Problem 34 - Algebra 1: Concepts and Skills

Question

Decide whether the number is in scientific notation. If not, write the number in scientific notation. $$ 10 \times 10^{-3} $$

Step-by-Step Solution

Answer

The given number in scientific notation is \(1 \times 10^{-2}\).

Step 1: Identify the requirements for scientific notation

Scientific notation requires a number to be represented as a decimal number between 1 and 10, multiplied by a power of 10. The number given is \(10 \times 10^{-3}\) and it can be simplified in the next step.

Step 2: Simplify the given number

By simplifying the number \(10 \times 10^{-3}\) we get \(10 \times (1/1000) = 0.01\).

Step 3: Convert simplified number to scientific notation

In scientific notation, the number 0.01 can be represented as \(1 \times 10^{-2}\). The number has been converted so that the decimal number is between 1 and 10 and it is multiplied by a power of 10.

Key Concepts

Decimal Representation

Decimal representation is a way of expressing numbers using a base-ten system, which is the standard in everyday mathematics. In this system, numbers are represented using digits from 0 to 9, along with a decimal point to indicate parts of a whole number. For example, the number 0.75 in decimal form tells us this is three-quarters of a whole.

When converting numbers into scientific notation, decimal representation plays a crucial role. The goal is to adjust the number so that its decimal portion falls between 1 and 10. This makes it easier to handle very large or very small numbers by simplifying their representation. For instance, while 0.01 is a valid decimal, for scientific notation it needs to be adjusted to 1 (a number between 1 and 10) by multiplying or dividing by appropriate powers of ten.

Power of Ten

Powers of ten are a way of expressing how many times the number ten is multiplied by itself. In mathematics, they help in easily representing large or small numbers by using exponential terms. The expression \(10^{-3}\) means 10 is multiplied by itself in inverse three times, simplifying it to a fraction: \(\frac{1}{1000}\).

Every time you shift the decimal point in a number during the conversion to scientific notation, you're essentially multiplying or dividing by a power of ten. To maintain equivalency, a move to the right divides by ten, producing negative exponents, while a move to the left multiplies by ten, producing positive exponents. In \(10 \times 10^{-3}\), this can be simplified and represented more succinctly in scientific notation as \(1 \times 10^{-2}\). The negative exponent shows we are dealing with a small number, specifically moving the decimal place to achieve the required format.

Number Simplification

Number simplification involves reducing a number to its most manageable form, especially when transitioning to scientific notation. It allows complex numbers to be expressed in straightforward terms, facilitating easier computations and understanding.

In the example of transitioning from \(10 \times 10^{-3}\), the multiplication unfolds as a simple decimal division: \(10 \times \frac{1}{1000}\) which simplifies down to 0.01. This step simplifies understanding by narrowing down calculations into manageable units of ten. Turning this into scientific notation involves determining the exponent needed to present this decimal in the required form of \(a \times 10^b\).

• If the number is greater than 10, divide and adjust the exponent up.
• If the number is smaller than 1, multiply and adjust the exponent down.

This process of simplification allows for the number to be neatly rewritten as \(1 \times 10^{-2}\), maintaining clarity and functionality in presenting the numerical value.