Problem 99 - Algebra 1

Question

SCIENTIFIC NOTATION Rewrite the number in scientific notation. $$1,229,000,000$$

Step-by-Step Solution

Answer

The scientific notation of 1,229,000,000 is \(1.229 \times 10^9\).

Step 1: Identify the decimal placement

In 1,229,000,000 the decimal is at the end of the number. In the absence of a decimal point, it is understood to be at the end of the number.

Step 2: Move the decimal

Move the decimal point 9 places to the left (which is the number of digits after the first digit in the number), until it is one space to the right of the first nonzero digit. This gives \(1.229\). Positive 9 indicates that the decimal point was moved to the left.

Step 3: Write in scientific notation

Write the number as the product of the digits and 10 to the power equal to the number of places you moved the decimal. Hence, \(1.229 \times 10^9\).

Key Concepts

Decimal Point Placement

When working with numbers, understanding where the decimal point is located is crucial. In many large whole numbers like 1,229,000,000, it may seem like the decimal point is missing. However, even when it's not visible, the decimal is assumed to be at the end of the number. This is because whole numbers implicitly have their decimal point positioned after the last digit.

For example, with 1,229,000,000, the implied decimal point is at the rightmost end, making it 1,229,000,000.0. This understanding is your starting point when converting to scientific notation, as it helps to rearrange the figure correctly by aligning digits relative to the decimal.

Moving the Decimal

Once you have identified the current placement of the decimal, the next step involves moving the decimal to transform the number into scientific notation. This representation requires the decimal point to be placed immediately to the right of the first non-zero digit.

In the case of 1,229,000,000, we need to move the decimal point 9 places to the left. Count each digit as you shift leftwards: start at the end, move past all zeroes, and stop just after the first non-zero digit, which is '1'. This results in 1.229. Your count determines how many places you've moved the decimal, and this count is essential for the next step.

• Each move of the decimal maintains the number’s value by multiplying with powers of ten.
• Ensure the number is in the form where the significant digit is a non-zero followed by a decimal and other figures.

Powers of Ten

Powers of ten play a key role in scientific notation. They make expressing large or small numbers much more manageable by representing them as a product of a number between 1 and 10 and a power of ten.

For 1,229,000,000, after moving the decimal 9 places to the left, it becomes 1.229. To convey that we've moved the decimal, we multiply by 10 raised to the power of the number of moves. As we moved the decimal 9 places to the left, this is represented by a positive exponent of 9 on the 10, yielding the notation as \(1.229 \times 10^9\).

This method highlights the role of the exponent: it conveys how many places the decimal was shifted, establishing whether the original number was large or small. A positive exponent indicates that the original number was large (greater than 1), while a negative exponent is used for numbers less than one in scientific notation.