In algebra, the four processes of addition, subtraction, multiplication, and division, are frequently required; and when the required process cannot be actually performed upon the letters themselves, a certain method has been adopted by which the process is indicated. For example, in additon, when it is required to add a to b, the two letters cannot be intermingled as numerals may be, and their sum presented; but the process of addition is simply indicated by placing between the two letters this sign, +, which is called plus, meaning added to; therefore, to add a to b we have -
a + b,
which is read a plus b, or the sum of a and b. When the quantities represented by a and b are substituted for them - and not till then - they can be condensed into one sum. For example, let a equal 4 and b equal 3, then for -
a+b we have -
and we may at once write their sum 7, instead of 4 + 3.
So, likewise, in the process of subtraction, one letter cannot be taken from another letter so as to show how much of this other letter there will be left as a remainder; but the process of subtraction can be indicated by a sign, as this, - , which is called minus, less, meaning subtracted from. For example, let it be required to subtract b from a. To do this we have -
a - b;
which is read a minus b, and when the values of a and b are substituted for them, we have, when a equals 4, and b equals 3 -
a - b,
and now, instead of 4 - 3, we may put the value of the two, which is unity, or 1.
The algebraic signs most frequently used are as follows:
+ ,plus, signifies addition, and that the two quantities between which it stands are to be added together; as a + b, read a added to b.
- , minus, signifies subtraction, or that of the two quantities between which it occurs, the latter is to be subtracted from the former; as a - b, read a minus b.
X, multiplied by, or the sign of multiplication. It denotes that the two quantities between which it occurs are to be multiplied together; as a x b, read a multiplied by b, ' or a times b. This sign is usually omitted between symbols or letters, and is then understood, as a b. This has the same meaning as ax b. It is never omitted between arithmetical numbers; as 9x5, read nine times five.
÷, divided by, or the sign of division, and denotes that of the two quantities between which it occurs, the former is to be divided by the latter; as a÷b, read a divided by b. Division is also represented thus:
a/b, in the form of a fraction. This signifies that a is to be divided by b. When more than one symbol occurs above or below the line, or both, as anr/cm, it denotes that the product of the symbols above the line is to be divided by the product of those below the line.
=, is equal to, or sign of equality, and denotes that the quantity or quantities on its left are equal to those on its right; as a - b = c, read a minus b is equal to c, or equals c; or, 9 - 5 = 4, read nine minus five equals four. This sign, together with the symbols on each side of it, when spoken of as a whole, is called an equation.
a2 denotes a squared, or a multiplied by a, or the second power of a, and a3 denotes a cubed, or a multiplied by a and again multiplied by a, or the third power of a. The small figure, 2, 3, or 4, etc., is termed the index or exponent of the power. It indicates how many times the symbol is to be taken. Thus, a'2 = a a, a3 = a a a, a4 - a a a a,
3√ denotes that the cube root of the quantity following it is to be extracted. Thus, √9 = 3, and √7 = 3. The extraction of roots is also denoted by a fractional index or exponent, thus -
a1/2 denotes the square root of a,
a1/3 denotes the cube root of a,
a2/3 denotes the cube root of the square of a, etc.
402. - Example in Addition and Subtraction: Cancelling. - Let there be some question which requires a statement to represent it, like this -
a + d = c - b,
which indicates that if the quantity represented by a be added to the quantity represented by d, the sum will be equal to the quantity represented by c, after there has been subtracted from it the quantity represented by b; or, as it is usually read, a plus d equals c minus b; or the sum of a and d equals the difference between c and b. For illustration, take in place of these four letters, in the order they stand, the numerals 4, 2, 9, 3, and we shall have by substitution -
a + d = c - b,
44+2=9 - 3, or adding and subtracting - 6 = 6.
If it be required to add to each member of the equation the quantity represented by b, this will not interfere with the equality of the members. For a + d are equal to c - d, and if to each of these two equals a common quantity be added, the sums must be equal; therefore -
a + d+ b = c - b+ b, or by numerals -
4 + 2 + 3 = 9-3 + 3, or -
9 = 9.
It will be observed that the right hand member contains the quantity - b and + b. This shows that the quantity b is to be subtracted and then added. Now, if 3 be subtracted from 9, the remainder will be 6, and then if 3 be added, the sum will be 9, the original quantity. Thus it is seen that when in the same member of an equation a symbol appears as a minus quantity and also as a plus quantity, the two cancel each other, and may be omitted. Therefore, the expression -
a + d+ b = c - b + b becomes -
a + d+ b =c.