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Feb 13, 2026

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2 pages

Understanding the Properties of Real Numbers

Chelsea

@zuncho

Real numbers form the foundation of mathematics, describing any number... Show more

Day 2
HW/Notes

1.1 & 1.2 PROPERTIES OF REAL NUMBERS

OBJECTIVES: 1) Classify real numbers and use properties of real numbers to evaluate ex

Properties of Real Numbers: Classification

Ever wonder where all the numbers you use fit in the bigger picture? Real numbers include any number that can be placed on a number line. They're organized into several important categories that help us understand their behavior.

Natural numbers (also called counting numbers) start at 1 and continue infinitely {1, 2, 3...}. Add zero to this set and you get whole numbers {0, 1, 2...}. When you include negative whole numbers, you're working with integers {...-3, -2, -1, 0, 1, 2...}.

Rational numbers can be written as fractions (ratios of integers), like $\frac{2}{3}$ , -2.75, or even 9.9. Irrational numbers cannot be expressed as fractions and have non-repeating, non-terminating decimals, such as $\sqrt{6}$ or $\pi$ .

Quick Check: When classifying a number like $\sqrt{121}$ , first simplify it (equals 11) to determine it belongs to the real, rational, integer, whole, and natural number sets.

Remember that every number belongs to multiple sets. For example, all natural numbers are also whole numbers, integers, rationals, and reals. But an irrational number like $\sqrt{6}$ only belongs to the set of real numbers.

Number Properties and Domains

The way numbers behave follows specific patterns called properties. These rules help us manipulate expressions and solve equations efficiently.

For addition, we have the commutative property $$a+b=b+a$$ , which means order doesn't matter, and the associative property, which lets us regroup numbers $((a+b)+c=a+(b+c))$ . The identity property shows that adding zero leaves a number unchanged $$a+0=a$$ , while the inverse property tells us that adding opposites gives zero $$a+(-a)=0$$ .

Multiplication has similar properties, with 1 as its identity element $$a\cdot1=a$$ and reciprocals as inverses $a·\frac{1}{a}=1$ . The distributive property connects multiplication and addition: $a(b+c)=ab+ac$ .

Remember: When solving in specific domains, check if your solution fits the required set. For example, $x^2=8$ has solutions $±2\sqrt{2}$ in the real number domain, but no solutions in the rational domain!

When solving equations like $(x+5)(2x-3)=0$ , set each factor equal to zero to find $x=-5$ or $x=\frac{3}{2}$ . If the domain is integers, only $x=-5$ works since $\frac{3}{2}$ isn't an integer.

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Algebra 2

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115

•

Feb 13, 2026

•

2 pages

Understanding the Properties of Real Numbers

Chelsea

@zuncho

Real numbers form the foundation of mathematics, describing any number that can be plotted on a number line. Understanding how to classify numbers and apply their properties is essential for solving equations and working with expressions in algebra and beyond.

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Properties of Real Numbers: Classification

Quick Check: When classifying a number like $\sqrt{121}$ , first simplify it (equals 11) to determine it belongs to the real, rational, integer, whole, and natural number sets.

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Access to all documents

Improve your grades

Join milions of students

Number Properties and Domains

The way numbers behave follows specific patterns called properties. These rules help us manipulate expressions and solve equations efficiently.

Remember: When solving in specific domains, check if your solution fits the required set. For example, $x^2=8$ has solutions $±2\sqrt{2}$ in the real number domain, but no solutions in the rational domain!

When solving equations like $(x+5)(2x-3)=0$ , set each factor equal to zero to find $x=-5$ or $x=\frac{3}{2}$ . If the domain is integers, only $x=-5$ works since $\frac{3}{2}$ isn't an integer.