1-8A Number Systems Add closure property? Algebra 1 Glencoe McGraw-Hill Linda Stamper

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

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,…

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.

Two numbers that are the same distance from 0 on a number line but on opposite sides of 0 are opposites. • • -2 -1 0 1 2 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.

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

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

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

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.

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.

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.

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 (3.4641016…)2 (3.4641016…) not an integer (irrational number) 12 is not a perfect square. (3.4641016…)

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?

Evaluate the expression. Example 9 Example 10 Example 11 Example 12 undefined

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. • • • -4 -3 -2 -1 0 1 2 3 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.

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. • • • • -6 -5 -4 -3 -2 -1 0 1 2 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.

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

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?

• • 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?

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.

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 •

Homework 1-A12 Pages