The Number System
GradesGrade 8The Number SystemProperties of Real Numbers

Properties of Real Numbers

∞ The Number System
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Properties of Real Numbers

The real number system obeys fundamental properties that underpin all algebraic reasoning.

The set of real numbers includes all rational and irrational numbers. Together they form a complete, ordered number line with no gaps. Operations on real numbers obey properties that hold for any real values a, b, and c.

Key properties of real numbers:

Commutative (+ and ×): a + b = b + aAssociative (+ and ×): (a + b) + c = a + (b + c)Distributive: a(b + c) = ab + acIdentity: a + 0 = a and a × 1 = aInverse: a + (−a) = 0 and a × (1/a) = 1 (a ≠ 0)Closure: sum or product of two reals is always a real number
🔄Closure and irrational numbers
The sum of two irrationals is not always irrational:
√2 + (−√2) = 0 ← rational!

But the product of two irrationals can be rational:
√2 × √2 = 2 ← rational!

However, √2 × √3 = √6 ← still irrational.
2×2=2but2×3=6\sqrt{2} \times \sqrt{2} = 2 \quad \text{but} \quad \sqrt{2} \times \sqrt{3} = \sqrt{6}
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Remember This!

The density property: between any two distinct real numbers, there is always another real number. The real number line has no "gaps" or "jumps."

✏ī¸ Try It!

Which property is illustrated by: 5(x + 3) = 5x + 15?

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