Test: Chrystal’s Algebra Aug 29, 2020

The Fundamental Laws and Processes of Algebra as exhibited in ordinary Arithmetic.

§ 1.] The student is already familiar with the distinction between abstract and concrete arithmetic. The former is concerned with those laws of, and operations with, numbers that are independent of the things numbered; the latter is taken up with applications of the former to the numeration of various classes of things.

Confining ourselves for the present to abstract arithmetic, let us consider the following series of equalities:—

\begin{align*} \frac{2623}{61} + \frac{1023}{3} &= \frac{2623\times3+1023\times61}{61\times3}\\ &= \frac{70272}{183} = 384 \end{align*}

The first step is merely the assertion of the equivalence of two different sets of operations with the same numbers. The second and third steps, though doubtless based on certain simple laws from which also the first is a consequence, nevertheless require for their direct execution the application of certain rules, of a kind to which the name arithmetical is appropriated.

We have thus shadowed forth two great branches of the higher mathematics:— one, algebra, strictly so called, that is, the theory of operation with numbers, or, more generally speaking, with quantities; the other, the higher arithmetic, or theory of numbers. These two sciences are identical as to their fundamental laws, but differ widely in their derived processes. As is usual in elementary text-books, the elements of both will be treated in this work.

§ 2.] Ordinary algebra is simply the general theory of those operations with quantity of which the operations of ordinary abstract arithmetic are a particular case.

The fundamental laws of this algebra are therefore to be sought for in ordinary arithmetic.

However various and complex the operations of arithmetic may seem, it appears on consideration that they are merely the result of the application of a very small number of fundamental principles. To make this plain we return for a little to the very elements of arithmetic.

Addition, and the general laws connected therewith

§ 3.] When a group of things, no matter how unlike, is considered merely with reference to the number of individuals it contains, it may be represented by another group, the individuals of which are all alike, provided only there be as many individuals in the representative as in the original group. The members of our representative group may be merely marks ($$1$$’s say) on a piece of paper. The process of counting a group may therefore be conceived as the successive placing of $$1$$’s in our representative group, until we have as many $$1$$’s as there are individuals in the group to be numbered. This process of adding a 1 is represented by writing $$+ 1$$. We may thus have

$+1\mbox{,}\quad+1+1\mbox{,}\quad+1+1+1\mbox{,}\quad+1+1+1+1\mbox{, &c.,}$

as representative groups or “numbers.” As the student is of course aware, these symbols in ordinary arithmetic are abbreviated into

$1\mbox{,}\quad2\mbox{,}\quad3\mbox{,}\quad4\mbox{, &c.}$

Hence using the symbol “$$=$$” to stand for “the same as,” or “replaceable by,” or “equal to,” we have, as definitions of $$1\mbox{, }2\mbox{, }3\mbox{, }4\mbox{, &c.,}$$

\begin{align*} 1 &= +1\mbox{,}\\ 2 &= +1+1\mbox{,}\\ 3 &= +1+1+1\mbox{,}\\ 4 &= +1+1+1+1\mbox{, &c.} \end{align*}

And there is a further arrangement for abridging the representation of large numbers, which the student is familiar with as the decimal notation. With numerical notation we are not further concerned at present, but there is a view of the above equalities which is important. After the group $$+1 +1 +1$$ has been finished it may be viewed as representing a single idea to the mind, viz. the number “three.” In other words, we may look at $$+1 +1 +1$$ as a series of successive additions, or we may think of it as a whole. When it is necessary for any purpose to emphasise the latter view, we enclose $$+1 +1 +1$$ in a bracket, thus $$(+1 +1 +1)$$; and it will be observed that precisely the same result is attained by writing the symbol $$3$$ in place of $$+1 +1 +1$$, for in the symbol $$3$$ all trace of the formation of the number by successive addition is lost. We might therefore understand the equality or equation

\begin{align*} && 3 &= +1+1+1\\ \text{to mean} && (+1+1+1) &= +1+1+1\mbox{,} \end{align*}

and then the equation is a case of the algebraical Law of Association.

The full meaning of this law will be best understood by considering the case of two groups of individuals, say one of three and another of four. If we wish to find the number of a group made up by combining the two, we may adopt the child’s process of counting through them in succession, thus,

$+1 +1 +1 | +1 +1 +1 +1 = 7\mbox{.}$

But by the law of association we may write for $$+1 +1 +1$$

$(+1 +1 +1)\mbox{,}$

and for $$+1 +1 +1 +1$$

$(+1 +1 +1 +1)\mbox{,}$

and we have

\begin{align*} +(+1 +1 +1) + (+1 +1 +1 +1) &= 7\\ \mbox{or} +3 +4 &= 7\mbox{.} \end{align*}

It will be observed that we have added a $$+$$ in each case before the bracket, and it may be asked how this is justified. The answer is simply that setting down a representative group of three individuals is an operation of exactly the same nature as\ldots