Learning Math: Number and Operations
Number Sets, Infinity, and Zero Part A: Number Sets (35 minutes)
In This Part: Relating Number Sets
We will continue our focus on the number line and the relationships among the various types of numbers that make up the real number system. The following exercises will help you further understand the properties that hold true for each of the sets of numbers and the relationships among them.
As we saw in Session 1, the real number system is made up of many different sets. Some of these sets are quite large and contain other smaller sets. The integers, for example — made up of the whole numbers and their negatives — clearly contain the counting numbers (1, 2, 3, 4, …). But which sets contain which other sets, and how do they all relate to one another? Let’s explore.
Using the number line from Session 1 as a reference, draw a diagram that illustrates the relationships among the different sets of numbers that make up our number system — the real numbers plus imaginary and complex numbers. Include all of the sets we discussed in Session 1:
- Counting numbers (1, 2, 3, 4, …)
- Whole numbers (the counting numbers and 0)
- Integers (positive and negative whole numbers)
- Rational numbers (numbers that can be expressed as a ratio of two integers; when expressed in a decimal form, they will either terminate or repeat)
- Irrational numbers (numbers such as or e or square roots that can’t be expressed as a ratio of two integers; they can be expressed as infinite, non-repeating decimals)
- Transcendental numbers (numbers that cannot be the solution to a polynomial equation; e.g., and e)
- Real numbers (all rational and irrational numbers; numbers that can be represented on a number line)
- Algebraic numbers are solutions to polynomial equations with rational coefficients (e.g., 1/2x3 – 3x2 +17x +5/8). They include all integers, rational numbers, and some irrational numbers (e.g.,, the solutions to x2 – 2 = 0). Algebraic numbers also include some complex numbers (e.g.,-1, the solutions to x2 + 1 = 0).
- Pure imaginary numbers (multiples of i, a number such that when you square it, you get -1; e.g., 5i, 99i)
- Complex numbers (numbers created by the addition of imaginary and real number elements; e.g., 1 + 5i; 3/2 + 19i)
Use boxes or circles to represent each number set. Shapes that represent number sets should be placed within their larger set in the number system.
In This Part: Operations
Compare and contrast your diagram with the diagram below, which shows one way to illustrate the relationships among sets of numbers.
Which operations can we do within the following sets: counting numbers, whole numbers, integers, irrational numbers and rational numbers (i.e., for each set decide under which operations is it closed)?
Select a set and try adding, subtracting, multiplying, or dividing random numbers from that set. What happens?
a. Within each set, which operations require us to expand to a new set?
b. To go from one set to the next biggest, what new types of numbers do you need to include?
In this video segment, Donna and Susan contemplate the relationships between different sets of numbers in the real number system. They discuss the operations and how different operations require them to expand the number sets they’re using. Watch this segment after you’ve completed Problems A1-A3.
You can find this segment on the session video approximately 4 minutes and 7 seconds after the Annenberg Media logo.
These diagrams are known as Venn diagrams. To learn more about Venn diagrams, go to Learning Math: Number and Operations, Session 6, and Learning Math: Geometry, Session 3, Part B: Game.
The fact that all algebraic numbers lie within the complex numbers was proven by a German mathematician, Carl Friedrich Gauss, and is known as the Fundamental Theorem of Algebra.
For each set, you can only do operations for which that set is closed:
- Counting Numbers: This set is closed only under addition and multiplication. In other words, we can solve all addition and multiplication problems, but not all subtraction and division problems are solvable.
- Whole Numbers: This set is closed only under addition and multiplication.
- Integers: This set is closed only under addition, subtraction, and multiplication.
- Rational Numbers: This set is closed under addition, subtraction, multiplication, and division (with the exception of division by 0).
- Irrational Numbers: This set is closed for none of the operations (e.g., • = 2, a rational number).
- Real Numbers: This set is closed only under addition, subtraction, multiplication, and division (with the exception of division by 0).
a. Operations under which a particular set is not closed require new sets of numbers:
- Counting Numbers: Subtraction requires 0 and negative integers; division requires rational numbers.
- Whole Numbers: Subtraction requires negative integers; division requires rational numbers.
- Integers: Division requires rational numbers.
