Deck 2: Numeration Systems

Question 1

Is the old Greek numeration system ( $?$ = 1, $\beta$ = 2, $y$ = 3, etc.) a place-value system? Explain.

Answer 1

No. A place-value system requires that the placement of the symbol have meaning, which is not true of the Greek system.

Question 2

Most present-day societies use the Hindu-Arabic numeration system.

Answer 2

True

Question 3

How many tens are in 7654? How many whole tens are in 7654?

Answer 3

765.4; 765

Question 4

How many hundreds are in 23? How many whole hundreds?

Question 5

How many tenths are in 1.03? How many whole tenths?

Question 6

How many ones are in 4352.678? How many whole ones?

Question 7

In base ten, 3421 is exactly _____ ones, exactly _____ tens, exactly _____ hundreds, and exactly _____ thousands; also, 3421 is exactly _____ tenths and exactly _____ hundredths.

Question 8

In base ten, 215.687 is exactly _____ ones, exactly _____ tens, exactly _____ hundreds, and exactly _____ thousands; also, 3421 is exactly _____ tenths and exactly _____ hundredths.

Question 9

(Roman numerals) IX = ______ten and XI = _____ _ten.

Question 10

34,597 has 345 whole thousands in it.

Question 11

34.597 has 345 whole tenths in it.

Question 12

56 has 560 tenths in it.

Question 13

23 has 230 hundredths in it.

Question 14

45 has 4500 hundredths in it.

Question 15

632.1 has 632.1 ones in it.

Question 16

A soap factory packs 100 bars of soap in each box for shipment. If the factory makes 15,287 bars of soap, how many full boxes will they have for shipment? Explain.

Question 17

How many $10 bills could one get for $10 million?

A) 10,000,000
B) 1,000,000
C) 100,000
D) 10,000
E) 1000

Question 18

How many $100 bills could one get for a billion dollars?

A) 100,000,000
B) 10,000,000
C) 1,000,000
D) 100,000
E) 10,000

Question 19

How many $100 bills would make $45 billion?

Question 20

Judy says, "Well, hundredths are smaller than tenths. So 0.36 is smaller than 0.4." Comment on Judy's reasoning.

Question 21

Tyler thinks that 0.36 is bigger than 0.4 because 36 is bigger than 4. Comment on Tyler's reasoning.

Question 22

Four children are talking about different ways to think about the number 196.83. Following are their answers. For each answer, tell whether it is correct or incorrect. If it is incorrect, please explain.
Kim's answer: 196.83 could be thought about as 190 tens and 683 hundredths.
A) Is Kim's answer correct or incorrect?
B) If Kim's answer is incorrect, please explain the error.
Laurie's answer: 196.83 could be thought about as 1 hundred, 9 tens, and 6.83 hundredths.
A) Is Laurie's answer correct or incorrect?
B) If Laurie's answer is incorrect, please explain the error.
Mick's answer: 196.83 could be thought about as 1 hundred and 9683 hundredths.
A) Is Mick's answer correct or incorrect?
B) If Mick's answer is incorrect, please explain the error.
Nola's answer: 196.83 could be thought about as 18 tens, 15 ones, 17 tenths, and 13 hundredths.

A) Is Nola's answer correct or incorrect?
B) If Nola's answer is incorrect, please explain the error.

Question 23

For whole numbers, any two-digit numeral in base five represents a smaller number than the same two-digit numeral in base twenty.

Question 24

In base b, there are b - 1 different digits.

Question 25

These are the digits that are needed for a base seven place-value system: 0, 1, 2, 3, 4, 5, 6, 7.

Question 26

In base b, 3 + 2b³ + b would be written _____.

Question 27

A place-value, base-twenty system would require _____ digits.

Question 28

524_eight = ______ten.

Question 29

287_ten = ______four.

Question 30

1012_five = _____ in base ten.

Question 31

32_ten = _____ in base four.

Question 32

2.31_four = _____ as a mixed number in base ten.

Question 33

$6 \frac { 2 } { 3 }$
in base ten = _____ in base three.

Question 34

1_ten = ______twelve.

Question 35

214.3_five = _____ in base ten.

Question 36

29_ten = _____ in base three.

Question 37

7_ten = _____ in base nine.

Question 38

203.6_ten = ______five.

Question 39

2003_five = ______ten.

Question 40

200.3_five = ______ten.

Question 41

Write 49_ten in base seven.

Question 42

Do the "translations" in parts A-D. Show your work. A) $3102 _ {\text { five }}$ $=$ $\underline{\quad\quad}$ ten
B) $310.2 _ { \text {five } } =$ $\underline{\quad\quad}$ ten

C) 203.6_ten = ______five
D) (base-six pieces with small block as the unit)

Question 43

You are living and working on a planet that uses only base five. How many $5 bills can you get for $1234.20_five? Write your answer in base five because you are living on the planet. Write enough (numbers, words, etc.) to make your thinking clear.

