Observability in Mathematics was developed based on the denial of the concept of infinity. The book introduces  Observers into arithmetic, and arithmetic becomes dependent on Observers. And after that, the basic mathematical parts also become dependent on Observers. One of such parts is arithmetic itself and algebra. Arithmetic and Algebra play important roles not only in pure Mathematics but in contemporary Physics, for example, in Relativity theory and Quantum Mechanics. They will be called New Arithmetic and Algebra, both observers at the logical level and in arithmetic and algebra. The book reconsiders the foundations of classic arithmetic and algebra from this mathematical perspective. The relationships between numbers, polynomials, quaternions, groups, and algebras are discovered and exhibit new properties.  It is shown that almost all classic arithmetic and algebra theorems are satisfied in Mathematics with Observers' arithmetic and algebra, where probabilities are less than 1.