Real numbers are all the numbers that can be found on a number line.
That means real numbers include:
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whole numbers
-
integers
-
fractions
-
decimals
-
square roots
-
pi
Basically, if you can place it somewhere on the number line, it’s a real number.
🔢 Types of Real Numbers
Real numbers are split into two main groups:
1) Rational Numbers
A rational number is any number that can be written as a fraction:
a/b
where a and b are integers and b ≠ 0
Examples:
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3/4
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−5 (because −5 = −5/1)
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0.75 (because 0.75 = 3/4)
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0.333… (repeating, equals 1/3)
Rational decimals either:
✅ end (terminate)
OR
✅ repeat
2) Irrational Numbers
An irrational number is a number that cannot be written as a fraction.
These numbers have decimals that:
❌ never end
❌ never repeat
Examples:
-
π = 3.1415926535…
-
sqrt{2} = 1.41421356…
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sqrt{7}
📍 Real Numbers on a Number Line
All real numbers can be shown on a number line like this:
… -3, -2, -1, 0, 1, 2, 3 …
But it also includes decimals between them like:
-
1.5
-
-2.7
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0.333…
-
sqrt{2}
So real numbers fill up the entire number line with no gaps.
🧠 Quick Summary
✅ Real numbers = Rational + Irrational
Real Numbers Include:
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Natural numbers: 1, 2, 3, …
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Whole numbers: 0, 1, 2, 3, …
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Integers: … -2, -1, 0, 1, 2, …
-
Rational numbers: fractions, terminating decimals, repeating decimals
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Irrational numbers: non-repeating, non-terminating decimals like π, sqrt{2}