational numbers (ratios of integers) solve equations of the form <math display='block'> <mrow> <mrow><mo>(</mo> <mrow> <mn>3</mn><mi>x</mi><mo>+</mo><mn>5</mn> </mrow> <mo>)</mo></mrow><mrow><mo>(</mo> <mrow> <mn>2</mn><mi>x</mi><mo>&#x2212;</mo><mn>1</mn> </mrow> <mo>)</mo></mrow><mo>=</mo><mn>0.</mn> </mrow> </math> $$\left( {3x + 5} \right)\left( {2x - 1} \right) = 0.$$Irrational numbers, for example √7, solve equations of the form <math display='block'> <mrow> <msup> <mi>x</mi> <mn>2</mn> </msup> <mo>&#x2212;</mo><mn>2</mn><mi>x</mi><mo>&#x2212;</mo><mn>6</mn><mo>=</mo><mn>0.</mn> </mrow> </math> $${x^2} - 2x - 6 = 0.$$