Page:IntroductionToMathematicsWhitehead.pdf/66

portant symbols should have an important meaning. This is one of the merits of the Arabic notation for numbers; by means of ten symbols, ${\displaystyle 0,1,2,3,4,5,6,7,8,9,}$ and by simple juxtaposition it symbolizes any number whatever. Again in algebra, when we have two variable numbers ${\displaystyle x}$ and ${\displaystyle y}$, we have to make a choice as to what shall be denoted by their juxtaposition ${\displaystyle xy}$. Now the two most important ideas on hand are those of addition and multiplication. Mathematicians have chosen to make their symbolism more concise by defining ${\displaystyle xy}$ to stand for ${\displaystyle x\times y}$. Thus the laws ${\displaystyle (3),(4),}$ and ${\displaystyle (5)}$ above are in general written,

thus securing a great gain in conciseness. The same rule of symbolism is applied to the juxtaposition of a definite number and a vari- able: we write ${\displaystyle 3x}$ for ${\displaystyle 3\times x}$, and ${\displaystyle 30x}$ for ${\displaystyle 30\times x}$.

It is evident that in substituting definite numbers for the variables some care must be taken to restore the ${\displaystyle \times }$, so as not to conflict with the Arabic notation. Thus when we substitute ${\displaystyle 2}$ for ${\displaystyle x}$ and ${\displaystyle 3}$ for ${\displaystyle y}$ in ${\displaystyle xy}$, we must write ${\displaystyle 2\times 3}$ for ${\displaystyle xy}$, and not ${\displaystyle 23}$ which means ${\displaystyle 20+3}$.

It is interesting to note how important for the development of science a modest-looking symbol may be. It may stand for the emphatic presentation for an idea, often a very