Supplemental Worksheet Problems To Accompany: The Pre-Algebra Tutor: Volume 1 Section 1 Real Numbers
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1 Supplemental Worksheet Problems To Accompany: The Pre-Algebra Tutor: Volume 1 Please watch Section 1 of this DVD before working these problems. The DVD is located at: Page 1
2 Part 1: Real Numbers 1) Identify the real numbers below. 2) Identify the real numbers below. 3) Identify the real numbers below. Page 2
3 Part 2: Rational Numbers 4) Which of the following is a rational number? 5) Which of the following is a rational number? 6) Which of the following is a rational number? Page 3
4 Part 3: Irrational Numbers 7) Which of the following is an irrational number? 8) Which of the following is an irrational number? 10.10, 0.12, 18, π 9) Which of the following is an irrational number? 2, 30, , Page 4
5 Part 4: Integers 10) Which of the following is not an integer? 11) Which of the following is not an integer? 12) Which of the following is not an integer? Page 5
6 Part 5: Whole Numbers 13) Identify the whole number(s) below. 14) Identify the whole number(s) below. 15) Identify the whole number(s) below. Page 6
7 Part 6: Natural Numbers 16) Identify the natural number(s) below. 17) Identify the natural number(s) below. 18) Identify the natural number(s) below. Page 7
8 Part 7: Prime Numbers 19) Identify the prime number(s) below. 20) Identify the prime number(s) below. 21) Identify the prime number(s) below. Page 8
9 Part 8: Putting it all together For each of the numbers below, identify which group they belong to. (Real, Irrational, Rational, Integer, Whole, Natural and/or Prime number) Example: Is both a Real and a Rational number. 22) 23) 24) 25) Page 9
10 Question Answer 1) Identify the real numbers below. First, we need to remember the definition of a real number. This is any number that can be located on a number line. This excludes imaginary numbers. All of the numbers mentioned can be plotted onto a number line even if they are fractions or have numerous decimal places, they can still be plotted somewhere on a number line. All of them are real numbers Since all of the numbers meet the definition of a real number they are all by definition real numbers. Ans: All are considered real numbers Page 10
11 2) Identify the real numbers below. First, we need to remember the definition of a real number. This is any number that can be located on a number line. This excludes imaginary numbers. All of the numbers mentioned can be plotted onto a number line even if they are fractions or have numerous decimal places, they can still be plotted somewhere on a number line. All of them are real numbers Since all of these numbers meet the definition of a real number they are all by definition real numbers. Ans: All are considered real numbers Page 11
12 3) Identify the real numbers below. First, we need to remember the definition of a real number. This is any number that can be located on a number line. This excludes imaginary numbers. All of the numbers mentioned can be plotted onto a number line even if they are fractions or have numerous decimal places, they can still be plotted somewhere on a number line. All of them are real numbers Since all of these numbers meet the definition of a real number they are all by definition real numbers. Ans: All are considered real numbers Page 12
13 4) Which of the following is a rational number? First, we need to remember the definition of a rational number. This is any number that can be expressed as a fraction. We see as we express the numbers in fraction form that all but one can be expressed as a fraction. If we type the square root of two into a calculator we notice that the result is a non repeating decimal pattern. The rest of the numbers can be expressed as a fraction and therefore are rational numbers. Ans: Page 13
14 5) Which of the following is a rational number? First, we need to remember the definition of a rational number. This is any number that can be expressed as a fraction. We see as we express the numbers in fraction form that all but one can be expressed as a fraction. We notice right away that has a non repeating infinite pattern. Therefore, there is no way to express this number as a fraction. The rest of the numbers can be expressed as a fraction and therefore are rational numbers. Ans: Page 14
15 6) Which of the following is a rational number? First, we need to remember the definition of a rational number. This is any number that can be expressed as a fraction. We see as we express the numbers in fraction form that all but one can be expressed as a fraction. If we type into a calculator we notice that pi has a non repeating decimal pattern that goes on forever. Therefore, there is no way to express this number as a fraction. The rest of the numbers can be expressed as a fraction and therefore are rational numbers. Ans: Page 15
16 7) Which of the following is an irrational number? First, we need to remember the definition of an irrational number. This is any number that can t be expressed as a fraction. All of the numbers mentioned can be written as a fraction. Since all of these numbers can be expressed as fractions, none of these numbers are considered irrational Ans: None of the numbers listed are irrational numbers Page 16
17 8) Which of the following is an irrational number? 10.10, 0.12, 18, π First, we need to remember the definition of an irrational number. This is any number that can t be expressed as a fraction. The first three numbers mentioned can be written as a fraction. Since Pi = is an infinite non repeating decimal, it cannot be written as a fraction. π = Ans: Pi is an Irrational Number Page 17
