5 The set of real numbers is all numbers that can be written on a number line. It consists of the set of rational numbers and the set of irrational numbers.Irrational numbersRational numbersReal NumbersIntegersWholenumbers
6 Recall that rational numbers can be written as the quotient of two integers (a fraction) or as either terminating or repeating decimals.4523= 3.8= 0.61.44 = 1.2
7 Irrational numbers can be written only as decimals that do not terminate or repeat. They cannot be written as the quotient of two integers. If a whole number is not a perfect square, then its square root is an irrational number. For example, 2 is not a perfect square, so is irrational.A repeating decimal may not appear to repeat on a calculator, because calculators show a finite number of digits.Caution!
8 Reals Make a Venn Diagram that displays the following sets of numbers: Reals, Rationals, Irrationals, Integers, Wholes, and Naturals.RealsRationals-2.65Integers-3-19WholesIrrationalsNaturals1, 2, 3...
9 Additional Example 1: Classifying Real Numbers Write all classifications that apply to each number.A.55 is a whole number that is not a perfect square.irrational, realB.–12.75–12.75 is a terminating decimal.rational, real162= = 24216C.whole, integer, rational, real
10 Write all classifications that apply to each number. Check It Out! Example 1Write all classifications that apply to each number.A.99 = 3whole, integer, rational, realB.–35.9–35.9 is a terminating decimal.rational, real813= = 39381C.whole, integer, rational, real
11 A fraction with a denominator of 0 is undefined because you cannot divide by zero. So it is not a number at all.
12 Additional Example 2: Determining the Classification of All Numbers State if each number is rational, irrational, or not a real number.A.21irrational33= 0B.rational
13 Additional Example 2: Determining the Classification of All Numbers State if each number is rational, irrational, or not a real number.4 0C.not a real number
14 State if each number is rational, irrational, or not a real number. Check It Out! Example 2State if each number is rational, irrational, or not a real number.A.2323 is a whole number that is not a perfect square.irrational9B.undefined, so not a real number
15 State if each number is rational, irrational, or not a real number. Check It Out! Example 2State if each number is rational, irrational, or not a real number.648189=6481C.rational
16 Square Number Also called a “perfect square” A number that is the square of a whole numberCan be represented by arranging objects in a square.
24 Square NumbersOne property of a perfect square is that it can be represented by a square array.Each small square in the array shown has a side length of 1cm.The large square has a side length of 4 cm.4cm4cm16 cm2
25 Square Numbers The large square has an area of 4cm x 4cm = 16 cm2. The number 4 is called the square root of 16.We write: 4 =4cm4cm16 cm2
26 Square RootA number which, when multiplied by itself, results in another number.Ex: 5 is the square root of 25.5 = 25
27 Finding Square Roots 4 x 9 = 4 x 9 36 = 2 x 3 6 = 6 We can use the following strategy to find a square root of a large number.4 x 9 = x 9= x 3=
28 Finding Square Roots 4 x 9 = 4 9 36 = 2 x 3 6 = 6 =We can factor large perfect squares into smaller perfect squares to simplify.
29 Finding Square Roots 256 = 4 x 64 = 2 x 8 = 16 Activity: Find the square root of 256256= x64= 2 x 8= 16
36 Estimating Square Roots 27 = ?Since 27 is not a perfect square, wehave to use another method tocalculate it’s square root.
37 Estimating Square Roots Not all numbers are perfect squares.Not every number has an Integer for a square root.We have to estimate square roots for numbers between perfect squares.
38 Estimating Square Roots To calculate the square root of a non-perfect square1. Place the values of the adjacent perfect squares on a number line.2. Interpolate between the points to estimate to the nearest tenth.
39 Estimating Square Roots Example:What are the perfect squares on each side of 27?25303536