Worksheet 1
We define a relation ∼ on N × N as follows: (p,q)∼(p′,q′)⇔p+q′ =p′ +q.
idk lol
Division Algorithm
....
Compute the base-9 representation of 17.
easy peasy
Expressions may look irrational yet still be rational.
a. Write v3 + V7 — V/8 — 2V7 as the quotient of two
integers.
b. Make up another example of the same type as part a.
Bryan did this!
Find the decimal for each rational number by repeated
applications of the Division Algorithm.
817/37
Did someone do this?
It then makes sense to define equivalence classes (p,q)∼:
(p,q)∼ :={(p′,q′)∈N×N|(p′,q′)∼(p,q)}. The set of all these equivalence classes is denoted by (N × N)∼.
Find all elements in the equivalence class (2,5)∼. What do all these pairs of natural numbers have in common?
Find all elements in the equivalence class (4,2)∼. What do all these pairs of natural numbers have in common?
uh?
real, whole, natural, integers, complex numbers
all the numbers
Check your answer using a geometric series.
repetitive process
Prove or disprove the claim of a student that the sides of any
right triangle can be written in the form Va, Vb, and Va + b.
student is right
4. Consider those reciprocals of primes that have simple periodic
decimal representations. Using the theorems of the
section, prove that, of these:
a. There is exactly 1 with period 1. What is it?
b. There is exactly 1 with period 2. What is it?
c. There is exactly 1 with period 3. What is it?
d. There is exactly 1 with period 4. What is it?
i think bryan did these ?
One can then identify the set of integers Z with this set (N × N)∼. Which equivalence class corresponds to the integer 0? What about the equivalence classes corresponding to the integers 1 and -3, respectively?
help!
periodic and terminating
easy peasy x3
Compute other bases !
We got this yall
Let s be a nonzero rational number and v be irrational,
a. Prove that s - v is irrational.
Steve did this!
Consider those reciprocals of integers that have simple periodic
decimal representations. Using the theorems of the
section, prove that, of these:
a. There are exactly 2 with period 1. What are they?
b. There are exactly 3 with period 2. What are they?
c. There are exactly 5 with period 3. What are they?
idk who did these ??
How can one define addition of two integers? More precisely, what should be the meaning of
(p, q)∼ + (p′, q′)∼?
Is your definition well-defined1?
??
cardinality
...
Long division/Division Algorithm
easy peasy 2x
Although V2 + V3 does not equal the square root of
an integer, V27 + V48 does.
a. What integer’s square root equals V27 + V48, and why?
b. Make up another example like V27 + V48.
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a. Find the decimal representations for 1/27 and 1/37
b. Explain the peculiar relationship between these decimals,
and find other pairs of decimals with the same relationship.
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What is the neutral element in (N × N)∼ with respect to addition?
Given (p, q)∼ ∈ (N×N)∼, what is the inverse element of (p, q)∼ with respect to addition?
How can one define multiplication of two integers? More precisely, what should be the meaning of
(p, q)∼ · (p′, q′)∼?
knaust did this
algebraic?
when knaust asked bryan
Wu's Principals?
idk
Find decimals representing the rational numbers 21/20 and20/21
Kate did this!
Identify a rational number and an irrational number
between the two given real numbers.
a. -86 and-87
c. pi and pi-1/2^7
daf did these?