This is true because the definition of a rational number is that it can be turned into another fraction so if you were to add two fractions together your outcome would be another fraction which would be a rational number
a c e ad
-- + -- = --- = --- + Rational
b d f bd
By finding the common denominator and multiplying it by a whole fraction the denominators will be equal to one another.
a c a + c
-- + -- = -------
d d d
Chloe, Sophia, Nathaniel, Cary
For each of the following conjectures, determine whether you think it is always true, sometimes true, or never true. If it is always true or never true, prove or explain why. If it is sometimes true, determine the conditions under which it is true.
The sum of a rational number and an irrational number is an irrational number is true, but to prove our assumption, we are going to assume that it is false
Contradiction method: Assume that i+r₁=r₂.
r₁+ i=
5+√7=r (r is a rational number)
5+√7=r (subtract 5 from both sides isolate the irrational number)
-5 -5
√7=r-5
For each of the following conjectures, determine whether you think it is always true, sometimes true, or never true. If it is always true or never true, prove or explain why. If it is sometimes true, determine the conditions under which it is true.
l..],. ]= ]-n.,0c].’- ]’
\=nm,-=.;’�n irrational + an irrational= an irrat].’/=ional. this is only true sometimes because in order to prove this we did an irrational +an irrational=a rational. because we cannot disprove this it means that an irrational + an irrational= a rational or irrational.
For each of the following conjectures, determine whether you think it is always true, sometimes true, or never true. If it is always true or never true, prove or explain why. If it is sometimes true, determine the conditions under which it is true.
The product of two rational numbers is a rational number.
Always True
a/b where a and b are integers and x/y where
x and y are integers (and b and y are not 0)
a/b(x/y)=(ax)/(by)
a is an integer and x is an integer so ax is an integer
b is an integer and y is an integer so by is an integer
The final result is represented exactly as a fraction were both the numerator and denominator are integers
Louise
Ian
Nadia
Kenneth
(1/2)(4/9)=4/18=2/9
2/9 is rational
1 is an integer and 4 is an integer so 4 is an integer
2 is an integer and 9 is an integer so 18 is an integer
For each of the following conjectures, determine whether you think it is always true, sometimes true, or never true. If it is always true or never true, prove or explain why. If it is sometimes true, determine the conditions under which it is true.
Cole Atwell, Ana Orloff, Pablo Jativa, Elizabeth Heller
For each of the following conjectures, determine whether you think it is always true, sometimes true, or never true. If it is always true or never true, prove or explain why. If it is sometimes true, determine the conditions under which it is true.
The product of a rational number and an irrational number is an irrational number.
This conjecture is sometimes is true, because it works when you multiply by 1, but when you use any other rational it is not true.
R-Rational
I-Irrational
R x I = R, divide by R and then I = R/R
Tyler,Emily, Nyla
For each of the following conjectures, determine whether you think it is always true, sometimes true, or never true. If it is always true or never true, prove or explain why. If it is sometimes true, determine the conditions under which it is true.
The diagonal of a square is irrational.
Always true.
-To find the diagonal of a square you use the Pythagorean Theorem : a^2+b^2=c^2
-A square is a quadrilateral with all equal sides. Therefore, by using the theorem and plugging in two sides of the square as a and b, and saying that the diagonal of the line is the hypotenuse, we can use the Pythagorean Theorem to discover the diagonal of a square.
-Example:
a=2 b=2
2^2+2^2=c^2
4+4=c^2
Square root of 8=irrational
If we assume that it was true, then 8=c^2
Since the square of 8 has no perfect squares,
you can only justify it as 2root2
Bentley, Sam, Edward
The hypotenuse is also seen as
c = √a^2+b^2
For each of the following conjectures, determine whether you think it is always true, sometimes true, or never true. If it is always true or never true, prove or explain why. If it is sometimes true, determine the conditions under which it is true.
The circumference of a circle is irrational.
Kaitlin Birnbaum & Katherine Perrine
Formula for a Circumference:
C = 2πr
What does that mean?
The circumference is equal to 2 pi times the radius.
This conjecture is sometimes true because with most problems, if you do not approximate pi to 3.14, then you will almost always end up with pi somewhere in the problem. Because we know pi is an irrational number, and if it is left somewhere else in a problem, that makes one side irrational, as as they must equal each other, both sides must be irrational. However, there are some exceptions. For example, if C or r were 0, then the other variable will also be 0. Also, if you made 3 the circumference, then r would be 3/2π, which makes C, the circumference, rational.
For each of the following conjectures, determine whether you think it is always true, sometimes true, or never true. If it is always true or never true, prove or explain why. If it is sometimes true, determine the conditions under which it is true.
Between two rational numbers, there is an irrational number.
For each of the following conjectures, determine whether you think it is always true, sometimes true, or never true. If it is always true or never true, prove or explain why. If it is sometimes true, determine the conditions under which it is true.
Between two irrational numbers, there is an irrational number.