Base Conversion and Number Systems: Examples and Practice Questions

Part I: Base Conversion and Number Systems

Part I:

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

2.A place-value, base-20 system would require _____ digits.

3.If you have been counting in base five, what would have been the five numerals before 2314five?

4.Do the “translations” in parts A–D.  Show your work.

A) 3102five =  _____________ ten

B) 310.2five = _____________ ten

C) 203.6ten = ____________ five

D) (base six pieces with small block (not pictured) as the unit and the stick/rod as 10)

= ______________ten

5.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.)

6.Sketch the base blocks that show 1203seven and identify your 1 whole unit.  Give the base ten value of each different-sized piece.  Convert 1203seven to a base ten numeral based on your sketch.

7.Sketch and solve: 24five + 33five= _______ five.  

8.Sketch and solve: 3five  2five = _______ five.  

9.Sketch and solve: 34five ÷ 4five= _______ five.  

10.In what base does the following addition not work: 13 + 13 = 26?

a. Base six

b. Base seven

c. Base eight

d. Base ten

e. It works in all of these bases.

11.The blocks are base four.

(a) What base-four numeral is illustrated with the long as the unit? 

(b) Suppose the small block is the unit, give the base ten value.

12.Write the “basimal” (expanded) place values and then the usual base-ten fraction or mixed number for 20.11three.

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

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

15.Circle T or F.

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

T F In base b, there are b – 1 different digits.

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

T F Every two different prime numbers are relatively prime.

16.  2.31four = ________________ as a mixed number in base ten.

Part II

1.If the following is false, correct it so that it is true in some nontrivial way:

If D = , then 100D = 74.200.

2.What are the “rational” numbers?  How are they different from the “irrational” numbers?

3.Comment on Tommy’s work below.

Put the following in order from least to greatest without using common denominators or converting everything into decimals. 

Tommy: Those are the only two ways I know, but I will try. I notice that both fractions are missing two pieces to make them whole. I think that makes them alike somehow. I think since 9ths are smaller than 5ths and you take out two pieces each, 80% is missing way more. It is missing 20 pieces to be whole, so it has to be way smaller than both of those. It should be 80%, 

4.14.Is the following sentence all right? If it is, explain why. If it is not, give a counterexample. If a number has n factors (n > 1), then the square of the number has 2n factors.

5.Prove that an odd number plus an odd number is an even number.