Introducing complex numbers into basic growth functions - (VII) differences between '1' and '-1' in their hypothetic breakdown into complex numbers and application to definite integral of exp(t) expanded into infinite series

The present study was designed to investigate differences between '1' and '-1' in their hypothetic breakdown into complex numbers and application to definite integral of exp(t) expanded into infinite series. The results obtained were as follows. Complex numbers that were left by the hypothetic breakdown of product form in the complex representation of '1' and '-1' were classified into four groups according to properties of complex numbers; plus or minus, anti-clockwise or clockwise rotation on a circumference, hypothetic right-handed or left-handed spiral. Three groups or less showing an incomplete appearance of properties of complex numbers were derived from both '1' and '-1', but four groups that showed a complete appearance of all properties of them were derived from '1' only. This difference between '1' and '-1' might be due to the presence or absence of a minus sign. Complex numbers that were left by the hypothetic breakdown of product form in the complex representation of '1+(-1)' disappeared immediately in the calculation of growth using the definite integral of exp(t) expanded into infinite series.