CHAMBERS'S INFORMATION FOR THE PEOPLE. 



thus the square root of 2 is I and an indefinite 

 number of decimals ; but as the exact value can 

 never be determined, the name of irrational is 

 given to such quantities, to distinguish them from 

 all numbers whatever, whether whole or fractional, 

 of which the value can be found, and which are 

 therefore termed rational. Irrational numbers are 

 generally called surds, from the Latin surdus, deaf 

 or senseless. 



Equations. 



When two quantities are equal to each other, 

 the algebraical expression denoting their equality 

 is called an equation. Thus x 2 = 4 + 3 is an 

 equation denoting that if 2 be deducted from some 

 unknown quantity represented by x, the remainder 

 will be equal to 4 + 3, that is, to 7 ; therefore the 

 value of x in this equation is evidently 7 + 2, 

 or 9. 



The doctrine of equations constitutes by far the 

 most important part of algebra, it being one of 

 the principal objects of mathematics to reduce all 

 questions to the form of equations, and then to 

 ascertain the value of the unknown quantities by 

 means of their relations to other quantities of 

 which the value is known. 



When the unknown quantity x appears in the 

 equation only in the first power, the equation is 

 said to be a simple one. When x appears in the 

 second power, the equation is a quadratic j and 

 when x appears in the third power, the equation is 

 cubic. 



One side of an equation remains equal to the 

 other : 



1. After the same quantity is added to both. 

 Thus if A = B, then A + m = B + m. 



2. After the same quantity is subtracted from 

 both. Thus if A = B, then A - m = B -m. 



3. After both sides have been multiplied by the 

 same number. Thus A = B, then mA = mE. 



4. After both sides have been divided by the 

 same number. Thus if A = B, then 



A = J5 

 m m 



5. After both sides have been raised to the 

 same power. Thus if A = B, then 



A 2 = B'. 



6. After the extraction of the same root of both 

 sides. Thus if A = B, then JA = JB. 



Solution of a Simple Equation with one Unknown Quantity. 



Example i. Solve the equation yc 5 = 2x-\- i. 



In the first place, I get the ;rs all on one side, 



and arrange it so that nothing but xs will appear 



on this side. Now, for this purpose, I subtract 



ix from both sides ; we then have 



3* 2x 5 = i. 



I next add 5 to both sides, and get 



yc - 2.x = i + 5, 



or x = 6, which is the solution of the given 

 equation. 



Example 2. Solve the equation x 5 = + 3. 



Here I multiply both sides by 3 ; there results 

 then yc 15=^ + 9- 



Subtract x from both sides : 



.-. 3*-* -15 = 9. 



608 



Add 15 to both sides : 



.-. 3*-* = 9 + 15; 



.'. 2X = 24. 



Divide both sides by 2 : 



.'. X = 12. 

 Solution of an Equation with two Unknown Quantities. 



When two unknown quantities are to be deter- 

 mined, we must have two separate and independ- 

 ent equations ; this is evident, for if we wish to 

 know, for example, what two numbers added 

 together make 20, we have evidently as many pairs 

 as we please. 



Example. Solve the equations 

 + 2y = 22 ) 



- J=n} 



Here, if we multiply the second equation by 2, we 

 get 



&r 2y == 22. 



If this last equation be added to the first, we 

 have 



Substituting this value of x in either of the given 

 equations, say the first, we have 



12 + 2y = 22, 

 2y = 10, 



y = s; 



.'. x = 4, y = 5, is the solution. 



In a similar manner we can find x,y, and z from 

 a system of three equations between these three 

 unknowns. 



Solution of a Quadratic Equation. 



First, if the unknown quantity x appears only in 

 the second power, the equation may be solved for 

 x 3 in the same way as we solved for x ; it will then 

 reduce to an equation, having x 3 on the one side 

 and a number or given quantity on the other side ; 

 as, for example, 



x* 16; 

 /. *=4, 



x being obtained by extracting the square root of 

 both sides. 



Again, suppose x appears in the equation both 

 in the second power and also in the first power, 

 as in the following example : Solve the equation 

 x*i2x = 35- We should proceed to extract the 

 square root of both sides, but we see at once that 

 the left-hand side of the equation is not an exact 

 square ; and hence, before solving, we must com- 

 plete the square, and this we do by adding 36 to it ; 

 for this reason, that in a complete square the 

 third term is always the square of half the coeffi- 

 cient or multiplier oix in the second term, which 

 is here 1 2 ; but if we add 36 to the one side, we 

 must also add 36 to the other. We then have 



x 3 - I2x + 36 = - 35 + 36. 

 Or x 1 - \2x + 36 = i. 



Taking the square root of both sides, we have 



x- 6= i ; 

 .:x= 1 + 6, 

 = 7 or 5. 

 Hence x = 7, or 5, which is the required solution. 



