Question

For each of the following numbers, by how many places does the decimal point have to be moved to express the number in standard scientific notation? In each case, is the exponent positive or negative? a. 102 b. 0.00000000003489 c. 2500 d. 0.00003489 e. 398,000 f. 1 g. 0.3489 h. 0.0000003489

Short answer

a. Decimal point movement: 2 places, Exponent: positive b. Decimal point movement: 11 places, Exponent: negative c. Decimal point movement: 3 places, Exponent: positive d. Decimal point movement: 5 places, Exponent: negative e. Decimal point movement: 5 places, Exponent: positive f. Decimal point movement: 0 places, Exponent: neither positive nor negative (zero) g. Decimal point movement: 1 place, Exponent: negative h. Decimal point movement: 7 places, Exponent: negative

Step-by-step solution

a. 102

To express 102 in standard scientific notation, move the decimal point two places to the left: \(1.02 \times 10^{2}\). Decimal point movement: 2 places Exponent: positive

b. 0.00000000003489

To express 0.00000000003489 in standard scientific notation, move the decimal point 11 places to the right: \(3.489 \times 10^{-11}\). Decimal point movement: 11 places Exponent: negative

c. 2500

To express 2500 in standard scientific notation, move the decimal point three places to the left: \(2.5 \times 10^{3}\). Decimal point movement: 3 places Exponent: positive

d. 0.00003489

To express 0.00003489 in standard scientific notation, move the decimal point five places to the right: \(3.489 \times 10^{-5}\). Decimal point movement: 5 places Exponent: negative

e. 398,000

To express 398,000 in standard scientific notation, move the decimal point five places to the left: \(3.98 \times 10^{5}\). Decimal point movement: 5 places Exponent: positive

f. 1

The given number 1 is already in standard scientific notation and does not require the decimal point to be moved: \(1 \times 10^{0}\). Decimal point movement: 0 places Exponent: neither positive nor negative (zero)

g. 0.3489

To express 0.3489 in standard scientific notation, move the decimal point one place to the right: \(3.489 \times 10^{-1}\). Decimal point movement: 1 place Exponent: negative

h. 0.0000003489

To express 0.0000003489 in standard scientific notation, move the decimal point seven places to the right: \(3.489 \times 10^{-7}\). Decimal point movement: 7 places Exponent: negative

Key concepts

Decimal Point Movement

Understanding decimal point movement is crucial when converting numbers into scientific notation. Scientific notation is a way to express very large or very small numbers compactly and is typically formatted as a product of a number between 1 and 10 and a power of 10. This form is especially convenient in scientific and engineering work to easily comprehend and calculate with unwieldy numbers.

When converting a number into scientific notation, the decimal point must be moved to create a new number that is between 1 and 10. If the original number is greater than 10, the decimal point is moved to the left, and for numbers less than 1, it is moved to the right. The number of places the decimal point moves determines the exponent on the 10 in scientific notation. With each move to the left, the exponent increases by 1. Conversely, with each move to the right, the exponent decreases by 1.

For example, in the case of the number 2500, moving the decimal point three places to the left gives us 2.5, so we write this as 2.5 times 10 to the power of 3. Here are some points to remember:

• Moving the decimal point to the left turns the exponent positive.
• Moving the decimal point to the right turns the exponent negative.
• If the number is already between 1 and 10, the decimal point does not move, and the exponent is 0, as with the number 1.

It's beneficial for students to practice this step to gain confidence in recognizing how many places the decimal point needs to be moved for different numbers.

Exponent in Scientific Notation

The exponent in scientific notation signifies how many places the decimal point has been moved from its original position in a standard number to convert it to a number between 1 and 10. An important thing to understand is that the exponent provides a shorthand for multiplication or division by 10, which is why this format is so powerful in scientific calculations.

A positive exponent indicates that the decimal point has moved to the left, suggesting the original number is large and we are scaling down to get our coefficient between 1 and 10. For example, the number 398,000 expressed in scientific notation as \(3.98 \times 10^{5}\) indicates that the decimal point moved 5 places to the left.

A negative exponent, conversely, indicates that the decimal point has moved to the right, revealing the original number is a small fraction. For example, the number 0.00003489 becomes \(3.489 \times 10^{-5}\) which tells us the decimal point moved 5 places to the right.

The zero exponent is a special case and indicates that the number is already between 1 and 10; thus, the decimal does not move. For instance, the value 1 remains as \(1 \times 10^{0}\). Here, the exponent shows that we have not shifted the magnitude of the original number.

Expressing Numbers in Scientific Notation

Expressing numbers in scientific notation can initially seem challenging, but it’s essentially about simplifying how we write and calculate with very large or very small numbers. The aim is to express numbers as the product of a number from 1 up to, but not including, 10 and a power of 10.

To express a number in scientific notation, follow these general steps:

• Determine if the number is greater than or less than 1.
• Move the decimal point in the original number to obtain a new number between 1 and 10.
• Count the number of places you moved the decimal point; this number will be your exponent.
• Remember to assign a positive exponent if you moved the decimal to the left and a negative exponent if you moved it to the right.
• Write out the new expression with the coefficient (the number between 1 and 10) and the power of 10 with the determined exponent.

For example, to convert the number 0.0000003489, you would move the decimal 7 places to the right, resulting in \(3.489 \times 10^{-7}\). It's especially helpful for students to write numbers in both the original and scientific notation to clearly see the relationship between the two forms.