An algorithm is a set of rules for solving a math problem which, if done properly, will give a correct answer each time.
翻譯成中文：算法是指解決數學問題的壹組規則，如果操作適當，每壹次都可以得到正確的結果。
更加通俗的解釋是：
It is a step-by-step procedure for calculations. Algorithms are used for calculation, data processing, and automated reasoning.and arithmetic.
算法是壹步壹步的計算過程，可以用作計算，數據處理，自動推理和算術。
而Arithmetic在Wikipedia中解釋是：
Arithmetic or arithmetics is the oldest and most elementary branch of mathematics, used very popularly, for tasks ranging from simple day-to-day counting to advanced science and business calculations. It involves the study of quantity, especially as the result of operations that combine numbers.
翻譯成中文：Arithmetic是數學中最古老和最原始的應用非常廣泛的壹個分支。它可以用來解決從簡單的日常計數到先進的科學和商業計算各種任務。它涉及到數量的研究，特別是結合數字的操作的結果。
從上面的分析可以基本上厘清他們之間的關系。
Algorithm研究解決難題思路和方法的科學，他屬於計算機科學領域。而Arithmetic解決如何更快更精確計算的科學，屬於數學領域。涉及到常用的運算：+ - x ÷。