A quaternion is an ordered combination of four real numbers. It can also be expressed as a sum of a scalar and a vector in three dimensional Euclidian space. Quaternion algebra allows the division of vectors. Since the current vector algebra does not allow the division of vectors; in many branches of mechanics, when a vectoral quantity falls to the denominator of an expression, the general tendency is to use its magnitude rather than its vector character. Once the vector division is defined by the quaternion algebra, it becomes possible to redrive the equations that have vector quantities in their denominators. In this study, basic equations of stress in strength of materials are reviewed according to the rules of quaternion algebra. It is shown that this new point of view brings a more powerful and consistent system. The normal stress becomes a scalar quantity. Area and moment of inertia of a cross-section becomes a vectoral quantity. Direction of shearing stresses change and becomes a more consistent convention especially in torsion problems. The most striking results are obtained in shear flow.