Question

In Problems 129-136, write each number as a decimal. \(1.214 \times 10^{-3}\)

Short answer

0.001214

Step-by-step solution

Understand the Scientific Notation

The given number is written in scientific notation as \(1.214 \times 10^{-3}\). This means that 1.214 needs to be multiplied by \(10^{-3}\).

Interpret the Exponent on 10

The exponent \(-3\) indicates that the decimal point in 1.214 needs to be moved 3 places to the left.

Move the Decimal Point

Start with 1.214 and move the decimal point 3 places to the left. This transforms 1.214 into 0.001214.

Key concepts

decimal notation

Decimal notation is one of the most common ways to represent numbers. It uses a base-10 system, which includes digits from 0 to 9. This means that each position in a number has a value depending on its place relative to the decimal point.
For example, in the number 123.45, the digits to the left of the decimal point represent whole numbers, and the digits to the right represent fractions of a whole.
Decimal notation is particularly useful for representing very large or very small numbers when combined with scientific notation. The scientific notation like the one given in the problem, allows us to work with these numbers more easily by expressing them as a product of a decimal and a power of ten.

exponents

Exponents are a way to express repeated multiplication of the same number. For example, in the expression \(10^{-3}\), the \(10\) is the base and \(-3\) is the exponent.
Exponents can be positive or negative:
- Positive exponents indicate multiplication. For example, \(10^3\) means. \(10 \times 10 \times 10\), which equals 1000.
- Negative exponents indicate division or moving the decimal point to the left. For example, \(10^{-3}\) means \(\frac{1}{10 \times 10 \times 10}\) or moving the decimal point three places to the left.
In our problem, \(10^{-3}\) means we need to move the decimal point three places to the left in the number \(1.214\), making it \(0.001214\).

moving decimal points

Moving decimal points is an essential step in converting numbers from scientific notation to decimal notation. To move the decimal point:
- Positive exponent: Move the decimal point to the right.
- Negative exponent: Move the decimal point to the left.
For example, given \(1.214 \times 10^{-3}\):
1. Start with \(1.214\).
2. Because the exponent is \(-3\), move the decimal point three places to the left.
3. The result is \(0.001214\).
This method ensures that the number is accurately converted from scientific notation to a readable decimal form.