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**1-8A Number Systems Add closure property?**

Algebra Glencoe McGraw-Hill Linda Stamper

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What are real numbers? Pretend you are in the first grade. Your teacher asks you to count. What would you say?

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**Natural or Counting Numbers**

REAL NUMBERS Rational Numbers: Any number that can be written in the form of As a decimal they repeat or terminate. ex: Repeats ex: Terminates Irrational Numbers: ex: and These must be represented by a symbol (ex: ), or as a rounded number, or in radical form because the decimal doesn’t repeat or terminate (stop). Integers: Whole numbers and their opposites (this means positive and negative whole numbers). ex: … ־ 4, 3, 2, 1, 0, 1 ־, 2 ־, 3 ־, 4 … Whole Numbers: Natural Numbers and zero. ex: 0,1,2,3… Natural or Counting Numbers ex: 1,2,3,4,…

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**When you divide by zero and get no solution ( ), and = i (imaginary numbers).**

So what isn’t a real number? A rational number is any number you can write as a quotient of two integers, where b is not zero.

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Two numbers that are the same distance from 0 on a number line but on opposite sides of 0 are opposites. • • The numbers –2 and 2 are opposites because each is 2 units from zero. Integers are the whole numbers, including zero, and their opposites. Zero is neither positive nor negative, and zero has no opposite.

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**Name the set of numbers to which each real number belongs.**

Example 1 Example 2 Example 3 Example 4 rational irrational natural rational whole number integer integer rational

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Square Roots You will be allowed to use a calculator for tomorrow’s lesson but NOT on the CHAPTER 1 test! NO GRAPHING CALCULATORS!

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**You know how to find the square of a number**

You know how to find the square of a number. For instance, the square of 3 (written as 32) is 9. 3 3 The square of –3 is also equal to 9 because (–3)2 = 9. The inverse of a square number is the square root. Square roots are written with a radical symbol The number or expression inside a radical symbol is called the radicand. radical symbol radicand

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**All positive real numbers have two square roots:**

positive square root (principal square root) What two identical factors = 9? read as the positive square root of 9 is 3 negative square root What two identical factors = 9? read as the negative square root of 9 is –3 This may be written together: read as plus or minus the square root of 9 is plus or minus 3.

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**What two identical factors = – 9?**

All negative real numbers do NOT have square roots because two negative numbers multiplied produce a positive number. = The square root of a negative radicand is undefined! undefined Zero has only one square root and that is zero! What two identical factors = – 9? When two negatives are multiplied the result is positive.

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**The square of an integer is called a perfect square.**

Therefore 9 is a perfect square. 32 3 3 is an integer 3 is an integer 3 3.52 3.5 is not an integer (integers are whole numbers) The figure is a square but it is not composed of square sections.

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**The square of an integer is called a perfect square.**

therefore 9 is a perfect square. 32 3 3 is an integer What two identical factors = 12? 3 ( …)2 ( …) not an integer (irrational number) 12 is not a perfect square. ( …)

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**Determine whether the number is a perfect square.**

Example 5 Example 6 Example 7 Example 8 yes no no yes What two identical factors = the given number? Is your answer an integer?

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**Evaluate the expression.**

Example 9 Example 10 Example 11 Example 12 undefined

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To graph a set of numbers means to draw, or plot, the points named by those numbers on a number line. The number that corresponds to a point on a number line is called the coordinate of that point. Graph –1, 2 and – 3 on a number line. Order the numbers from least to greatest. • • • Draw a number line. Label the number line. Plot the points on the number line. –3, –1, 2 List the integers from least to greatest.

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**Example 13 Graph – 4, , – 6 and 0 on a number line**

Example 13 Graph – 4, , – 6 and 0 on a number line. Order the numbers from least to greatest. • • • • Draw a number line. – 6, – 4, 0, Label the number line. Plot the points on the number line. List the integers from least to greatest.

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**Graphing Inequalities**

For this part of the lesson, you will need a ruler and a colored pencil.

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The graph of an inequality in one variable is the set of points on a number line that represent all solutions of the inequality. 4 endpoint • ray O If the endpoint on the graph is not a solution, draw an open dot. If the endpoint on the graph is a solution, draw a solid dot. Then draw an arrowhead to show that the graph continues to infinity. What is the name for the geometric figure that represents the solution?

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**• • Reading and Graphing an Inequality in One Variable x > 2**

All real numbers greater than or equal to 2. • a < 0 All real numbers less than 0. O – 5 > y Rewrite as y < – 5 – 5 All real numbers less than or equal to – 5. • When the variable is before the inequality symbol, what do you notice about the direction of the ray and the direction of the inequality symbol?

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**You do NOT need to draw in the tick marks.**

Graphing an Inequality in One Variable 1. Write inequality. 7 > x Rewrite with variable first. x < 7 7 2. Draw a line (use arrowheads). • 3. Draw open or solid dot and label the endpoint. 4. Draw the ray in the direction of the inequality symbol. You do NOT need to draw in the tick marks.

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**Example 14 Graph the solutions of each inequality on a number line.**

a) x > – 4 –4 • b) y < 15 15 O c) –3 > x Rewrite as x < – 3 –3 •

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Homework 1-A12 Pages

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