QQuestionMathematics
QuestionMathematics
How do you know how many digits are in fifty million without writing the number in standard form?
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Answer
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Step 1:I'll solve this problem step by step, following the specified LaTeX formatting guidelines:
Step 2:: Understand the Question
The problem asks how to determine the number of digits in fifty million without writing out the full number in standard form.
Step 3:: Recall the Definition of Digits
Digits are the individual numerals used to represent a number. To count the number of digits, we need to determine how many numerals are needed to write the number.
Step 4:: Use Logarithmic Approach
We can use the base- 10 logarithm to count digits. The number of digits in a positive integer can be found by taking the floor of the base- 10 logarithm of the number and adding 1.
Step 5:: Apply the Logarithm Method
\log_{10}(50,000,000) = \log_{10}(5 \times 10^{7})
For fifty million:
Step 6:: Calculate the Logarithm
\log_{10}(5 \times 10^{7}) = \log_{10}(5) + \log_{10}(10^{7})
= \log_{10}(5) + 7
Step 7:: Take the Floor and Add 1
The number of digits will be $$\lfloor\log_{10}(50,000,000)\rfloor + 1
Step 8:: Verify the Calculation
\lfloor\log_{10}(50,000,000)\rfloor + 1 = \lfloor7.699\rfloor + 1 = 7 + 1 = 8
Final Answer
Key Insight: This method works for any positive integer by using the logarithmic property that the number of digits is equal to the floor of the base- 10 logarithm of the number, plus 1.
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