Learning Math: Number and Operations
Place Value Part B: Exponents and Logarithms (35 minutes)
As you saw earlier in this session, one way to represent each power of two is to write the base (two) raised to a power (another number). This other number is known as the exponent. An exponent tells us how many times the base is used as a factor. Exponents can simplify the calculations for such operations as multiplication and division. For example, rather than multiply 16 • 32, we can multiply 24 • 25. Let’s look at how this is done.
To compute with numbers that have exponents, you need to understand how exponents work. Here are some basic rules to begin with:
- To add or subtract numbers with exponents, the base numbers must be the same, and the exponents must also be the same:
x4 + x4 + x3 + x3 = 2x4 + 2x3
- To multiply numbers with exponents, the base numbers must be the same; then we simply add the exponents. For example:
x4 • x3
x4 = x • x • x • x and x3 = x • x • x
Because multiplication is both associative and commutative, we can solve these equations as one:
x4 • x3 = x • x • x • x • x • x • x = x7
So the end result, x7, is equivalent to x4 + 3:
x4 • x3 = (x • x • x • x) • (x • x • x) = x4 + 3 = x7
This presumes that both the bases are the same. In other words, for example, we couldn’t multiply 22 by 33 because the bases are not the same.
- To divide numbers with exponents, the base numbers must be the same; then we simply subtract the exponents. For example:
These examples illustrate the meaning of positive-integer exponents. But what does an exponent of 0 represent?
Use the rule xa xb = xa-b to figure out the value of x when the exponent is 0.
If a – b = 0, what does that tell you about a and b?
What happens if the exponent is a negative integer like -1? Solve x3÷x4 to find out. Explain why x cannot be equal to 0.
Now let’s look at what happens when the exponent is a fraction or a decimal. We know that for positive numbers greater than or equal to 1, x2 x3 x4. Is this true for exponents between 0 and 1?
a. Use the rules for multiplying exponents to determine the meaning of x1/2.
b. How about x1/3?
c. Which value is greater for positive numbers greater than 1?
Express (x3)2 as a multiplication problem and then simplify it as much as possible.
Consider your answers to Problems B1-B4. In each case, can 0 be a valid base? Explain why (or why not).
In This Part: Scientific Notation
When doing complicated computations with very large or very small numbers, people often write the numbers in scientific notation so that they can more easily estimate the magnitude of the number. In scientific notation, every number is written as a decimal number greater than or equal to 1 but less than 10, multiplied by a power of 10. Thus, the decimal always has exactly one non-zero digit to the left of the decimal point. For example, 6,253 would be written 6.253 • 103, and .06253 would be written 6.253 • 10-2. The best way to remember how to find the correct power of 10 is to write the number with one digit to the left of the decimal point and then think about what to do to that decimal to make it equal to the original number.
Write the following numbers using scientific notation:
Scientific notation can help us quickly perform operations on large numbers. Calculating 2,300,000 • 3,000,000,000 is much easier to think of as the following:
(2.3 • 106) • (3.0 • 109) = 6.9 • 1015
Perform the following calculations using scientific notation:
a. 2,300,000 + 790,000
b. 10,000,000 • 678,000,000,000
c. 1,490,000,000 7,000
In This Part: Logarithms
John Napier from Scotland invented logarithms in the early 17th century. Napier was not a professional mathematician, but he made many important contributions to mathematics. He invented not only logarithms, but the decimal point as well, and he carved multiplication tables on sticks to simplify the multiplication of multi-digit numbers.
What are logarithms? Basically, they are exponents. In order to use logarithms, you must stipulate both the base and the exponent. Here is one example: Since 23 is equal to 8, we can write in symbols log2 8 = 3, which we read as “log to the base two of 8 is 3.” This means that the exponent needed on the base two to get to 8 is 3. When working in base ten, it is not necessary to write the base. For example, log 50 is the same as log10 50.
Before the advent of calculators and computers, logarithms were extremely important because they simplified complex multiplication and division by turning them into simple addition or subtraction, and reduced powers to multiplication. See
Most calculators are programmed with values for base ten (abbreviated LOG) and base e(abbreviated LN) logarithms. The letter e represents the transcendental number that is the base of natural logarithms. The value of e is found by taking the limit of (1 + 1/n)nas n approaches infinity. This gives the value of about 2.718.
a. What is log 100?
This question is asking, what exponent is needed in base ten to get 100? In other words, if 10x = 100, what is x?
b. What is log3 81?
c. What is the base ten number whose log is 4?
d. What is logb b?
a. Estimate the value of log5 50.
b. Estimate the value of log3 100.
These logarithms will be decimals. Try to figure out between which integers these fractions will fall.
One way to do this division is to use the rule that xa xb = xa – b. Here, this means that x3 divided by x3 equals x0. The other way to do this is to recognize that we are dividing a number by itself and that any non-zero number divided by itself equals 1:
Since the answers must be the same, this means that x0 = 1 for any value of x except 0.
