Chapter 1, Problem 1.1P

PRACTICE 1.1 Express the following quantities in scientific no-
tation and then express the number and unit with the most appro-
priate prefix.

(a) The diameter of an insulin molecule, 0.000000005 m
(b) The circumference of the Earth at the Equator, 40,075,017 m

Interpretation Introduction

To determine: The scientific notation of following quantities and to express the number and unit with the most appropriate prefix.

1. The diameter of an insulin molecule, 0.000000005m
2. The circumference of the earth at the equator, 40075017m

Answer to Problem 1.1P

Solution:

  $5\times {10}^{-9}\text{m,5nm}$

  $4.0075017\times {10}^7\text{m},40.075017\text{Mm}$

Explanation of Solution

The best way to write any scientific notation is in the form of $N\times {10}^a$. Here, N is a number between 1 and 10 and “a” is an exponent.

The prefix multipliers are the fixed numeral values used in SI system to reduce the difficulties in measurement. For example, one nanometre: 1 nm=10-9 m, here Nano is a prefix used for a value of 10-9.

Given:

a. The diameter of an insulin molecule, 0.000000005m

Step 1: To write the given value in scientific notation, the decimal should be shifted right to find a value of N and an exponent for 10.

The value becomes as: $0.000000005\text{m}=5\times {10}^{-9}\text{m}$

Step 2: The unit “meter” is written with proper prefix as follows:

Since, $\text{1 nm}={10}^{-9}\text{m}$

Hence,

  $\begin{array}{c}5\times {10}^{-9}\text{m}=\frac{1\text{ nm}}{{10}^{-9}\text{m}}\times 5\times {10}^{-9}\text{m}\\ =\text{5 nm}\end{array}$

b. The circumference of the earth at the equator, 40075017m

Step 1: To write the given value in scientific notation, the above value should be converted into decimal form to find a value of N and an exponent for 10.

The value becomes as:

  $4007501\text{7m}=4.0075017\times {10}^7\text{m}$

Step2: The unit “meter” is written with proper prefix as follows:

Since, the prefix mega (M) has the value of $\text{1 Mm}={10}^6\text{m}$

Now, the value of circumference should be converted to the comparative value as:

  $4.0075017\times {10}^7\text{m}=40.075017\times {10}^6\text{m}$

Hence,

  $\begin{array}{c}40.075017\times {10}^6\text{m}=\frac{\text{1Mm}}{{\text{10}}^{\text{-6}}\text{m}}\times 40.075017\times {10}^{-6}\text{m}\\ =40.075017\text{Mm}\end{array}$

Conclusion

1. The diameter of an insulin molecule, 0.000000005m is represented as $5\times {10}^{-9}\text{m, 5nm}$.
2. The circumference of the earth at the equator, 40075017m is represented as $4.0075017\times {10}^7\text{m,}40.075017\text{Mm}$.