In many subjects, we deal with very small or very large numbers. To present such numbers in an easy way, we use a certain method called scientific notation.

Using this way, we can write the distances between galaxies or the mass of an atom in forms that can be read easily. In Table 6.3, some measurements are shown in both standard and scientific notations.

When is a Number in Scientific Notation?

We can write positive numbers in scientific notation. A number in the form of a × 10ⁿ is in scientific notation. In this form, 1< a < 10 and n is an integer. Any number in scientific notation has two parts; a number between 1 and 10, and a power of 10.

To write a large number in scientific notation follow the steps below.

Move the decimal point to the left until you have only one digit to the left of it. Count the number of places the decimal point has moved to the left. Take the number that you have and multiply it by 10 with an exponent. This exponent is equal to the number of places the decimal point was moved to the left. Now look at your number again. Normally, you want to round the actual number to the nearest tenth, leaving only a number with one decimal place multiplied by 10 to a power.

Example 1: 657,000,000,000 = 6.6 x 10¹¹

To write a small number in scientific notation follow the steps below.

Move the decimal point to the right until you have only one digit to the left of it. Count the number of places the decimal point has moved to the right. Take the number that you have and multiply it by 10 with an exponent. This time, the exponent will be negative so you can tell the difference between very large numbers and very small numbers. This exponent is equal to the number of places the decimal point was moved to the right. Now look at your number again. Normally, you want to round the actual number to the nearest tenth, leaving only a number with one decimal place multiplied by 10 to a power.

Example 2: 0.000000004589 = 4.6 x 10^-9

Practice 8. Which numbers are in the form of scientific notation?

(a)   13 × 10²²

(b)   3.24 × 10¹⁶

(c)   –3.9 × 10¹²

(d)   3.2 × 10¹⁷

(e)   –1.2 × 10¹⁷

(f)     –6.2 × 10¹⁴

(g)   0.3 × 10¹²

(h)   3.9 × 10¹⁶

(i)     3.6 × 10^–12

Answer. b, c, d, e, f, h, i

Practice 9. Write each number in scientific notation.

(a)   3200000

(b)   3245600

(c)   121200200

(d)   0.04325

(e)   14.5678

(f)     0.2345

(g)   1235009

Answer.

(a)   3.2 × 10⁶

(b)   3.2 × 10⁶

(c)   1.2 × 10⁸

(d)   4.3 × 10^–2

(e)   1.5 × 10  (note: when we would normally have 10¹, we omit the exponent. When there is no exponent, it is always assumed to be 1)

(f)     2.3 × 10^–1

(g)   1.2 × 10⁶
  

Operations with Scientific Notations.

1.      Addition and Subtraction. To add or subtract two numbers in scientific notation, do the steps below.

a. Express both numbers in a common power of 10.

b. Add or subtract the decimal parts of two numbers. The result will be the decimal part of the answer. Choose the common power of 10 as the power part of the solution.

Example 1. 4.5 × 10¹¹ – 3.2 × 10¹¹ = (4.5 – 3.2)(10¹¹)

                                                            = 1.3 × 10¹¹

Example 2. 6.1 × 10⁸ + 3.2 × 10⁵ = 6.1 × 10⁸ + 0.0032 × 10⁸ (note: when we move the decimal point to the left, we add to the power of 10 by the number of places that we have moved the decimal point.)

                                                            = (6.1 + 0.0032) × 10⁸

                                                            = (6.1032) × 10⁸

                                                            = 6.1 × 10⁸

Example 3.  4.5 x 10³ + 7.6 x 10³ = (4.5 + 7.6)x = 12.1 x 10³
To return the answer to scientific notation, we will have to move the decimal point one place to the left, which means that we will have to round off our result to only one decimal place, and add one to the exponent of the 10. Thus our answer becomes: 1.2 x 10⁴

2.      Multiplication. Multiply the decimal parts of two numbers first. Then add the exponents.

Example.  (3.1 × 10⁷)(1.1 × 10²¹) = (3.11 × 1.1)(10^{7 + 21})

                                                            = (3.41)(10²⁸)

= 3.4 × 10²⁸

3.      Division. Divide the decimal parts first. Then subtract the exponents.

Example. (6.4 × 10¹⁷) ÷ (1.2 × 10⁵) = (6.4 ÷ 1.2)(10^{17 –  5})

                                                              = (5.33)(10¹²)

                                                              = 5.3 × 10¹²

Practice 10. Calculate.

(a) 6.3 × 10¹² + 3.2 × 10¹⁴

(b) 3.3 × 10¹⁰ – 2.2 × 10⁷

(c) (5.8 × 10¹⁶)(1.5 × 10¹²)

(a)   (7.6 × 10¹⁸) ÷ (4.5 × 10¹¹)

Solution.

(a) 6.3 × 10¹² + 3.2 × 10¹⁴ = 3.3 × 10¹⁴

(b) 3.3 × 10¹⁰ – 2.2 × 10⁷ = 3.3 × 10¹⁰

(c) (5.8 × 10¹⁶)(1.5 × 10¹²) = 8.7 × 10²⁸

(b)   (7.6 × 10¹⁸) ÷ (4.5 × 10¹²) = 1.7 × 10⁶

Practical Exercise 3. Calculate.

(a)   4.3 × 10¹⁶ + 6.4 × 10¹⁶

(b)   6.9 × 10¹⁶ – 3.2 × 10¹⁴

(c)   (3.6 × 10¹⁷)(1.9 × 10¹⁰)

(d)   (9.4 × 10¹⁵) ÷ (1.6 × 10⁸)

Answer