Arithmetic
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Arithmetic is an elementary branch of mathematics that studies numerical operations like addition, subtraction, multiplication, and division. In a wider sense, it also includes exponentiation, extraction of roots, and taking logarithms.
Calculations in mental arithmetic are done exclusively in the mind without relying on external aids.
Abacuses are tools to perform arithmetic operations by moving beads.
Some historians interpret the Ishango bone as one of the earliest arithmetic artifacts.
Image: Vlčí radius
Calculus
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Calculus is the mathematical study of continuous change, in the same way that geometry is the study of shape, and algebra is the study of generalizations of arithmetic operations.
Isaac Newton developed the use of calculus in his laws of motion and universal gravitation.
Maria Gaetana Agnesi
The logarithmic spiral of the Nautilus shell is a classical image used to depict the growth and change related to calculus.