Question

By how many places must the decimal point be moved, and in which direction, to convert each of the following to standard scientific notation? a. 5993 b. -72.14 c. 0.00008291 d. 62.357 e. 0.01014 f. 324.9

Short answer

a. 3 places left b. 1 place left c. 5 places right d. 1 place left e. 2 places right f. 2 places left

Step-by-step solution

a. 5993

To convert 5993 to scientific notation, we need to write it in the form a x 10^n where 1 ≤ |a| < 10. Move the decimal point 3 places to the left, resulting in: 5.993 x 10^3 The decimal point must be moved 3 places to the left.

b. -72.14

To convert -72.14 to scientific notation, we need to write it in the form a x 10^n where 1 ≤ |a| < 10. Move the decimal point 1 place to the left, resulting in: -7.214 x 10^1 The decimal point must be moved 1 place to the left.

c. 0.00008291

To convert 0.00008291 to scientific notation, we need to write it in the form a x 10^n where 1 ≤ |a| < 10. Move the decimal point 5 places to the right, resulting in: 8.291 x 10^-5 The decimal point must be moved 5 places to the right.

d. 62.357

To convert 62.357 to scientific notation, we need to write it in the form a x 10^n where 1 ≤ |a| < 10. Move the decimal point 1 place to the left, resulting in: 6.2357 x 10^1 The decimal point must be moved 1 place to the left.

e. 0.01014

To convert 0.01014 to scientific notation, we need to write it in the form a x 10^n where 1 ≤ |a| < 10. Move the decimal point 2 places to the right, resulting in: 1.014 x 10^-2 The decimal point must be moved 2 places to the right.

f. 324.9

To convert 324.9 to scientific notation, we need to write it in the form a x 10^n where 1 ≤ |a| < 10. Move the decimal point 2 places to the left, resulting in: 3.249 x 10^2 The decimal point must be moved 2 places to the left.

Key concepts

Decimal Point Movement

Understanding decimal point movement is essential for working with large or small numbers, especially in scientific contexts. Imagine you have a long number line that extends both to incredibly large and incredibly small numbers. To navigate this line more efficiently, scientists use a shorthand notation—scientific notation—which requires moving the decimal point to create a new form for the number.

Moving the decimal point to the left makes the number smaller, while moving it to the right makes the number larger. For example, when we take the number 5993 from our exercise, we move the decimal point three places to the left to get 5.993. Think of it as compressing the original number into a more manageable, yet equivalent, form. Each movement of the decimal point is counted and then used to calculate the power of ten that will accompany the compressed number in scientific notation.

Standard Scientific Notation

Standard scientific notation is a method of writing numbers to make them easier to read, compare, and compute especially when dealing with very large or very small values. It expresses numbers as a product of two factors: a coefficient and a power of ten. The coefficient must be a number greater or equal to 1 and less than 10. For instance, in the given solution, the number -72.14 becomes -7.214 when we move the decimal one place to the left, making it fit the criteria for the coefficient.

Why is standardizing important? It provides a universal way to express numbers, ensuring that scientists and mathematicians around the world can communicate their findings without ambiguity. This standardized form helps in performing mathematical operations such as multiplication and division with numbers of different magnitudes. When instructing on scientific notation, emphasize the role of this convention in scientific communication and numerical analysis.

Exponent Representation

The exponent in scientific notation represents the number of places the decimal point has been moved. It's positive when the original number is greater than 1, indicating the decimal has moved to the left. Conversely, it is negative when the original number is less than 1, showing the decimal has moved to the right. For example, with the number 0.00008291, we move the decimal point five places to the right, leading to a negative exponent: 8.291 x 10^-5.

The exponent is a crucial part of scientific notation as it provides information about the original magnitude of the number. In education, emphasizing how the exponent reflects the decimal point movement can give students a clear mental image of the number's size. Additionally, the sign of the exponent instantly tells whether the number is big or small compared to one, which is an essential concept to grasp when diving into the realms of scientific measurements and calculations.