How do you use scientific notation to compare very large or very small numbers in your field?

1 Why use scientific notation?

Scientific notation is useful for many fields that deal with numbers that span several orders of magnitude, such as astronomy, physics, chemistry, biology, engineering, and economics. For example, the mass of the sun is about 2 x 10^30 kg, while the mass of an electron is about 9 x 10^-31 kg. Using scientific notation, you can quickly see that the sun is about 22 orders of magnitude heavier than an electron. Without scientific notation, you would have to count 61 zeros to compare these numbers.

2 How to write scientific notation?

To write a number in scientific notation, you need to move the decimal point until you have one non-zero digit before it, and then multiply by 10 raised to the power of how many places you moved the decimal point. For example, to write 0.0000012 in scientific notation, you need to move the decimal point six places to the right, and then multiply by 10^-6. To write 3,400,000,000 in scientific notation, you need to move the decimal point nine places to the left, and then multiply by 10^9.

3 How to compare scientific notation?

To compare two numbers in scientific notation, you need to look at the powers of 10 first. The number with the higher power of 10 is larger than the number with the lower power of 10. For example, 5 x 10^8 is larger than 2 x 10^7, because 8 is greater than 7. If the powers of 10 are equal, then you need to compare the coefficients, which are the numbers before the x. The number with the larger coefficient is larger than the number with the smaller coefficient. For example, 6 x 10^5 is larger than 4 x 10^5, because 6 is greater than 4.

4 How to use significant figures?

Significant figures are the digits in a number that are meaningful and accurate. For example, in 1.234 x 10^3, there are four significant figures: 1, 2, 3, and 4. In 0.0056 x 10^-2, there are two significant figures: 5 and 6. The zeros before or after the decimal point are not significant, unless they are between non-zero digits or indicate precision. For example, in 0.00300 x 10^4, there are three significant figures: 3 and the two zeros after it. The zeros before the 3 are not significant, but the zeros after it indicate that the measurement is precise to three decimal places.

5 How to compare significant figures?

When comparing numbers with different significant figures, you need to round them to the same number of significant figures before comparing. For example, if you want to compare 1.234 x 10^3 and 1.23 x 10^3, you need to round the first number to two significant figures, which is 1.2 x 10^3. Then you can compare them and see that they are equal. If you want to compare 0.0056 x 10^-2 and 0.0057 x 10^-2, you need to round them to one significant figure, which is 0.006 x 10^-2 and 0.006 x 10^-2. Then you can compare them and see that they are also equal.

6 How to apply scientific notation and significant figures in your field?

Depending on your field, you may encounter different situations where you need to use scientific notation and significant figures to compare numbers. For example, if you are an astronomer, you may need to compare the distances between stars or planets, which are often expressed in light-years or astronomical units. If you are a physicist, you may need to compare the energies or frequencies of particles or waves, which are often expressed in joules or hertz. If you are a chemist, you may need to compare the concentrations or masses of substances, which are often expressed in moles or grams. In any case, you need to pay attention to the powers of 10 and the coefficients, and round them to the appropriate number of significant figures before comparing.