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3 years ago

Can you find a rational number that is exactly √2

Mathematics

1 answer:

Gnom [1K]3 years ago

8 0

9514 1404 393

Answer:

no, √2 is irrational
7/5, 99/70, 19601/13860, any truncation of the decimal value

Step-by-step explanation:

1.√2 is not the square root of a perfect square integer, so is irrational. There is no rational number that is exactly √2.

The proof usually looks something like this. Let p/q represent the reduced square root. Then 2 = p²/q², or 2q² = p², which means p must be even. If p is even, then p² will have at least 2 factors of 2. This requires that q² be even, hence q must be even. This contradicts our assumption that p/q is a reduced fraction, so there cannot be any p/q that is equal to 2.

2. Any truncation or rounding of the decimal representation of √2 will be a rational number: 1.4, 1.41, 1.414, and so on. In addition, there are some fractions that represent √2 fairly well. A couple are 99/70 and 19601/13860.

Additional fractions can be found by evaluating the continued fraction ...

1 +1/(2 +1/(2 +1/(2+ ...)))

at some level of truncation. A few of these values are 7/5, 17/12, 41/29, 99/70, 239/169, 577/408.

The Babylonian method can also be used to find rational approximations. If you start with any rational approximation p/q, then a better approximation will be (2q² +p²)/(2pq). This formulation can be iterated as many times as you like. It very rapidly converges to very good approximations.

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There are 11 boys and 13 girls in a class. Write the ratio of boys to girls.

marishachu [46]

Answer:

for every 1 boy theres 2 girls

Step-by-step explanation:

7 0

2 years ago

Read 2 more answers

Significant Figures PracticeName________________________ Date_____________ Section________________

balu736 [363]

Answer:

1. State the number of significant digits in each measurement.

1) 2804 m = 4 significant digits

2) 2.84 km = 3 significant digits

3) 5.029 m = 4 significant digits

4) 0.003068 m = 4 significant digits

5) 4.6 x 10^5 m = 2 significant digits

6) 4.06 x 10^-5 m = 3 significant digits

7) 750 m = 2 significant digits

8) 75 m = 2 significant digits

9) 75,000 m = 2 significant digits

10) 75.00 m = 4 significant digits

11) 75,000.0 m = 6 significant digits

12) 10 cm = 1 significant digit

2. Round the following numbers as indicated:

To four figures:

3.6824172 = 3.682

1.86005137 = 1.860

5.652311 = 5.652

2.5114 = 2.511

5.4673 = 5.467

To one decimal place:

1.3511 = 1.4

2.473 = 2.5

5.687524 = 5.7

7.555 = 7.6

8.235 = 8.2

To two decimal places:

22.494 = 22.49

79.258 = 79.26

80.0306 = 80.03

23.4125 = 23.41

41.866323 = 41.87

Solve the following problems and report answers with appropriate number of significant digits.

1) 6.201 cm + 7.4 cm + 0.68 cm +12.0 cm = 26.281 cm (5 significant digits)

2) 1.6 km + 1.62 km +1200 km = 1203.22 km (6 significant digits)

3) 8.264 g - 7.8 g = 0.464 g (3 significant digits)

4) 10.4168 m - 6.0 m = 6.4168 (5 significant digits)

5) 12.00 kg+15.001 kg= 27.001 kg (5 significant digits)

6) 1.31 cm x 2.3 cm = 3.013 cm² (4 significant digits)

7) 5.7621 m x 6.201 m = 35.7307821 (9 significant digits)

8) 20.2 cm / 7.41 s = 2.726046 (7 significant digits)

Explanation

The significant digits of a number are the digits that have meaning or contribute to the value of the number. Sometimes they are also called significant figures.

Which digits are significant?

There are some basic rules that tell you which digits in a number are significant:

All non-zero digits are significant

Any zeros between significant digits are also significant

Trailing zeros to the right of a decimal point are significant

Which digits aren't significant?

The only digits that aren't significant are zeros that are acting only as place holders in a number. These are:

Trailing zeros to the left of the decimal point (note: these zeros may or may not be significant)

Leading zeros to the right of the decimal point

4 0

3 years ago

Read 2 more answers

Volume of a oblique cone height 1= 6in height2 = 10in diameter = ?

