Exploring, Adding, Subtracting, Multiplying, and Dividing

1. Real Numbers Exploring, Adding, Subtracting, Multiplying, and Dividing

2. Each of the graphs below shows a set of numbers on a number line. The number below a point is its coordinate on the number line. • As you can see, the set of integers has negative numbers as well as zero and positive numbers. There are also numbers that are not integers, such as 0.37 or , which are rational numbers

3. A rational number is any number that you can write in the form of , where a and b are integers and b ≠ 0. A rational number in decimal form is either terminating, such as 6.27, or repeating, such as 8.222…, which you can write as 8. • All integers are rational numbers because you can write any integer n as .

4. Examples • Name the set(s) of numbers to which each number below belongs: • rational numbers • 23 natural #s, whole #s, integers, rational #s • 0 whole #s, integers, rational #s • 4.581 rational numbers • Your turn: • -12 • 5/12 • -4.67 • 6

5. I • An irrational number cannot be expressed in the form , where a and b are integers. Here are three irrational numbers. • Together, rational numbers and irrational numbers form the set of real numbers.

6. The Venn Diagram below shows the relationships of the sets of numbers that make up real numbers.

7. Inequality • An inequality is a mathematical sentence that compares the value of two expressions using an inequality symbol, such as < or >.

8. The number line below shows how values of numbers increase as you go to the right on a number line. • To compare fractions, you may find it helpful to write the fractions as decimals and then compare the decimals.

10. Two numbers that are the same distance from zero on a number line but lie in opposite directions are opposites.

11. The absolute value of a number is its distance from 0 on a number line. Both -3 and 3 are 3 units from zero. Both have an absolute value of 3. You write “the absolute value of -3” as |-3|.

12. Adding Real #s • Identity Property of Addition • For every rational number n, n + 0 = n and 0 + n = n. • Examples: -5 + 0 = -5 and 0 + 5 = 5 • The opposite of a number is its additive inverse. The number line shows the sum of 4 + (-4).

13. The additive inverse of a negative number is a positive number. The number line below shows the sum of -5 and 5. • Inverse Property of Addition • For every real number n, there is an additive inverse –n such that n + (-n) = 0. • Examples: 17 + (-17) = 0 and -17 + 17 = 0

17. Matrices • You can use matrices to add real numbers. A matrix is a rectangular arrangement of numbers in rows and columns. The plural of matrix is matrices. The matrix below shows the data in the table. • You identify the size of a matrix by the number of columns. The matrix above has 3 rows and 2 columns, so it is a matrix. Each item in a matrix is an element.

18. Matrices are equal if the elements in corresponding positions are equal. • You add matrices are the same size by adding the corresponding elements.

20. Subtracting Numbers • To subtract a number, add its opposite.

21. Subtracting Real Numbers

22. Subtracting and Absolute Value

23. Evaluating Expressions

24. Subtracting Matrices

26. Multiplication Properties • Identity Property of Multiplication • For every real number n, and • Multiplication Property of Zero • For every real number n, and • Multiplication Property of -1 • For every real number n, and

27. Multiplying Numbers with the Same Sign • The product of two positive numbers or two negative numbers is positive. • Multiplying Numbers with Different Signs • The product of a positive number and a negative number, or a negative number and a positive number, is negative.

28. Multiplying Numbers

29. Division Properties • Dividing Numbers with the Same Sign • The quotient of two positive numbers or two negative numbers is positive. • Dividing Numbers with Different Signs • The quotient of a positive number and a negative number, or a negative number and a positive number, is negative.

30. Dividing Numbers • Simplify each expression. • Your turn!

31. Inverse Property of Multiplication • For every nonzero real number a, there is a multiplicative inverse such that . • The multiplicative inverse, or reciprocal, of a nonzero rational number is . Zero does not have a reciprocal. Division by zero is undefined.

32. Division Using a Reciprocal

33. The Distributive Property

35. Combining Like Terms

38. Multi-Step Equations

41. Steps for Solving a Multi-Step Equation

42. Variables on Both Sides

44. Practicing Identity and No Solution • Determine whether each equation is an identity or whether it has no solution.

45. Group Work Time! • Together at your table, figure out what the value of each variable is in both matrices below.

46. Let’s go over your homework! • We will be looking at the homework you turned in on Monday of this week! • If you have not turned it in yet, now is the time to do so! • Make sure that you work on MATH homework AT HOME or in time allotted in class—NOT in Mr. Suralik’s English class!!!!! (Yep—He saw you do it!) • I will pass your homework back out tomorrow (still going through them!) • Write down the answers on a sheet of paper (if you wish) so that you will know the correct answers to any you missed. • Today we will go over any questions you have!