One way to do this division is to use the rule that xa xb = xa – b. Here, this means that x3 divided by x4 equals x-1. The other way to do this is to write out the numerator and denominator:
If x is not 0, we can cancel x three times from the numerator and denominator to leave 1/x as the final answer. Since the answers must be the same, this means that x-1 = 1/x for any value of x except 0 (1/x is undefined for x = 0). Therefore, in any division problem involving a negative exponent, we must restrict the base to a non-zero number.
a. If x1/2 follows the same rules as xm for integer m, then x1/2 multiplied by x1/2 must be x. This means that x1/2 is the number we multiply by itself to make x. This is the definition of a square root, so x1/2 represents the square root of x.
b. Similarly, x1/3 needs to be multiplied by itself three times to make x, so it is the cube root of x.
c.If x is a positive number greater than 1, then x1/2 will be greater than x1/3. One way to think about this is to look at x1/6, the sixth root of x. Since x is greater than 1, x1/6 is also greater than 1. If x1/6 were positive but less than 1, multiplying it by itself would give a smaller number. Since x1/6 is greater than 1, multiplying it by itself produces a larger number each time.
If you multiply x1/6 by itself, or x1/6, you get x2/6, or x1/3. Meanwhile, x1/6multiplied by itself three times produces x3/6, or x1/2. Therefore, x1/2 must be larger than x1/3 if x is a positive number greater than 1:
Therefore, we know that x0 < x1/6 < x1/3 < x1/2 < x2/3 < x1.
(The points on the line above are not drawn to scale.)
To do this, we write out exponentiation as repeated multiplication: (x3)2 = x3 • x3 = (x • x • x) • (x • x • x). According to the rules given earlier, we add these exponents when multiplying, so the result is x6, the same value as x3 • 2.
In general, consider(xa)b. Writing this as a multiplication problem, we’d see xa • xa • xa… • xa, where there are b occurrences of xa. Adding these exponents, we get a + a + a … + a; that is, a added to itself a total of b times. Repeated addition is multiplication, so the result is ab, the product of a and b. So we see that, in general, (xa)b = x(a • b) = xab
Zero cannot be used as a base for several reasons. The base can only be 0 when working with rules involving multiplication and exponentiation of positive exponents. However, all positive powers of 0 equal 0, and products and sums of 0 are all 0, thus making a one-value system. Since we cannot divide by 0, we cannot define 00 as 0n0n for some n (see Problem B1 for more information). Additionally, we cannot define 0n for any negative n (see Problem B2).
a. 43,007 = 4.3007 • 104. The lead digit (4) was originally in the 10,000, or 104position, so when we move it to the 100 position, we must multiply by 104. Multiplying 4.3007 by 104 gives us 43,007.
b. 0.00245 = 2.45 • 10-3. The lead digit (2) was originally in the 1/1000, or 10-3position, so when we move it to the 100 position, we must multiply by 10-3. Multiplying 2.45 by 10-3 gives us 0.00245.
c. -675 = -6.75 • 102. The lead digit (6) was originally in the 100, or 102 position, so when we move it to the 100 position, we must multiply by 102. Multiplying -6.75 by 102 gives us -6.75.
a. 2,300,000 + 790,000 = (2.30 • 106) + (.79 • 106) = 3.09 • 106
b. 10,000,000 • 678,000,000,000 = (1 • 107) • (6.78 • 1011) = 6.78 • 107+ 11 = 6.78 • 1018
c. 1,490,000,000 7,000 = (1.49 • 109) (7 • 103) = (1.49 7) • 109 – 3 = 0.213 • 106
a. In exponential form, log 100 is 10x = 100. Since 102 = 100, log 100 = 2.
b. In exponential form, log3 81 is 3x = 81. Since 3 • 3 • 3 • 3 = 34 = 81, the solution is 4.
c. If log x = 4, the equation in exponential form is 104 = x, so x = 10,000.
d. The equation is bx = b. This is solved by x = 1 for all valid bases for b (with the convention that b must be positive and not equal to 1).
a. Log5 50 = x. We can also write this expression as 5x = 50, i.e., 5 raised to what power equals 50? We know that 52 = 25 and 53 = 125, so we can estimate x as somewhere between 2 and 3. To bring our estimate even closer, we can see what happens for x = 2.5, which, converting it into a fraction, can be written as 25/10, or 5/2. So we write 55/2 = = 55.9. Thus, we can further narrow down our estimate by saying that x is slightly under 2.5.
b. Similarly, log3 100 = y can be written as 3y = 100. We know that 34 = 81 and 35 = 243, so we can estimate y as somewhere between 4 and 5. For y = 4.5 we can convert 4.5 into a fraction by writing 4.5 = 45/10 = 9/2, we get 39/2 = = 140.3. Thus, we can further narrow down our estimate by saying that y is somewhere between 4 and 4.5.
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.