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BACKGROUND: Mental arithmetic analysis based on electroencephalogram (EEG) signals can help to understand some disorders such as attention deficit hyperactivity disorder, arithmetic disorder, or autism spectrum disorder in which learning is difficult. Most mental computation detection and classification systems rely on the characteristics of a single channel, however, the understanding of the connections between EEG channels, which certainly contains valuable information, is still evolving. The methods presented in this paper are the result of a research project that introduces an alternative method for better and faster receipt of information from the EEG signals of individuals, which are generally complex and nonlinear. METHODS: The EEGs of 66 healthy individuals were recorded in two rest modes and mental task a designed, with a sampling frequency of 500 Hz. To classify these two modes, we extracted features from our recordings to differentiate the EEG signals of these two groups in a single channel as well as combine possible channels. The new method that was proposed was the extraction of several geometric features from Poincaré design analysis, which used the necessary comparison t-test to determine brain differences, with a significance level of less than 0.05 in the state of mental calculations and facial rest. Also, an artificial neural network (ANN) has been used for automatic learning and diagnosis in the two mentioned modes. RESULTS: The results of this paper show that by using a combination of geometric properties (sides, angles, shortest distance, slope, and coefficients of the third-degree equation) using selected channels (FP1, F7, C4, O1) can achieve 100 % accuracy. The sensitivity reached 100 %. As well as 100 % feature. CONCLUSIONS: With the help of mental calculation, it is possible to diagnose, treat, rehabilitate and rehabilitation people who have lost the function of a part of their brain due to a disease in this field (Tab. 6, Fig. 15, Ref. 45). [ABSTRACT FROM AUTHOR]