Problem 33 - Prealgebra

Question

Express each number in scientific notation. A human blinks about 6.25 million times a year.

Step-by-Step Solution

Answer

6.25 million is expressed as \(6.25 \times 10^6\) in scientific notation.

Step 1: Understanding Scientific Notation

Scientific notation is a way of expressing numbers that are too big or too small in the form of a product of a number (called the coefficient) and a power of 10. The format is: \[ a \times 10^n \]where \(1 \leq \left| a \right| < 10\) and \(n\) is an integer.

Step 2: Identifying the Number

The given number is 6.25 million. The first step is to express 6.25 million as a numerical value, which is 6,250,000.

Step 3: Moving the Decimal Point

To express 6,250,000 in scientific notation, identify the position where you'd place the decimal to make the coefficient \(a\) fall between 1 and 10. For the number 6,250,000, the point needs to move to between 6 and 2, making it 6.25.

Step 4: Counting Decimal Places

Count the number of places the decimal point has moved. It originally is after the last zero in 6,250,000 (i.e., 6,250,000.0) and moves to between 6 and 2, so it moves 6 places left.

Step 5: Expressing in Scientific Notation

Since the decimal moves to the left, the exponent on the 10 will be positive, indicating the original number was big. Thus, the scientific notation for 6,250,000 is:\[6.25 \times 10^6\]

Key Concepts

Understanding Exponentiation

Exponentiation is a mathematical operation that involves raising a number, known as the base, to the power of another number, known as the exponent. For example, when we write \(10^6\), 10 is the base and 6 is the exponent, indicating that the base should be multiplied by itself 6 times. This operation can make calculations involving very large or very small numbers much simpler.

• Base: The number that is going to be raised to a power.
• Exponent: This denotes how many times the base is multiplied by itself.

In scientific notation, as seen in the expression \(6.25 \times 10^6\), the "10" is the base and the exponent reflects how many decimal places the original number is moved to achieve a number between 1 and 10. This notation is beneficial for simplifying expressions of grand or miniscule magnitudes, streamlining computation and comparisons.

Role of the Decimal Place

The decimal place plays a crucial role in converting numbers into scientific notation. To express a number correctly this way, the decimal must be placed such that the resulting coefficient is a number between 1 and 10. For the number 6,250,000 in its numerical form:

• Identify the numerical value, 6,250,000.
• Locate the decimal point (which is initially at the end).
• Move the decimal to form 6.25, creating a suitable coefficient.
• Count the number of places moved, which is crucial for determining the exponent.

In this example, we shift the decimal 6 places to the left, transforming the number into 6.25, which makes it simpler to handle in its abbreviated form as \(6.25 \times 10^6\). The movement of the decimal directly influences the exponent's value, enabling the compressing of extensive numbers.

The Mathematics of Scientific Notation

Scientific notation is a powerful technique in mathematics for dealing with very large or small numbers, allowing for more concise calculations and comparisons. This notational method is particularly prevalent in fields such as physics, chemistry, and engineering, where precision and the ability to manage complex calculations are essential.

• Concise Expression: Transforms cumbersome numbers into a succinct, standard form.
• Operation Ease: Simplifies multiplication and division of large values.
• Comparison Simplicity: Eases the direct comparison of significantly varied magnitudes.

By expressing a number like 6,250,000 as \(6.25 \times 10^6\), we gain the ability to quickly understand, communicate, and manipulate these values within mathematical operations. Grasping scientific notation is fundamental for students to navigate the breadth of mathematics more efficiently and effectively.