18 9) Which of the following is an irrational number? 2, 30, , First, we need to remember the definition of an irrational number. This is any number that can t be expressed as a fraction. All of the numbers can be written as a fraction except for the infinite non repeating decimal. This is the only irrational number in our list Ans: is irrational Page 18
19 10) Which of the following is not an integer? First, we need to remember the definition of what an integer is. This is, any number that is positive, negative or zero, but has no decimal point.,-5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5 Right away, we look at the list of numbers and notice that all but one falls under this definition. Ans: Page 19
20 11) Which of the following is not an integer? First, we need to remember the definition of what an integer is. This is, any number that is positive, negative or zero, but has no decimal place.,-5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5 Right away, we look at the list of numbers and notice that all but one falls under this definition. The negative fraction will provide a result with a decimal place when we divide 15 by 16. Since the square root of 9 is exactly 3, this is an integer as well. Ans: Page 20
21 12) Which of the following is not an integer? First, we need to remember the definition of what an integer is. This is, any number that is positive, negative or zero, but has no decimal place.,-5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5 Right away, we look at the list of numbers and notice that all but one falls under this definition. Ans: Page 21
22 13) Identify the whole number(s) below. First, we need to remember the definition of what makes a whole number. A whole number is any positive number including zero that does not have a decimal place. By simply applying the definition we see that the two negative numbers are not whole numbers. Zero and one hundred do fit the definition. Ans: Page 22
23 14) Identify the whole number(s) below. First, we need to remember the definition of what makes a whole number. A whole number is any positive number including zero that does not have a decimal place. First we see that the negative number does not meet our definition. Then we see that the fraction will give us a decimal place as well as the square root of 2. The number fifteen is the only whole number. Ans: Page 23
24 15) Identify the whole number(s) below. First, we need to remember the definition of what makes a whole number. A whole number is any positive number including zero that does not have a decimal place. We see that 640 is both positive and does not contain a decimal place. The negative 640 however does not meet our definition so it is not a whole number. The one half will give us a decimal place, 0.5 so it is not a whole number. The number 1.1 is positive but has a decimal place and therefore is not a whole number. The only whole number from the list is 640. Ans: Page 24
25 16) Identify the natural number(s) below. First, we need to remember the definition of what makes a natural number. This is any number that can be used to physically count something. This includes: 1,2,3,4,5,6,7,8,9.. Based on our definition, we see that the negative numbers are not natural numbers because I can t physically count something I don t have so the negative numbers are not natural numbers. The last number to consider from the list is zero. If I have zero of something I can t naturally count it so zero is not a natural number. None. Since none of the numbers meet our criteria of what makes a natural number none of these are natural numbers Ans: None Page 25
26 17) Identify the natural number(s) below. First, we need to remember the definition of what makes a natural number. This is any number that can be used to physically count something. This includes: 1,2,3,4,5,6,7,8,9.. Based on our definition, we see that the negative numbers are not natural numbers because I can t physically count something I don t have so the negative numbers are not natural numbers. The square root of 9 is the same as 3, so I can physically count something if I have 3 apples or 3 pencils so the square root of 9 is a natural number. The last number to consider from the list is 26. I can definitely physically have 26 of something and count it so 26 is a natural numbers. Ans: Are natural numbers Page 26
27 18) Identify the natural number(s) below. First, we need to remember the definition of what makes a natural number. This is any number that can be used to physically count something. This includes: 1,2,3,4,5,6,7,8,9.. Based on our definition, we see that the negative numbers are not natural numbers because I can t physically count something I don t have so the negative numbers are not natural numbers. We are left with 640 and 0. If I have zero of something I can t naturally count it so zero is not a natural number. The number 640 however can be used to physically count something like the 640 pennies in my piggy bank. Therefore the number 640 is a natural number. The only natural number is the number 640. Ans: Page 27
28 19) Identify the prime number(s) below. First, we need to remember the definition of a prime number. This is a whole number other than zero and 1, which can only be divided by 1 and itself. When we say divided only by 1 and itself, we mean that those are the only numbers that will not give you a decimal place when you divide. The first number we have is 32. It is by definition a whole number and it is not zero or 1. However, 1 and 32 are not the only numbers I can divide 32 by. I can divide it by 2, 4, 8, and 16. So therefore it is not a prime number. The number 2 is a whole number that is not zero or 1. And I can only divide it by 1 and 2, so it is a prime number. The number zero already violates our definition so it is not a prime number. The number 16 is a whole number that is not zero or 1. However 1 and 16 are not the only numbers I can divide 16 by. I can divide it by 2, 4 and 8. So it is not a prime number. Ans: Page 28
