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Chapter 0: Problem 5

\text { In scientific notation, } 1234.5678= ________________

Short Answer

Expert verified
1.2345678 \times 10^3

Step by step solution

01

Identify the significant figures

Determine the significant figures in the number 1234.5678. The significant figures in this number are 12345678.
02

Place the decimal point

Move the decimal point to the position just after the first significant figure, which is after 1. This changes the number to 1.2345678.
03

Count the number of decimal places moved

Count how many places the decimal point was moved to convert 1234.5678 to 1.2345678. The decimal point was moved 3 places to the left.
04

Express in scientific notation

Write the number as a product of the new number and 10 raised to the power of the number of places moved. The result is \(1.2345678 \times 10^3\).

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Key Concepts

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

headline of the respective core concept
Scientific notation is a method used to write very large or very small numbers conveniently. It expresses numbers as the product of two factors. The first factor is a number between 1 and 10 with significant figures. The second factor is a power of ten.
Significant Figures
Significant figures refer to the digits in a number that contribute to its accuracy. In the number 1234.5678, all the digits are significant.
Here are a few rules to identify significant figures:
  • All non-zero digits are significant.
  • Any zeros between significant digits are also significant.
  • Leading zeros (zeros before the first non-zero digit) are not significant.
  • Trailing zeros in a number without a decimal point are not significant, but they are significant if a decimal point is present.
So for 1234.5678, each digit from 1 to 8 is significant, giving us 8 significant figures.
Decimal Point
A decimal point separates the whole number part of a number from its fractional part.
Example: In 1234.5678, the decimal point separates 1234 from 5678.
When converting to scientific notation, we often need to move the decimal point. Here鈥檚 how:
  • Move the decimal point to just after the first significant digit.
  • Count how many places you moved the decimal point.
For 1234.5678, we move the decimal point 3 places left, resulting in 1.2345678.
Power of Ten
The power of ten represents how many times you multiply or divide by 10.
In scientific notation, after adjusting the decimal point, the number is multiplied by 10 raised to the number of places moved.
Example: For 1234.5678, moving the decimal 3 places to the left becomes: \[1.2345678 \times 10^3\] The exponent 3 indicates the decimal point was originally 3 places to the right.

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