- Rational Numbers: All four operations are okay here (with the exception of division by 0). However, solving problems with exponents would require us to expand from the rational numbers. For example, a problem like x2 = 3 can be solved using the real numbers, but not the rational numbers.
- Irrational Numbers: All operations require rational numbers.
- Real Numbers: All four operations are okay here (with the exception of division by 0).
b. To go from one set to the next requires new types of numbers:
- To go from counting numbers to whole numbers, we need the additive identity 0.
- To go from whole numbers to integers, we need the additive inverses — the opposites of the counting numbers.
- To go from integers to rational numbers, we need the multiplicative inverses of all non-zero counting numbers and their multiples. These are fractions with integer numerators and denominators, like 2/3 and -7/4.
- To go from rational numbers to real numbers, we need irrational numbers, such as and . Similarly, to go from irrational to real numbers, we need rational numbers.
- To go from real numbers to complex numbers, we need i (a number such that when squared it gives -1) and all its real multiples — the imaginary numbers. Adding any real number and any imaginary number then forms a complex number, for example, 2 + 3i and -2/3 + 2.718i.
Session 1 What Is a Number System?
Understand the nature of the real number system, the elements and operations that make up the system, and some of the rules that govern the operations. Examine a finite number system that follows some (but not all) of the same rules, and then compare this system to the real number system. Use a number line to classify the numbers we use, and examine how the numbers and operations relate to one another.
Session 2 Number Sets, Infinity, and Zero
Continue examining the number line and the relationships among sets of numbers that make up the real number system. Explore which operations and properties hold true for each of the sets. Consider the magnitude of these infinite sets and discover that infinity comes in more than one size. Examine place value and the significance of zero in a place value system.
Session 3 Place Value
Look at place value systems based on numbers other than 10. Examine the base two numbers and learn uses for base two numbers in computers. Explore exponents and relate them to logarithms. Examine the use of scientific notation to represent numbers with very large or very small magnitude. Interpret whole numbers, common fractions, and decimals in base four.
Session 4 Meanings and Models for Operations
Examine the operations of addition, subtraction, multiplication, and division and their relationships to whole numbers. Work with area models for multiplication and division. Explore the use of two-color chips to model operations with positive and negative numbers.
Session 5 Divisibility Tests and Factors
Explore number theory topics. Analyze Alpha math problems and discuss how they help with the conceptual understanding of operations. Examine various divisibility tests to see how and why they work. Begin examining factors and multiples.
Session 6 Number Theory
Examine visual methods for finding least common multiples and greatest common factors, including Venn diagram models and area models. Explore prime numbers. Learn to locate prime numbers on a number grid and to determine whether very large numbers are prime.
Session 7 Fractions and Decimals
Extend your understanding of fractions and decimals. Examine terminating and non-terminating decimals. Explore ways to predict the number of decimal places in a terminating decimal and the period of a non-terminating decimal. Examine which fractions terminate and which repeat as decimals, and why all rational numbers must fall into one of these categories. Explore methods to convert decimals to fractions and vice versa. Use benchmarks and intuitive methods to order fractions.
Session 8 Rational Numbers and Proportional Reasoning
Begin examining rational numbers. Explore a model for computations with fractions. Analyze proportional reasoning and the difference between absolute and relative thinking. Explore ways to represent proportional relationships and the resulting operations with ratios. Examine how ratios can represent either part-part or part-whole comparisons, depending on how you define the unit, and explore how this affects their behavior in computations.
Session 9 Fractions, Percents, and Ratios
Continue exploring rational numbers, working with an area model for multiplication and division with fractions, and examining operations with decimals. Explore percents and the relationships among representations using fractions, decimals, and percents. Examine benchmarks for understanding percents, especially percents less than 10 and greater than 100. Consider ways to use an elastic model, an area model, and other models to discuss percents. Explore some ratios that occur in nature.
Session 10 Classroom Case Studies, K-2
Watch this program in the 10th session for K-2 teachers. Explore how the concepts developed in this course can be applied through case studies of K-2 teachers (former course participants) who have adapted their new knowledge to their classrooms.
Session 11 Classroom Case Studies, 3-5
Watch this program in the 10th session for grade 3-5 teachers. Explore how the concepts developed in this course can be applied through case studies of grade 3-5 teachers (former course participants) who have adapted their new knowledge to their classrooms.