Module 9 Irrational Numbers PDF

IRRATIONAL NUMBERS GRADE 7 - MATHEMATICS  OBJECTIVES Determine the square roots of perfect squares and the cube roots of perfect cubes Identify irrational numbers involving square roots and cube roots, and their locations on the number line.  Engage The disc...

IRRATIONAL NUMBERS GRADE 7 - MATHEMATICS  OBJECTIVES Determine the square roots of perfect squares and the cube roots of perfect cubes Identify irrational numbers involving square roots and cube roots, and their locations on the number line.  Engage The discovery of irrational numbers is usually attributed to Pythagoras, more specifically the Pythagorean Hippasus of Metapontum; the first irrational number discovered was the square root of 2 ( 2). The most famous irrational number is pi 𝜋 , the symbol 𝜋 was devised by British mathematician William Jones in 1706. the other famous irrational numbers are Phi 𝜙 and the number 𝑒 or Euler’s number (e=2.718). Exploration Pythagorean Theorem In a right triangle the square of the hypotenuse is equal to the sum of the square of the legs. 𝑐 2 = 𝑎2 + 𝑏 2 or 𝑎2 + 𝑏 2 where 𝑐 is the hypotenuse and 𝑎 and 𝑏 are legs  Extension Square Roots An exponent can be used to show that a number has been multiplied by itself one or more times. 32 = 3 × 3 25 = 2 × 2 × 2 × 2 × 2 2 3  Extension Perfect Square – is the square of a whole number. 9 is a perfect square number. 9 = 32 7 is not a perfect square number A square root of a given number is a number which square is the given number. In symbols, if 𝑎2 = 𝑏, the number of 𝑎 is called a square root of 𝑏. When 2 is used as an exponent, the base is squared. When 3 is used as an exponent, the base is cubed.  Irrational Numbers is used to indicate the positive square root and is known as the radical sign Combination of the radical sign with the number is called a radical The number under the radical sign is known as the radicand.  Irrational Numbers -5 is also a square root of 25 because −5 2 = 25 The square root of a negative number doesn’t exist −9 When 𝑛 is an integer, the number 𝑛 is called a perfect square. Irrational Numbers Irrational Numbers  Irrational Numbers Cube Roots n as 3 𝑛, then is satisfies the equation 3 𝑛 3 =𝑛 3 The cube root of 8 ( 8)is 2 because 23 = 8 3 The cube root of 27 ( 27)is 3 because 33 = 27 3 The cube root of −8 ( −8)is -2 because (−2)3 = −8 3 The cube root of −27 ( −27)is -3 because (−3)3 = −27  Irrational Numbers Note: The cube root of a positive number “n” is a positive number, and the cube root of a negative number “n” is a negative number.  Ordering Radicals Between which two consecutive integers does each number lie? a. 12 b. 300 b. 300 lies between 289 and 324 a. 12 lies between 9 and 16 Thus, 289 < 300 < 324, Thus, 9 < 12 < 16, 289 < 300 < 324 9 < 12 < 16 Thus, 17 < 300 < 18 Thus, 3 < 12 < 4  Ordering Radicals Between which two consecutive integers does each number lie? a. 27 b. 215  Ordering Radicals Between which two consecutive integers does each number lie? a. 27 b. 215 b. 215 lies between 196 and 225 a. 27 lies between 25 and 36 Thus, 196 < 215 < 225, Thus, 25 < 27 < 36, 196 < 215 < 225 25 < 27 < 36 Thus, 14 < 215 < 15 Thus, 5 < 27 < 6  Ordering Radicals 3 Plot 4, 3, 20, 14, 𝑎𝑛𝑑 130 on the number line. 3 20 is between 4 and 5, 14 is between 3 and 4, and 130 is between 5 and 6. Thus, on the number line, these numbers can be represented as shown below. 3 3 14 4 20 5 130 6  Ordering Radicals Plot 10, 85, 105, 𝑎𝑛𝑑 11 on the number line.  Approximating Square Roots Divide-and-average method can be used to approximate square roots. This method works as follows: 𝑎 If 𝑎 = 𝑏 then 𝑎 = 𝑏 ∙ 𝑏 and =𝑏 𝑏  Approximating Square Roots Approximate 40 to the tenths place a. Find two integers between which 40 lies. 36 < 40 < 49 36 < 40 < 49 6 < 40 < 7 Because 40 is closer to 36 than to 49, we may try 6.3 as an estimate to 40  Approximating Square Roots Approximate 40 to the tenths place b. Divide 40 by the estimate 6.3 c. Get the average of the divisor and the quotient. 6.3 + 6.34 ≈ 6.32 2 d. Use the average as the next estimate. Repeat steps b and c until your divisor and quotient agree in the tenths place.  Approximating Square Roots Approximate 40 to the tenths place d. Use the average as the next estimate. Repeat steps b and c until your divisor and quotient agree in the tenths place. Note that 40 ≈ 6.3 to the tenths place. Since 40 is a nonterminating, non-repeating decimal, we say that 40is an irrational number.  Approximating Square Roots Approximate 50 to the tenths place Irrational Number An irrational number is a number that cannot be expressed in 𝑎 the form , where a and b are integers, and b is not equal to 0. 𝑏 IRRATIONAL NUMBERS GRADE 7 - MATHEMATICS