Question 44

In base five, the two whole numbers immediately BEFORE 2001_five are ______five and ______five.

Question 45

If you are counting in base five, what would be the next six numerals AFTER 2314_five?

Question 46

If you have been counting in base five, what would have been the five numerals BEFORE 2314_five?

Question 47

Write how many fingers you have in base five. In base two. In base ten.

Question 48

Which is LARGER? 2_four or 2_five? Explain.

Question 49

Consider: x = 81765_fifteen and y = 81765_thirteen. Which of x and y is GREATER? Explain.

Question 50

Consider: x = 74213_sixteen and y = 74213_fourteen. Which is GREATER, x or y (or are they equal)? Explain.

Question 51

Consider x = 0.3147_eight and y = 0.3147_nine. Which of x and y is GREATER? Explain. (Be careful.)

Question 52

Write an algebraic expression for 204_b.

Question 53

If a base-eight flat represents 1, the numeral _____ would give the numerical value of the small cube. (You may give your answer either in base eight or in base ten-just make clear which.)

Question 54

Write the base-ten numeral for the following base-eight pieces.
Base-eight pieces, with the small cube (a dot here) as the unit:

Question 55

Sketch the base blocks that show 1203_seven, and give the English words for the base-ten value of each different-sized piece.

Question 56

Write the base b numeral for 2b⁴ + b² + 3b + 1.

Question 57

Write out 32004_m in the algebraic form of the last item.

Question 58

The best coins to use in thinking about the first three whole-number place values in base five would be the penny, the nickel, and the quarter.

Question 59

The best coins to use in thinking about the first three whole-number place values in base ten would be the penny, the dime, and the half-dollar.

Question 60

If 10000_ten + 10_b = 10023_ten, what is base b?

Question 61

Define your unit and sketch base blocks to represent 32.67_eight.

Question 62

Define your unit, sketch the base blocks that show 1203_nine, and give the English words for the base-ten value of each piece of wood.

Question 63

53_six is the same number as which of these base-ten numerals?

A) 18
B) 183
C) 12
D) 85
E) 33

Question 64

In base ten, 111_five would be written:

A) 421.
B) 155.
C) 31.
D) 21.
E) 555.

Question 65

The base b numeral 321_b means:

A) 3 ^. b² + 2 ^. b¹ + 1.
B) 3 ^. b³ + 2 ^. b² + 1 ^. b¹.
C) 6b.
D) 3 ^. b + 2 ^. b + 1.

Question 66

In base five, 32_ten would be written:

A) 152_five.
B) 112_five.
C) 62_five.
D) 17_five.

Question 67

The base two numeral 100_two equals the base ten numeral:

A) 1100100.
B) 1011100.
C) 8.
D) 4.

Question 68

In base ten, 32_four would be written:

A) 400.
B) 200.
C) 122.
D) 14.
E) 8.

Question 69

The base-four numeral 11.1_four could be written in base ten as:

A) $33 \frac { 1 } { 4 }$
B) $33 \frac { 1 } { 10 }$
C) $11 \frac { 1 } { 4 }$
D)5 $\frac { 1 } { 4 }$

Question 70

The base-ten decimal 18.5 could be written in base six as:

A) 10.5_six.
B) 20.3_six.
C) 30.3_six.
D) 128.5_six.

Question 71

The base-ten fraction 1/4 equals which base-eight numeral?

A) 0.2_eight
B) 0.14_eight
C) 0.02_eight
D) 1.4_eight

Question 72

If 31_b = 28_ten, then b =

A) 4.
B) 5.
C) 7.
D) 9.
E) This is impossible for any whole number b.

Question 73

In what base does the following counting work:
1, 2, 3, 4, 10, 11, 12, 13, 14, 20, 21, …

A) base two
B) base four
C) base five
D) base six

Question 74

What is the base-ten fraction representation for 1.21_four?

A) 1 $\frac { 9 } { 16 }$
B) 1 $\frac { 3 } { 4 }$
C) 1 $\frac { 21 } { 100 }$
D) 1 $\frac { 3 } { 5 }$

Question 75

Write an addition equation for (# fingers) + (# toes) = (answer) in some base other than base ten.

Question 76

Three-fourths in base ten is what in base two?

Question 77

Seven-eighths in base ten is what in base two?

Question 78

Suppose you visited an alien planet and the creatures wrote numbers in order as follows: "obi, fin, mus, obi-na, obi-obi, obi-fin, obi-mus…" These creatures must be using which base?

A) base ten
B) base three
C) base four
D) base nine
E) base twelve

Question 79

A) Add 24_five + 33_five in base five. (The numbers are already written in base five, so no conversions are necessary.)
B) How would you illustrate this with the base-five blocks using drawings and showing the intermediate steps?

Question 80