Question

Rewrite each of the following as an "ordinary" decimal number. a. \(2.789 \times 10^{3}\) b. \(2.789 \times 10^{-3}\) c. \(9.3 \times 10^{7}\) d. \(4.289 \times 10^{1}\) e. \(9.999 \times 10^{4}\) f. \(9.999 \times 10^{-5}\)

Short answer

a. \(2.789 \times 10^{3} = 2789\) b. \(2.789 \times 10^{-3} = 0.002789\) c. \(9.3 \times 10^{7} = 93,000,000\) d. \(4.289 \times 10^{1} = 42.89\) e. \(9.999 \times 10^{4} = 99,990\) f. \(9.999 \times 10^{-5} = 0.00009999\)

Step-by-step solution

Identify A and B for each number

For each given number, identify the coefficient (A) and the power of 10 (B). We have the following: a. \(A = 2.789, B = 3\) b. \(A = 2.789, B = -3\) c. \(A = 9.3, B = 7\) d. \(A = 4.289, B = 1\) e. \(A = 9.999, B = 4\) f. \(A = 9.999, B = -5\)

Move the decimal point

For each number, move the decimal point of the coefficient to the right or left according to the value of B, following the procedure outlined in the analysis. a. Move the decimal point 3 places to the right: \(2789\) b. Move the decimal point 3 places to the left: \(0.002789\) c. Move the decimal point 7 places to the right: \(93,000,000\) d. Move the decimal point 1 place to the right: \(42.89\) e. Move the decimal point 4 places to the right: \(99,990\) f. Move the decimal point 5 places to the left: \(0.00009999\)

Write the decimal numbers

Finally, write down the decimal numbers resulting from the previous step. a. \(2.789 \times 10^{3} = 2789\) b. \(2.789 \times 10^{-3} = 0.002789\) c. \(9.3 \times 10^{7} = 93,000,000\) d. \(4.289 \times 10^{1} = 42.89\) e. \(9.999 \times 10^{4} = 99,990\) f. \(9.999 \times 10^{-5} = 0.00009999\)

Key concepts

Decimal Conversion

When we talk about decimal conversion, we're looking at the process of changing a number from scientific notation to its 'ordinary' decimal form. Scientific notation is a useful tool for expressing very large or very small numbers in a more compact form by using a coefficient and an exponent.

To convert a number written in scientific notation to a decimal, we follow the coefficient (which is a number usually between 1 and 10) with the decimal point and then multiply it by 10 to the power of the exponent. In the case of a positive exponent, we move the decimal point to the right, which effectively multiplies the coefficient by 10 for each step we move. Conversely, a negative exponent means we move the decimal to the left, thus dividing the coefficient by 10 with each step. This process allows us to see the 'ordinary' value of incredibly large or small numbers.

Exponents

In the concept of exponents, we deal with the power to which a number is raised. It's a succinct way to represent how many times a number, known as the base, is multiplied by itself. For instance, in scientific notation, the exponent indicates how many places the decimal point should be moved.

Understanding Positive and Negative Exponents

Positive exponents indicate a large number by moving the decimal point to the right, while negative exponents represent small numbers by moving it to the left. Remember, when the exponent is negative, it's like taking the reciprocal of the number with a positive exponent, which is why we're actually dividing when moving the decimal point to the left. Exponents are fundamental in scientific notation because they make it manageable to write out and work with numbers that are otherwise unwieldly.

Coefficient

The coefficient in scientific notation is the number that comes before the multiplication symbol (×). It's crucial because it gives us the significant figures of the number we're dealing with—the meat of the number, so to speak.

In scientific notation, the coefficient must be a number greater than or equal to 1 and less than 10. If it's not, we need to adjust it until it is, and amend the exponent accordingly to keep the value of the number consistent. This criterion is a vital rule to ensure that scientific notation remains standard and easy to compare across different figures.

• If the coefficient needs to be increased, the exponent decreases (and vice versa).
• When converting to 'ordinary' decimal form, the coefficient is what you physically move, according to the rules of exponents laid out above.

Proper understanding of coefficients and how they interact with exponents is essential for accurate and effective use of scientific notation.