# Equal sign or approximation sign?

• shuxue
The example you gave was not an exact answer, so it should have been written 686.122 ≈ 690. That's why the approximation sign is used instead of the equal sign.
shuxue
The authors of a physics textbook want to determine the number of grains, N in a beach of 500 m long, 100 m wide, and 3 m deep. They assumed that each grain is 1-mm-diameter sphere. They also assumed that the grains are so tightly packed that the volume of the space between the grains is negligible compared to the volume of the sand itself. The authors found the answer as follows:

Volume of the beach = N X Volume of each grain
N = (Volume of the beach) / (Volume of each grain)
$$N=\frac{(500)(100)(3)}{(\frac{4}{3}\pi)(0.5\times10^{-3})^3}=2.9\times10^{14}\approx3\times10^{14}$$

Why equal sign is used between $$\frac{(500)(100)(3)}{(\frac{4}{3}\pi)(0.5\times10^{-3})^3}$$ and $$2.9\times10^{14}$$ instead of approximation sign? Why approximation sign is used between $$2.9\times10^{14}$$ and $$3\times10^{14}$$ instead of equal sign?

You are giving too much significance to the particular sign used. The computation in itself is approximative.

shuxue said:
$$N=\frac{(500)(100)(3)}{(\frac{4}{3}\pi)(0.5\times10^{-3})^3}=2.9\times10^{14}\approx3\times10^{14}$$
The first equals sign should really be ≈, IMO, since the 2.9 x 1014 is an approximation. This value, in turn, is approximately equal to 3.0 x 1014.

The example is taken from "Physics: For Scientists and Engineers", by Paul A. Tipler and Gene Mosca.

I found another similar example in another textbook, "Fundamental of physics" by Halliday, Resnick, and Walker. The example is as follows:

A pirate ship is 560 m from a fort defending a harbor entrance. A defense cannon, located at sea level in front of the fort fires balls at initial speed of v = 82 m/s. Thus the maximum range, R of the cannonballs is

$$R=\frac{v^2}{g}=((82 \enspace m/s)^2)/(9.8\enspace m \enspace s^{-2})=686 \enspace m\approx690 \enspace m$$

Again, why equal sign is used between ((82 m/s)^2)/(9.8 m s^(-2)) and 686 m instead of approximation sign? Why approximation sign is used between 686 m and 690 m instead of equal sign?

shuxue said:
The example is taken from "Physics: For Scientists and Engineers", by Paul A. Tipler and Gene Mosca.

I found another similar example in another textbook, "Fundamental of physics" by Halliday, Resnick, and Walker. The example is as follows:

A pirate ship is 560 m from a fort defending a harbor entrance. A defense cannon, located at sea level in front of the fort fires balls at initial speed of v = 82 m/s. Thus the maximum range, R of the cannonballs is

$$R=\frac{v^2}{g}=((82 \enspace m/s)^2)/(9.8\enspace m \enspace s^{-2})=686 \enspace m\approx690 \enspace m$$

Again, why equal sign is used between ((82 m/s)^2)/(9.8 m s^(-2)) and 686 m instead of approximation sign? Why approximation sign is used between 686 m and 690 m instead of equal sign?
The = isn't appropriate because ##\frac{82^2}{9.8}## is not equal to 686. A better approximation is 686.122.

I would have written the calculation this way:
$$\frac{82^2}{9.8} \doteq 686 \approx 690$$

For your second question, since 686 is not equal to 690 (obviously), it would be incorrect to write 686 = 690.

Mark44 said:
For your second question, since 686 is not equal to 690 (obviously), it would be incorrect to write 686 = 690.

But then g is not equal to 9.8 ms^-2 either, unless this has been specifically mentioned earlier, even with the restraint of "at sea level"

I want to solve the equation ##2 \sin^2\theta+2 \sin \theta-1=0##, ##0\leq\theta<2\pi##. By using the quadratic formula. I found one of the solution to be ##\theta=\sin^{-1}( \frac{-1+\sqrt{3}}{2})##. Which of the following ways of writing the numerical value of ##\theta## is correct? (i) or (ii)?

(i) ##\theta=0.3747##, (ii) ##\theta\approx 0.3747##

Are both of them acceptable? I noticed that, frequently, in most trigonometric textbooks the numerical solution of a trigonometric equation is written using the equal sign (as in (ii)) even the numerical answer has been rounded. Why equal sign is used instead of approximation sign?

shuxue said:
I want to solve the equation ##2 \sin^2\theta+2 \sin \theta-1=0##, ##0\leq\theta<2\pi##. By using the quadratic formula. I found one of the solution to be ##\theta=\sin^{-1}( \frac{-1+\sqrt{3}}{2})##. Which of the following ways of writing the numerical value of ##\theta## is correct? (i) or (ii)?

(i) ##\theta=0.3747##, (ii) ##\theta\approx 0.3747##

Are both of them acceptable?
Not in my opinion. The first is only an approximation, but using = doesn't indicate that.
shuxue said:
I noticed that, frequently, in most trigonometric textbooks the numerical solution of a trigonometric equation is written using the equal sign (as in (ii)) even the numerical answer has been rounded.
As I recall, most of the trig books I've used (either as a student or as a teacher), the authors took pains to distinguish between exact answers (using = ) and answers that were rounded (using ≈ or ##\doteq##).
shuxue said:
Why equal sign is used instead of approximation sign?

You keep asking this, so apparently you aren't understanding. The = sign should be used for exact answers, and the ≈ should be used if you are writing only an approximate value.

## 1. How is the equal sign different from the approximation sign?

The equal sign (=) is used to show that two quantities or expressions are exactly equal to each other, while the approximation sign (≈) is used to show that two quantities or expressions are approximately equal to each other. The equal sign is used when the values are known or can be calculated precisely, while the approximation sign is used when the values are only estimated or rounded.

## 2. Can the equal sign be used interchangeably with the approximation sign?

No, the equal sign and the approximation sign cannot be used interchangeably. They have different meanings and are used in different contexts. The equal sign is used for exact equality, while the approximation sign is used for approximate equality. Using them interchangeably can lead to incorrect mathematical statements and calculations.

## 3. What are some examples of situations where the equal sign is used?

The equal sign is commonly used in equations, where it is used to show that the values on both sides of the sign are equal. For example, in the equation 2x + 5 = 11, the equal sign is used to show that the expressions 2x + 5 and 11 are equal to each other. The equal sign is also used in identities, where it is used to show that two expressions are equal for all values of the variables involved.

## 4. In what situations would the approximation sign be used?

The approximation sign is used in situations where the values are only estimated or rounded. For example, when calculating the value of pi (π), it is often approximated to 3.14 using the approximation sign (π ≈ 3.14). It is also commonly used in scientific and statistical calculations, where the values are not known precisely and can only be estimated.

## 5. Are there any other symbols that are similar to the equal sign or approximation sign?

Yes, there are other symbols that are similar to the equal sign and approximation sign. One example is the congruence sign (≡), which is used to show that two quantities or expressions are congruent or equivalent to each other. Another example is the approximately equal to or asymptotically equal to sign (≃), which is used to show that two values are very close to each other, but not necessarily exactly equal.

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