Fractions

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packages

Fractions

Fractions are packages.

Fractions are vertically arranged packages. There is a horizontal bar separating a top position called the numerator and a bottom position called the denominator.

\[ \frac {numerator}{denominator} \]

There is only one rule:

The denominator of a fraction cannot equal $0$.

We do not use “typewriter fractions”. We don’t write things like 3/4. We write $\frac {3}{4}$.

The reason is that typewriter fractions cause confusion.

\[ (x+1)(x+3)/2 e^x (x-4) = ??? \]

If you are going by the Order of Operations, then this says

\[ \frac {(x+1)(x+3)}{2} e^x (x-4) = ??? \]

But the author could mean any one of several fractions. We can’t tell.

This confusion often shows up in the middle of algebraic calculations and everything goes off the rails.

\[ \frac {numerator}{denominator} \]

Fractions are tools we use to represent many mathematical objects.

Numbers

We use fractions to represent numbers. Fractions give us an unlimited supply of representations for every number.

The number four can be represented with the following fractions:

\[ 4 = \frac {12}{3} = \frac {-24}{-6} = \frac {4}{1} = \frac {1}{\tfrac {1}{4}} = \frac {\tfrac {1}{5}}{\tfrac {1}{20}} = \frac {100\pi }{25\pi } \]

$\blacktriangleright $ 1

Having an endless supply of representations is very helpful when thinking about numbers, especially the number $1$.

\[ 1 = \frac {2}{2} = \frac {-7}{-7} = \frac {\pi }{\pi } = \frac {\sqrt {2}}{\sqrt {2}} \]

All of these options for the number $1$, help us find alternatives for another numbers.

\[ 8 = \frac {8}{1} = \frac {8}{1} \cdot 1 = \frac {8}{1} \cdot \frac {3}{3} = \frac {24}{3} \]

The number $1$ can be represented by any fraction of the form $\frac {N}{N}$, where $N$ is any number, except $0$.

Multiplication by the number $1$ is one of our most important way to compare numbers.

How do we compare $\frac {17}{23}$ and $\frac {27}{37}$ ?

We multiply both by $1$.

\[ \frac {17}{23} = \frac {17}{23} \cdot 1 = \frac {17}{23} \cdot \frac {37}{37} = \frac {629}{851} \]

\[ \frac {27}{37} = \frac {27}{37} \cdot 1 = \frac {27}{37} \cdot \frac {23}{23} = \frac {621}{851} \]

\[ \frac {17}{23} > \frac {27}{37} \]

All of the representations for $1$ follow the same pattern. Both the numerator and the denominator are the same number.

$\blacktriangleright $ 0

Similar to the number $1$, All of our fractional representations of $0$ follow a pattern.

The number $0$ can be represented by any fraction of the form $\frac {0}{N}$, where $N$ is any number, except $0$.

Note: the reason $1$ and $0$ cannot be represented with a fraction whose denominator equals $0$ is because fractions cannot have denominators equal to $0$.

$\frac {N}{0}$ is NOT a fraction, no matter what $N$ is.

For what values of $A$ is $\frac {(A-3)(A-5)}{(A+2)(A-5)} = 0$ ?

$\frac {(A-3)(A-5)}{(A+2)(A-5)} = 0$ when $A = 3$.

If $A = 5$, then $\frac {(A-3)(A-5)}{(A+2)(A-5)} = \frac {2\cdot 0}{7\cdot 0} = \frac {0}{0}$, which isn’t a fraction.

Ratios and Rates

We use fractions to represent ratios and rates between measurements.

The rate “$24$ hours per day” can be represented with the fraction $\frac {24 hours}{1 day}$.

In this context, $\frac {24 \, hours}{1 \, day} = 1$, since $24 \, hours = 1 \, day$.

We use fractions to represent rates when thinking about dimensional analysis.

\[ 4 \, days = 4 \, days \cdot \frac {24 \, hours}{1 \, day} \cdot \frac {60 \, mins}{1 \, hour} \cdot \frac {60 \, secs}{1 \, min} = 345600 \, seconds \]

Quotient Functions

We use fractions to represent quotient functions, which we will study in this course.

\[ \frac {x+1}{\sqrt {x}} \]

\[ \frac {e^{2t}}{\cos (t)} \]

\[ \frac {\ln (2k+1)-5}{7 - |3k+6|} \]

Arithmetic

No matter what you are representing with fractions, they all follow the same arithmetic.

\[ \frac {A}{B} + \frac {C}{B} = \frac {A + C}{B} \]

\[ \frac {A}{B} \cdot \frac {C}{D} = \frac {A \cdot C}{B \cdot D} \]

Create a single fraction equivalent to the sum $\frac {4x+1}{x-2} + \frac {1}{x}$.

\[ \frac {4x+1}{x-2} + \frac {3}{x} \]

\[ \frac {4x+1}{x-2} \cdot 1 + \frac {3}{x} \cdot 1 \]

\[ \frac {4x+1}{x-2} \cdot \frac {x}{x} + \frac {3}{x} \cdot \frac {x-2}{x-2} \]

\[ \frac {(4x+1)x}{(x-2)x} + \frac {3(x-2)}{x(x-2)} \]

\[ \frac {(4x+1)x + 3(x-2)}{(x-2)x} \]

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more examples can be found by following this link
More Examples of Real-Valued Functions

2025-09-27 14:12:53