Algebra Fractions

Algebra Help: Section 3.1

Fraction

Fractions express division. Fractional forms have precisely the same meaning in algebra as in arithmetic.

The word Fraction relates to a certain mode or form of indicating division; and fractional forms have the same signification in algebra as in arithmetic.

A Fraction is a quotient expressed by writing the dividend above a horizontal line, and the divisor below, as follows:

a
-
b

Read the fraction as a divided by b. You can also write a fraction with either of the following symbols: ÷ or /.

Express Fractions in One of Three Ways

÷: Use the division symbol, with a point above and below; ÷. Write the dividend to the left and the divisor to the right. For example 8 ÷ 6 means 8 divided by 6.
/: Use the division symbol with a right slanted line. Write the dividend to the left and the divisor to the right. For example 8/6 means 8 divided by 6.
-: Use the division symbol as a horizontal line. Write the dividend above and the divisor below. For example
```
8
-
6
```
also means 8 divided by 6.

Denominator

The Denominator of a fraction is the quantity below the line, or the divisor.

8
-
6

In the preceding fraction, 6 is the denominator.

Numerator

The Numerator of a fraction is the quantity above the line, or the dividend.

8
-
6

In the preceding fraction, 8 is the numerator.

Since a quantity is divided by dividing any one of its factors, we have the following equations.

a   1 * a   1
- = ----- = - * a
b     b     b

Therefore a fraction is equal to the reciprocal of its denominator multiplied by its numerator.

Entire Quantity

An Entire Quantity is an algebraic expression which has no fractional part, such as 3a, or x - 3y².

Mixed Quantity

A Mixed Quantity is one which has both entire and fractional parts such as,

    c
a + -
    a

     3c
m - -----
    1 - b

NOTE - Define a fraction. A denominator. A numerator. A fraction is equal to the reciprocal of what? Define an entire quantity. A mixed quantity.

Signs

Each term in the numerator and denominator of a fraction has its own particular sign, distinct from the real sign of the fraction.

For example, see the following fraction:

x² - 2xy + y²
------------
x²y - xy²

The signs of a part of the terms only are expressed. If no sign is prefixed to the first term of a numerator or denominator, the plus sign is understood.

Apparent Sign

The Apparent Sign of a fraction is the sign written before the dividing line to indicate whether the fraction is to be added or subtracted.

For example, see the following fraction:

    x² - ax
m + -------
    a - x

The apparent sign of the fraction is plus, and indicates that the fraction should be added to m.

Real Sign

The Real Sign of a fraction is the sign of its numerical value, when reduced to a mononomial, and shows whether the fraction is essentially a positive or a negative quantity. Therefore in the previous fraction:

x² - ax
-------
a - x

Let x = 2 and a = 12. Then

x² - ax
------- =
a - x 

4 - 12 * 2
---------- =
 12 - 2

-20
--- =
 10

-2

The real sign of the fraction is minus yet the apparent sign of the fraction is plus.

General Principles of Fractions

Since fractions indicate division, all changes in the numerator and denominator of a fraction will affect the value and sign of that fraction according to the laws of division. We have only to modify the language of the General Principles of Division, by substituting the words numerator, denominator, and fraction for the words dividend, divisor, and quotient. The substitution expresses the laws governing the changes in the value and sign of a fraction.

NOTE - Define the apparent sign of a fraction. The real sign. Fractions always indicate what? Adapt the general principles of division, to fractions. Repeat the principles that govern change of value.

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