Problem 58 - Beginning and Intermediate Algebra

Question

Convert to scientific notation. $$0.00000000514$$

Step-by-Step Solution

Answer

The given number, 0.00000000514, can be represented in scientific notation as \(5.14 \times 10^{-9}\).

Step 1: Identify "a" in the scientific notation

We have the number 0.00000000514, and we want to represent it as \(a \times 10^b\). To find the value of \(a\), we need to move the decimal point until we get a number between 1 and 10.

Step 2: Moving the decimal point

Move the decimal point 9 places to the right in 0.00000000514: \[ 0.00000000514 \rightarrow 5.14 \] Now the number is a decimal between 1 and 10, which is what we need for the \(a\) value.

Step 3: Identify "b" in the scientific notation

Since we moved the decimal point 9 places to the right, the value of the exponent, \(b\), will be -9 (negative because we moved the decimal point to the right).

Step 4: Write the number in scientific notation

Now that we have the values of \(a\) and \(b\), write the number in scientific notation: \[5.14 \times 10^{-9}\] The given number, 0.00000000514, can be represented in scientific notation as \(5.14 \times 10^{-9}\).

Key Concepts

Decimal Notation

Decimal notation is the system of writing numbers where digits are arranged in a sequence. This sequence is based on powers of ten. Each digit's position gives it a specific value, which is why this system is called base-10. For example, the number 345.67 has two parts: before and after the decimal point. The digits 3, 4, and 5 represent the hundreds, tens, and ones place respectively. After the decimal point, 6 and 7 represent tenths and hundredths.

• Decimals are numbers expressed in parts smaller than one. These use the decimal point to separate whole numbers from fractions.
• Moving the decimal point changes the value of the number itself. For example, moving the point in 0.1 to become 1 means increasing its value 10 times.
• The decimal system is easy to use because it aligns with our counting system using tens.

Understanding decimal notation helps in converting numbers into scientific notation by moving the decimal point to its required position.

Exponents

Exponents are a mathematical way to express repeated multiplications of a number by itself. They are written as a small number above and to the right of a base number, signifying how many times the base is used in a multiplication.
For example, with scientific notation, exponents represent the scaling factor for converting very large or very small numbers to a compact form. In the number \[10^{-9}\], the "-9" is the exponent showing the decimal point moved 9 places. Exponents can be positive or negative:

• Positive exponents (e.g., \(10^3\)) mean multiplying the base number (10) by itself three times (10 x 10 x 10 = 1000).
• Negative exponents (e.g., \(10^{-3}\)) mean dividing 1 by the base number raised to a positive exponent (1/1000 or 0.001).

Using exponents helps simplify handling numbers that are either very small or very large, making calculations easier.

Place Value

Place value is the value of where the digit is in the number. It's a cornerstone concept of the decimal notation system. In a number, each digit has a position representing a different value. This system helps us break down a number into simpler parts, making it easier to understand and work with.
For instance: In 0.00000000514, each zero fills a place that signifies a different degree of smallness.

• Whole numbers (1 to infinity) have place values like ones, tens, hundreds, etc.
• Decimals (less than one) have place values like tenths, hundredths, thousandths, etc.
• In scientific notation, understanding place value is key to knowing how many places to move the decimal point. For example, to move \(5.14\) from 0.00000000514, you counted 9 places where zeros were before the digit 5.

By using place value, we understand how to position digits correctly in numbers, crucial for conversions like those involving scientific notation.