Elements of Algebra

60 ELEMENTS SECT. I. 26. What multiplier will render $a + \sqrt{3}$ rational? Ans. $a - \sqrt{3}$. 27. What multiplier will render $\sqrt{a} - \sqrt{b}$ rational? Ans. $\sqrt{a} + \sqrt{b}$. 28. What multiplier will render the denominator of the fraction $\frac{\sqrt{6}}{\sqrt{7}+\sqrt{3}}$ rational? Ans. $\sqrt{7} - \sqrt{3}$. CHAP. XX. *Of the different Methods of Calculation, and of their mutual Connexion.* 206. Hitherto we have only explained the different methods of calculation: namely, addition, subtraction, multiplication, and division; the involution of powers, and the extraction of roots. It will not be improper, therefore, in this place, to trace back the origin of these different methods, and to explain the connexion which subsists among them; in order that we may satisfy ourselves whether it be possible or not for other operations of the same kind to exist. This inquiry will throw new light on the subjects which we have considered. In prosecuting this design, we shall make use of a new character, which may be employed instead of the expression that has been so often repeated, *is equal to*; this sign is $=$, which is read *is equal to*: thus, when I write $a = b$, this means that $a$ is equal to $b$: so, for example, $3 \times 5 = 15$. 207. The first mode of calculation that presents itself to the mind, is undoubtedly addition, by which we add two numbers together and find their sum: let therefore $a$ and $b$ be the two given numbers, and let their sum be expressed by the letter $c$, then we shall have $a + b = c$; so that when we know the two numbers $a$ and $b$, addition teaches us to find the number $c$. 208. Preserving this comparison $a + b = c$, let us reverse the question by asking, how we are to find the number $b$, when we know the numbers $a$ and $c$. It is here required therefore to know what number must be added to $a$, in order that the sum may be the number $c$: suppose, for example, $a = 3$ and $c = 8$; so that we must have $3 + b = 8$; then $b$ will evidently be found by sub-