Problem 116 - Algebra A Combined Function

Question

Write each number in standard form. Then write the number in scientific notation. The English version of Wikipedia has more than 2.3 million articles.

Step-by-Step Solution

Answer

2,300,000 in standard form; \(2.3 \times 10^6\) in scientific notation.

Step 1: Write in Standard Form

The number 2.3 million is expressed as 2,300,000 in standard form. This is done by recognizing that 'million' means 1,000,000. So, 2.3 multiplied by 1,000,000 equals 2,300,000.

Step 2: Convert to Scientific Notation

To express 2,300,000 in scientific notation, we identify the first non-zero digit and place the decimal point after it, resulting in 2.3. We then count how many places we moved the decimal to the left, which is 6 places. Thus, in scientific notation, it is written as \(2.3 \times 10^6\).

Key Concepts

Standard Form

In mathematics, standard form typically refers to the way in which numbers are written to showcase all their digits. This method clearly displays each digit according to its value. For example, the number 2.3 million should be represented in its expanded form: 2,300,000.

When dealing with large numbers, each digit’s position is crucial. In 2,300,000:

• '2' is the millions position, representing 2,000,000
• '3' occupies the hundred thousands place, equal to 300,000
• Following this are zeros in the thousands down to the units place

Writing numbers in standard form makes them easier to read and understand, especially when conveying the magnitude of large values.

Place Value

Place value is the concept of assigning value to digits based on their positions in a number. In any number, each place has a value that is a power of 10.

Consider the number 2,300,000 again. Here:

• The millions place holds the value in millions, i.e., 10 extsuperscript{6}.
• The hundred thousands place has a value equivalent to 10 extsuperscript{5}.

Recognizing place values helps in breaking down the number into more manageable parts, paving the way for easier operations such as addition, subtraction, and conversion to other forms like scientific notation. It's a foundational skill in mathematics that aids in understanding the structure of numbers.

Mathematical Notation

Mathematical notation encompasses symbols and conventions used to represent numbers and equations concisely. One of the most powerful tools within this realm is scientific notation.

Scientific notation appears in the form of a number times a power of 10, like the number 2,300,000, which can be rewritten as 2.3 × 10 extsuperscript{6}.

• The initial part is a number between 1 and 10 (inclusive), 2.3 in this instance.
• The exponent on 10 indicates how many places the decimal point was moved. A positive exponent means the decimal moves to the right to reach the larger original number.

This form is particularly useful for expressing very large or very small numbers efficiently, making calculations easier and providing clarity in scientific and mathematical contexts.