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Numbers, Algebra, Discrete vs Continuous
Rational Numbers
Rational numbers are numbers that can be written as a ratio of two integers.
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Rational Numbers
Definition
A rational number is a number that can be written as a ratio of two integers.
Basic form:
p / q
where:
p is an integer
q is an integer
q is not 0
Rational number = number expressible as an integer ratio.
Simple Examples
Examples:
1 / 2
3 / 4
-5 / 2
7 / 1
0 / 3
The number:
7
is rational because it can be written as:
7 / 1
Every integer is rational, but not every rational number is an integer.
Why The Denominator Cannot Be Zero
The denominator cannot be zero.
This is invalid:
5 / 0
Division by zero does not define a valid rational number.
Rational numbers require a nonzero denominator.
Fractions And Exactness
Rational numbers are useful when a quantity can be divided exactly into parts.
Examples:
1 / 2 of a value
3 / 4 of a path
2 / 5 of a group
Fractions preserve exact relationships.
The decimal 0.5 and the fraction 1 / 2 represent the same value, but the fraction shows the ratio directly.
Rational numbers make part-whole relationships explicit.
Decimal Form
Rational numbers have decimal forms that either terminate or repeat.
Examples:
1 / 2 = 0.5
1 / 3 = 0.333...
1 / 4 = 0.25
Rational decimals eventually stop or repeat.
Final Summary
- Rational numbers can be written as
p / q. pandqare integers.qcannot be zero.- Every integer is rational.
- Rational numbers express exact ratios.
- Their decimal forms terminate or repeat.
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Continue the series.
Real Numbers