Chapter 4

Express each expanded form as a Hindu-Arabic numeral. \(\left(6 \times 10^{-1}\right)+\left(8 \times 10^{-2}\right)+\left(1 \times 10^{-3}\right)+\left(2 \times 10^{-4}\right)\)

Look at the back of a U.S. one dollar bill. What date is written in Roman numerals along the base of the pyramid with an eye? What is this date's significance?

Express each expanded form as a Hindu-Arabic numeral. \(\left(7 \times 10^{-1}\right)+\left(2 \times 10^{-4}\right)+\left(3 \times 10^{-6}\right)\)

Express each expanded form as a Hindu-Arabic numeral. \(\left(8 \times 10^{-1}\right)+\left(3 \times 10^{-4}\right)+\left(7 \times 10^{-6}\right)\)

Explain how to determine the place values for a four-digit numeral in base six.

Express each expanded form as a Hindu-Arabic numeral. \(\left(5 \times 10^{3}\right)+\left(3 \times 10^{-2}\right)\)

Describe how to change a numeral in a base other than ten to a base ten numeral.

Express each expanded form as a Hindu-Arabic numeral. \(\left(7 \times 10^{4}\right)+\left(5 \times 10^{-3}\right)\)

Describe how a number is represented in the Egyptian numeration system.

Describe how to change a base ten numeral to a numeral in another base.

Express each expanded form as a Hindu-Arabic numeral. \(\left(3 \times 10^{4}\right)+\left(7 \times 10^{2}\right)+\left(5 \times 10^{-2}\right)+\left(8 \times 10^{-3}\right)\) \(+\left(9 \times 10^{-5}\right)\)

If you are interpreting a Roman numeral, when do you add values and when do you subtract them? Give an example to illustrate each case.

Express each expanded form as a Hindu-Arabic numeral. \(\left(7 \times 10^{5}\right)+\left(3 \times 10^{2}\right)+\left(2 \times 10^{-1}\right)+\left(2 \times 10^{-3}\right)\) \(+\left(1 \times 10^{-5}\right)\)

Describe how a number is represented in the traditional Chinese numeration system.

Determine whether each statement makes sense or does not make sense, and explain your reasoning. Base \(b\) contains \(b-1\) digit symbols.

Describe one disadvantage of the Ionic Greek numeration system.

Determine whether each statement makes sense or does not make sense, and explain your reasoning. Bases greater than ten are not possible because we are limited to ten digit symbols.

Determine whether each statement makes sense or does not make sense, and explain your reasoning. Because the binary system has only two available digit symbols, representing numbers in binary form requires more digits than in any other base.

Determine whether each statement makes sense or does not make sense, and explain your reasoning. In order to understand the early numeration systems presented in this section, it's important that I take the time to memorize the various symbols.

Determine whether each statement makes sense or does not make sense, and explain your reasoning. Because (6) represents 100 and \(\bigcap\) represents 10 in the Egyptian numeration system, (0) represents \(100+10\), or 110 , and (0) represents \(100-10\), or 90 .

Humans have debated for decades about what messages should be sent to the stars to grab the attention of extraterrestrials and demonstrate our mathematical prowess. In the \(1970 \mathrm{~s}\), Soviet scientists suggested we send the exponential message $$ 10^{2}+11^{2}+12^{2}=13^{2}+14^{2} . $$ The Soviets called this equation "mind-catching." Evaluate the exponential expressions and verify that the sums on the two sides are equal. What is the significance of this sum?

Determine whether each statement makes sense or does not make sense, and explain your reasoning. It takes far more space to represent numbers in the Roman numeration system than in the Egyptian numeration system.

Describe the difference between a number and a numeral.

Arrange from smallest to largest: $$ 11111011_{\text {two }}, 3 \mathrm{~A} 6_{\text {twelve }}, 673_{\text {eight }} \text {. } $$

Explain how to evaluate \(7^{3}\).

What is the base in our Hindu-Arabic numeration system? What are the digits in the system?

Why is a symbol for zero needed in a positional system?

Explain how to write a Hindu-Arabic numeral in expanded form.

Describe one way that the Babylonian system is similar to the Hindu-Arabic system and one way that it is different from the Hindu-Arabic system.

Take a moment to read the introduction to the group exercises on page 231. List some additional topics for individual or group research projects. Animals and Number Sense

Describe one way that the Mayan system is similar to the Hindu-Arabic system and one way that it is different from the Hindu-Arabic system.

Research activity. Write a report on the history of the Hindu-Arabic system of numeration. Useful references include history of mathematics books, encyclopedias, and the Internet.

Determine whether each statement makes sense or does not make sense, and explain your reasoning. I read that a certain star is \(10^{4}\) light-years from Earth, which means 100,000 light-years.

Determine whether each statement makes sense or does not make sense, and explain your reasoning. I write Babylonian numerals horizontally, using spaces to distinguish place values.

Your group task is to create an original positional numeration system that is different from the three systems discussed in this section. a. Construct a table showing your numerals and the corresponding Hindu-Arabic numerals. b. Explain how to represent numbers in your system, and express a three-digit and a four-digit Hindu-Arabic numeral in your system.