Real Numbers

Definition

Real numbers are the numbers that fill the number line.

They include:

• natural numbers
• integers
• rational numbers
• irrational numbers

Examples:

-2
0
1 / 2
3.14
sqrt(2)

Real numbers represent positions on a continuous number line.

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Rational And Irrational

Rational numbers can be written as ratios.

Example:

1 / 2

Irrational numbers cannot be written as ratios of integers.

Example:

sqrt(2)

Both rational and irrational numbers are real numbers.

Real numbers include both exact ratios and non-ratio values.

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Measurement

Real numbers are useful for measurement.

Examples:

length
weight
temperature
time
probability
position

Measurements often need values between whole numbers.

Example:

2.5 meters

Real numbers support quantities that vary between whole steps.

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Real Numbers Are Continuous

Between any two real numbers, there is another real number.

Example:

2 and 3

Between them:

2.5
2.25
2.125

and so on.

Real numbers form a continuous line, not isolated steps.

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Approximation In Computation

Computers usually represent real numbers approximately.

Example:

1 / 3 = 0.333...

A computer may store only a limited decimal-like approximation.

This does not change the mathematical idea.

It means implementation has limits.

Real numbers are mathematically continuous, but computers store finite approximations.

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Final Summary

• Real numbers fill the number line.
• They include rationals and irrationals.
• They are useful for measurement.
• Real numbers are continuous.
• Between two real numbers there is always another real number.
• Computers usually approximate real values.