Nataly_w [17]

376.99112
I hope i helped

7 0

4 years ago

This is a "water tank" calculus problem that I've been working on and I would really appreciate it if someone could look at my w

Sedaia [141]

Part A

Everything looks good but line 4. You need to put all of the "2h" in parenthesis so the teacher will know you are squaring all of 2h. As you have it right now, you are saying "only square the h, not the 2". Be careful as silly mistakes like this will often cost you points.

============================================================

Part B

It looks like you have the right answer. Though you'll need to use parenthesis to ensure that all of "75t/(2pi)" is under the cube root. I'm assuming you made a typo or forgot to put the parenthesis.

dh/dt = (25)/(2pi*h^2)
2pi*h^2*dh = 25*dt
int[ 2pi*h^2*dh ] = int[ 25*dt ] ... applying integral to both sides
(2/3)pi*h^3 = 25t + C
2pi*h^3 = 3(25t + C)
h^3 = (3(25t + C))/(2pi)
h^3 = (75t + 3C)/(2pi)
h^3 = (75t + C)/(2pi)
h = [ (75t + C)/(2pi) ]^(1/3)

Plug in the initial conditions. If the volume is V = 0 then the height is h = 0 at time t = 0
0 = [ (75(0) + C)/(2pi) ]^(1/3)
0 = [ (0 + C)/(2pi) ]^(1/3)
0 = [ (C)/(2pi) ]^(1/3)
0^3 = (C)/(2pi)
0 = C/(2pi)
C/(2pi) = 0
C = 0*2pi
C = 0

Therefore the h(t) function is...
h(t) = [ (75t + C)/(2pi) ]^(1/3)
h(t) = [ (75t + 0)/(2pi) ]^(1/3)
h(t) = [ (75t)/(2pi) ]^(1/3)

Answer:
h(t) = [ (75t)/(2pi) ]^(1/3)

============================================================

Part C

Your answer is correct.
Below is an alternative way to find the same answer

--------------------------------------

Plug in the given height; solve for t
h(t) = [ (75t)/(2pi) ]^(1/3)
8 = [ (75t)/(2pi) ]^(1/3)
8^3 = (75t)/(2pi)
512 = (75t)/(2pi)
(75t)/(2pi) = 512
75t = 512*2pi
75t = 1024pi
t = 1024pi/75
At this time value, the height of the water is 8 feet

Set up the radius r(t) function
r = 2*h
r = 2*h(t)
r = 2*[ (75t)/(2pi) ]^(1/3) .... using the answer from part B

Differentiate that r(t) function with respect to t
r = 2*[ (75t)/(2pi) ]^(1/3)
dr/dt = 2*(1/3)*[ (75t)/(2pi) ]^(1/3-1)*d/dt[(75t)/(2pi)]
dr/dt = (2/3)*[ (75t)/(2pi) ]^(-2/3)*(75/(2pi))
dr/dt = (2/3)*(75/(2pi))*[ (75t)/(2pi) ]^(-2/3)
dr/dt = (25/pi)*[ (75t)/(2pi) ]^(-2/3)

Plug in t = 1024pi/75 found earlier above
dr/dt = (25/pi)*[ (75t)/(2pi) ]^(-2/3)
dr/dt = (25/pi)*[ (75(1024pi/75))/(2pi) ]^(-2/3)
dr/dt = (25/pi)*[ (1024pi)/(2pi) ]^(-2/3)
dr/dt = (25/pi)*(1/64)
dr/dt = 25/(64pi)
getting the same answer as before

----------------------------

Thinking back as I finish up, your method is definitely shorter and more efficient. So I prefer your method, which is effectively this:
r = 2h, dr/dh = 2
dh/dt = (25)/(2pi*h^2) ... from part A
dr/dt = dr/dh*dh/dt ... chain rule
dr/dt = 2*((25)/(2pi*h^2))
dr/dt = ((25)/(pi*h^2))
dr/dt = ((25)/(pi*8^2)) ... plugging in h = 8
dr/dt = (25)/(64pi)
which is what you stated in your screenshot (though I added on the line dr/dt = dr/dh*dh/dt to show the chain rule in action)

8 0

3 years ago

- 5/8c = 20 A. -34 B. -32 C. 32 D. 34

lutik1710 [3]

The answer is B. -32 this is a really easy question

8 0

4 years ago

Can you find a rational number that is exactly √2​

Can you find a rational number that is exactly √2