Problem 41 - College Algebra with Modeling and Visualization

Question

Write the number in standard form. \(1 \times 10^{-6}\) (Wavelength in meters of visible light)

Step-by-Step Solution

Answer

The number in standard form is 0.000001.

Step 1: Understand Scientific Notation

Scientific notation, like \(1 \times 10^{-6}\), is a way of expressing very large or very small numbers. In general, \(a \times 10^{b}\) represents a number \(a\) multiplied by ten raised to the power of \(b\).

Step 2: Apply the Power of Ten

The number \(10^{-6}\) means we have a decimal point moved 6 places to the left from the number \(1\). This is because the exponent is negative, indicating a very small number.

Step 3: Convert to Standard Form

To convert \(1 \times 10^{-6}\) to standard form, start from 1 and move the decimal point 6 places to the left. This results in \(0.000001\).

Key Concepts

Standard Form

Standard form is a way of writing numbers that makes them easy to read and understand, especially when dealing with very large or very small numbers. In scientific terms, standard form typically refers to a number written with one digit to the left of the decimal point followed by a power of ten. This method streamlines numbers and reduces potential errors when comparing or calculating. For example, the number is expressed as \(1 \times 10^{-6}\) in scientific notation but written as \(0.000001\) in standard form. This representation clearly shows how tiny the number is in the context of standard measurements.

Negative Exponents

Negative exponents can seem intimidating, but they play a crucial role in expressing small numbers in scientific notation. An exponent indicates how many times to use a number in a multiplication. If the exponent is negative, it convinces us to think inversely: how many times to divide by the base number.
For instance, in \(1 \times 10^{-6}\), the exponent of -6 suggests moving the decimal six places left due to division by \(10\) six times. This process fundamentally reduces the value, transitioning it from a whole number to a fractional one. Using negative exponents allows for consistent expression of both large and small numbers, ensuring clarity and simplicity in computations.

Powers of Ten

The concept of powers of ten forms the foundation of scientific notation, enabling practitioners to express, understand, and manipulate vast numerical values easily. A power of ten is any number ten raised to an exponent, representing repeated multiplication of ten.

• Positive exponents, such as \(10^3\), amplify a number by indicating how many tens multiply to reach a larger magnitude, e.g., \(1000\).
• Negative exponents, like our example \(10^{-6}\), compress a number. They indicate division, reducing its size significantly for dealing with tiny quantities, similar to small wavelengths of visible light.

Understanding powers of ten and their use in scientific notation simplifies handling complex data. Whether large or small numbers, powers of ten maintain uniformity in our mathematical world.