This chapter examines a pivotal moment in the history of mathematics: the intellectual leap made by a young Isaac Newton (1642–1727), inspired by John Wallis’s Arithmetica Infinitorum. The core event is Wallis’s interpolation of Pascal’s triangle, which was the critical point at which fractional exponents gained common acceptance, a topic taught today in middle school algebra. The deeper justification for this notation, however, was the birth of the power series, typically reserved for advanced calculus courses. Newton, initially reluctant toward algebraic notation and believing that only geometry was true mathematics, was profoundly influenced by Wallis’s empirical methods. Using his own technique of solving large systems of linear equations to interpolate Wallis’s tables, Newton discovered the general binomial series for both negative and fractional exponents. This discovery provided Newton with an “analytical engine”—a method using infinite series to find the area under any algebraic curve, including the circle and hyperbola (logarithms). The chapter highlights Newton’s empirical methodology, evident in his private notebooks, where he continuously validated his new algebraic results against established geometric truths (like calculating π) and generalizations of arithmetic. His work effectively bridged the gap between discrete mathematics (like Pascal’s triangle) and the concept of continuity, laying the foundation for modern calculus and a powerful, utility-driven view of algebra.