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Chapter 1: Problem 12

Convert to scientific notation. 3,700,000

Short Answer

Expert verified
The scientific notation of 3,700,000 is \[ 3.7 \times 10^6 \].

Step by step solution

01

Identify the significant digits

The number is 3,700,000. The significant digits are 3.7.
02

Count the number of places to move the decimal point

Move the decimal point in 3,700,000 to the left until it is just to the right of the first non-zero digit, which is 3. This involves moving the decimal point 6 places to the left.
03

Express the number in scientific notation

Scientific notation is written as the significant digits followed by 'times 10' raised to the power of the number of places the decimal point was moved. Thus, the scientific notation for 3,700,000 is \[ 3.7 \times 10^6 \]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Significant Digits
To understand scientific notation, it's crucial to grasp the idea of significant digits. These are the digits in a number that are meaningful in terms of accuracy. When converting to scientific notation, only the non-zero digits are considered significant. For example, in the number 3,700,000, the significant digits are 3 and 7. This simplifies to 3.7 since trailing zeros do not count unless they appear after a decimal point. The significant digits tell us how precise the number is, which is vital for maintaining accuracy in scientific calculations.
Decimal Point Movement
Converting a large number into scientific notation involves moving the decimal point. This helps us express the number in a more compact and readable form. For instance, in 3,700,000, we need to move the decimal point 6 places to the left to place it just after the first significant digit (3). This movement creates a simpler number that still represents the same value. The number 3,700,000 becomes 3.7 when the decimal is moved 6 places to the left, effectively reducing the number to its significant digits while accounting for its magnitude.
Exponential Notation
Exponential notation is used to express large or small numbers in a concise format. It's essential in scientific communication to avoid unwieldy strings of digits. In the example of 3,700,000, once we have the significant digits (3.7) and have moved the decimal point 6 places left, we express this change as an exponent of 10. This gives us \( 3.7 \times 10^6 \). The exponent tells us how many places the decimal moved: a positive exponent for moves to the left and a negative exponent for moves to the right. Scientific notation thus combines significant digits and powers of 10 to convey numbers efficiently without losing precision.

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