Infinitesimals are quantities that are extremely small, approaching zero but never actually reaching it. They are used in mathematics, particularly in calculus, to help understand concepts like limits and derivatives. Infinitesimals allow mathematicians to analyze changes in functions and to calculate areas under curves. The concept of infinitesimals was notably developed by Gottfried Wilhelm Leibniz and Isaac Newton in the 17th century. Although they were initially controversial, infinitesimals have since been formalized in modern mathematics through the hyperreal numbers and non-standard analysis, providing a rigorous framework for their use in various mathematical applications.