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Properties of Real Numbers

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A series of free, online Intermediate Algebra Lessons or Algebra II lessons and solutions.
Videos, worksheets, and activities to help Algebra students.

In this lesson, we will learn

  • how to classify real numbers
  • how to analyze data
  • how to define the properties of real numbers

Introduction to Real Numbers

When analyzing data, graphing equations and performing computations, we are most often working with real numbers. Real numbers are the set of all numbers that can be expressed as a decimal or that are on the number line. Real numbers have certain properties and different classifications, including natural, whole, integers, rational and irrational.

How to classify different types of real numbers?
Rational numbers can be written as a fraction, as an integer over integer.
Irrational numbers cannot be written as a fraction, as an integer over integer.
Written as a decimal, irrational numbers never terminate nor repeat forever.
The real numbers consists of all rational numbers plus all irrational numbers.



Properties of Real Numbers

When analyzing data or solving problems with real numbers, it can be helpful to understand the properties of real numbers. These properties of real numbers, including the Associative, Commutative, Multiplicative and Additive Identity, Multiplicative and Additive Inverse, and Distributive Properties, can be used not only in proofs, but in understanding how to manipulate and solve equations.

How to define the properties of real numbers?
The properties that we'll see in this lesson can be used to manipulate any algebraic expression. They allow us to:
• Simplify Expressions
• Rewrite equations so they are easier to solve

Additive Identity Property
For any real number a, a + 0 = a

Multiplicative Identity Property
For any real number a, a • 1 = 1 • a = a

Additive Inverse Property
For any real number a, there is a real number -a such that
a + (-a) = (-a) + a = 0

Multiplicative Identity Property
For any real number a, where a ≠ 0
a • 1/a = 1/a • a = 1

Commutative Property of Addition
For any real number a and b,
a + b = b + a

Commutative Property of Multiplication
For any real number a and b,
a • b = b • a

Associative Property of Addition
For any real number a, b, and c,
a + (b + c) = (a + b) + c

Associative Property of Multiplication
For any real number a, b, and c,
a (bc) = (ab)c

Distributive Property
For any real number a, b, and c,
a(b + c) = ab + ac



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