29 20) Identify the prime number(s) below. First, we need to remember the definition of a prime number. This is a whole number other than zero and 1, which can only be divided by 1 and itself. When we say divided only by 1 and itself, we mean that those are the only numbers that will not give you a decimal place when you divide. The number 10 by definition a whole number and it is not zero or 1. However, 1 and 10 are not the only numbers I can divide 10 by. I can divide it by 2 and 5. So therefore it is not a prime number. The number 17 is a whole number and it is not zero or 1. When we try to divide 17 by something other than 1 or 17, we find that we can t do it without ending up with a decimal place. Therefore 17 is a prime number. The number 3 as well as 17 can only be divided by 1 and itself therefore it is a prime number. The number 1 violates our definition so it is not a prime number. Ans: Page 29
30 21) Identify the prime number(s) below. First, we need to remember the definition of a prime number. This is a whole number other than zero and 1, which can only be divided by 1 and itself. When we say divided only by 1 and itself, we mean that those are the only numbers that will not give you a decimal place when you divide. As we look at the first two numbers, -2 and 0.8, we notice they are not whole numbers which violates our definition of a prime number and therefore are not considered prime numbers. The number 15 is a whole number and it is not zero or 1. However, 1 and 15 are not the only numbers I can divide 15 by. I can divide it by the number 5 and still end up with a natural number (no decimal or zero). So it is not a prime number. The number 22 is a whole number and it is not zero or 1. However, 1 and 22 are not the only numbers I can divide 22 by. I can divide it by 2, and 11. So it is not a prime number. None. None of the numbers meet our definition of a prime number. Ans: None. Page 30
31 22) First, we need to remember our definitions and how each type of number is related to the other types of numbers. Remember that Real numbers is at the top of the umbrella. Everything else falls under it. Then it breaks off into Irrational and Rational numbers. Then under Rational numbers are Integers, Whole numbers, Natural numbers and Prime numbers. If I know a number is Irrational, then by definition it is not Rational or anything that falls under a Rational number. This number is a real number. Next we see if it is irrational or rational. Since I can write it as a fraction, then it is a rational number. Since it is has a decimal place, it is not an integer, a whole number, natural number or a prime number. Ans: Real number, Rational Page 31
32 23) number First, we need to remember our definitions and how each type of number is related to the other types of numbers. Remember that Real numbers is at the top of the umbrella. Everything else falls under it. Then it breaks off into Irrational and Rational numbers. Then under Rational numbers are Integers, Whole numbers, Natural numbers and Prime numbers. If I know a number is Irrational, then by definition it is not Rational or anything that falls under a Rational number. This number is a Real number. Next we see if it is Irrational or Rational. Just by looking at it, we see that it is Rational since we can write it as 7 over 1. It has no decimal place and it is positive so it is an Integer, a Whole number and a Natural number. We find that I can only divide this number by 1 and itself so it is a Prime number. Ans: Real number, Rational Number, Integer, Whole number, Natural number, and Prime Page 32
33 24) number First, we need to remember our definitions and how each type of number is related to the other types of numbers. Remember that Real numbers is at the top of the umbrella. Everything else falls under it. Then it breaks off into Irrational and Rational numbers. Then under Rational numbers are Integers, Whole numbers, Natural numbers and Prime numbers. If I know a number is Irrational, then by definition it is not Rational or anything that falls under a Rational number. This number is a Real number. Next we see if it is Irrational or Rational. Just by looking at it, we see that it has decimal places that are non repeatable and infer to go on forever. This means we can t represent it as a fraction and therefore is an Irrational number. Since it is an Irrational number it is not Rational and not any of the other types of numbers. Ans: Real number, Irrational Number Page 33
34 25) First, we need to remember our definitions and how each type of number is related to the other types of numbers. Remember that Real numbers is at the top of the umbrella. Everything else falls under it. Then it breaks off into Irrational and Rational numbers. Then under Rational numbers are Integers, Whole numbers, Natural numbers and Prime numbers. If I know a number is Irrational, then by definition it is not Rational or anything that falls under a Rational number. This number is a Real number. Next we see if it is Irrational or Rational. Just by looking at it, we see that it is Rational since we can write it as -2 over 1. We also notice that it has a negative sign but no decimal place. This number is still considered an Integer, but not considered a Whole number. Since it is not considered a whole number it can t be a Natural number or a Prime number by definition. Ans: Real number, Rational Number, and an Integer Page 34
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