General Principles of Division
Algebra Help: Section 2.23
The value of a quotient in division depends upon the relative values of the dividend and the divisor; and the sign of the quotient depends upon the signs of the dividend and divisor. Therefore any change in the value or the sign of either dividend or divisor must produce a change in the value or the sign of the quotient; though certain changes may be made in both dividend and divisor, at the same time, that will not affect the quotient. The laws that govern these changes are called General Principles of Division.
Any change in the value or sign of either dividend or divisor, produces a change in the value or sign of the quotient.
Change of Value
It's necessary to examine only those changes of value produced by multiplying and dividing the dividend and divisor.
Let us take abcd for dividend, and ab for a divisor;
the quotient will be cd, and the operations performed upon dividend
and divisor will affect this quotient as follows:
Dividend Divisor Quotient 1. abcde ÷ ab = cde 2. abc ÷ ab = c 3. abcd ÷ abc = d 4. abcd ÷ a = bcd 5. abcde ÷ abe = cd 6. bcd ÷ b = cd
ANALYSIS
- Multiplying the dividend by
eMultiplies the quotient bye. - Dividing the dividend by
ddivides the quotient byd. - Multiplying the divisor by
cdivides the quotient byc. - Dividing the divisor by
bmultiplies the quotient byb. - Multiplying both terms by
edoes not alter the quotient. - Dividing both terms by
adoes not alter the quotient.
In these six operations, the factors employed to operate with are literal quantities, and may represent any numbers whatever; hence the results are general truths; they may be stated as follows:
PRINCIPLES
-
Multiplying the dividend multiplies the quotient,
and dividing the dividend divides the quotient. (1 and 2)
Dividend * n ÷ Divisor = Quotient * n Dividend / n ÷ Divisor = Quotient / n
-
Multiplying the divisor divides the quotient, and dividing
the divisor multiplies the quotient. (3 and 4)
Dividend ÷ Divisor * n = Quotient / n Dividend ÷ Divisor / n = Quotient * n
-
Multiplying or dividing both dividend and divisor
by the same quantity does not alter the quotient. (5 and 6)
Dividend * n ÷ Divisor * n = Quotient Dividend / n ÷ Divisor / n = Quotient
GENERAL LAW
These principles may be embraced in one general law.
A change in the dividend produces a LIKE change in the quotient. A change in the divisor produces an OPPOSITE change in the quotient.
CHANGE OF SIGN
To investigate the relative changes of signs in division, remember that when the divisor and dividend have like signs, the quotient is plus. When divisor and dividend have unlike signs, the quotient is minus.
Then:
- Suppose the divisor and dividend have like signs; if either of the signs be changed, they will become unlike signs. The sign of the quotient will be changed from plus to minus.
- Suppose the divisor and dividend have unlike signs; if either of the signs be changed, they will become alike. The sign of the quotient will be changed from minus to plus.
- Suppose again that the divisor and dividend have like signs; if both signs be changed at once, they will still be alike. The sign of the quotient will remain plus.
- Suppose again that the divisor and dividend have unlike signs; if both signs be changed at once, they will still be unlike. The sign of the quotient will remain minus.
These results may be enbraced in two principles, as follows:
PRINCIPLES
- Changing the sign of either dividend or divisor, changes the sign of the quotient.
- Changing the sign of both dividend and divisor, does not alter the sign of the quotient.
NOTE - If the dividend or divisor is a polynomial, its entire value is changed by changing the signs of all its terms.
Page 62-64
Page 62-64
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