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Chapter 4: Problem 101

(a) Mark on the coordinate line all those points \(x\) in the interval [0,1) which have the digit "1" immediately after the decimal point in their decimal expansion. What fraction of the interval [0,1) have you marked? Note: " [0,1) " denotes the set of all points between 0 and 1 , together with \(0,\) but not including 1. [0,1] denotes the interval including both endpoints; and (0,1) denotes the interval excluding both endpoints. (b) Mark on the interval [0,1) all those points \(x\) which have the digit "1" in at least one decimal place. What fraction of the interval [0,1) have you marked? (c) Mark on the interval [0,1) all those points \(x\) which have a digit "1" in at least one position of their base 2 expansion. What fraction of the interval [0,1) have you marked? (d) Mark on the interval [0,1) all those points \(x\) which have a digit "1" in at least one position of their base 3 expansion. What fraction of the interval [0,1) have you marked?

Short Answer

Expert verified
(a) 0.1, (b) 1, (c) 1, (d) 1.

Step by step solution

01

Understand the Problem

We are asked to find what fraction of the interval \([0,1)\) we have marked by identifying numbers based on the presence of a digit '1' in various bases or positions. Each subquestion asks us to consider a different scenario about the presence of '1' in decimal, binary, or ternary expansions.
02

Solve Part (a)

In part (a), we are asked to mark numbers in \([0,1)\) that have '1' immediately after the decimal point in their decimal expansion. These numbers are of the form \(x = 0.1abcd\ldots\). Such numbers range from 0.1 (inclusive) to 0.2 (exclusive). The interval marked is \([0.1, 0.2)\). Therefore, the fraction marked is \(0.2 - 0.1 = 0.1\).
03

Solve Part (b)

For part (b), we must consider numbers that have at least one '1' in any decimal place. These numbers are \(x = 0.1\text{abc}\ldots\), \(x = 0.\text{a}1\text{bc}\ldots\), or any other form where '1' appears. The absence of '1' in all places means numbers consisting solely of the digits {0,2,...,9} except '1'. The probability of not having a '1' in the first digit after the decimal is \( \frac{9}{10} \), in the next place is \( \frac{9}{10} \), and so on. Thus, the fraction marked is \(1 - (0.9)^n\) for \(n \to \infty \), which approaches 1.
04

Solve Part (c)

In part (c), we mark fractions in \([0,1)\) with a '1' in the binary (base 2) representation. In binary, a number with no '1' is 0, but all other fractions in \([0,1)\) will have at least one '1' because binary is composed of digits 0 and 1. The fraction not having any '1's would be only 0, thus the entire interval except 0 is marked. Therefore, the fraction marked is 1.
05

Solve Part (d)

For part (d), we mark fractions with at least one '1' in the base 3 (ternary) representation. A number without a '1' consists of digits only \{0,2\}. The probability of a digit being '0' or '2' is \( \frac{2}{3}\). The probability of not having any '1's in any position converges to 0 as the number of places increases. Hence, the fraction marked is 1.

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Key Concepts

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

Decimal Expansion
Decimal expansion refers to expressing numbers in the base-10 system, which is the most familiar system used in our daily calculations. Here, each digit after the decimal point represents a fraction of a power of ten. For example, the number 0.1 is equivalent to \(\frac{1}{10}\), and 0.12 is \(\frac{12}{100}\).

When tasked to find numbers in the interval \[0,1)\] with '1' right after the decimal, we're looking for numbers like 0.1, 0.11, 0.12, etc. These numbers range from 0.1 (inclusive) to 0.2 (exclusive). This forms the interval \[0.1, 0.2)\], which represents a fraction of 0.1 of the interval \[0,1)\]. Understanding this helps visualize how small or large portions we are marking on the number line.
Binary Representation
The binary representation of numbers uses only two digits: 0 and 1. It's the basis of all modern digital computers. In base-2, any decimal number can be expressed using these two symbols. To convert a number from decimal to binary, we break it down into sums of powers of 2.

For instance, the decimal number 0.75 in binary is 0.11 because: \[0.75 = \frac{1}{2} + \frac{1}{4} = 0.11_2\]. Now, when considering the interval \[0,1)\], every number except 0 will have a '1' in its binary representation. Hence, almost the entire interval except the very point at 0 is marked, making the fraction marked in the interval equal to 1.
Ternary Representation
Ternary representation uses base-3, meaning numbers are expressed using digits 0, 1, and 2. Just like binary, we can convert decimal numbers into base-3 by using powers of 3.

In base-3, numbers can appear without the digit '1', using just 0 and 2. Numbers such as 0.02 (ternary) do not contain a '1'. However, the prevalence of '1's in ternary becomes significant over larger intervals.

Thus, while a number without any '1鈥� seems possible, as we extend the interval [0,1) infinitely, the chance of entirely evading '1's becomes negligible. This is why the marked fraction in base-3, similar to binary, approaches 1.
Coordinate Line Marking
When marking points on a coordinate line, it helps to visually represent intervals and segments. This is crucial in mathematics for understanding fractions of an interval such as \[0,1)\].

By marking sections like \[0.1, 0.2)\] or the full interval excluding points with certain characteristics, students gain a tangible grasp of abstract concepts. Such a method highlights how numbers and fractions can be segmentally represented and measured along a line.
  • It aids in visualizing numeric intervals.
  • Provides insight into how different base representations affect number positioning.
  • Enhances comprehension of fractional parts and their proportions on a number line.
Using coordinate markings to showcase number inclusion or exclusion solidifies abstract mathematical ideas in a visual and engaging format.

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Most popular questions from this chapter

(a)(i) Suppose \(a^{n}-1=p\) is a prime. Prove that \(a=2\) and that \(n\) must itself be prime. (ii) How many primes are there among the first five such numbers $$ 2^{2}-1,2^{3}-1,2^{5}-1,2^{7}-1,2^{11}-1 ? $$ (b)(i) Suppose \(a^{n}+1=p\) is a prime. Prove that either \(a=1\), or \(a\) must be even and that \(n\) must then be a power of 2 . (ii) In the simplest case, where \(a=2\), how many primes are there among the first five such numbers $$ 2^{1}+1,2^{2}+1,2^{4}+1,2^{8}+1,2^{16}+1 ? $$

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Dad took our new baby to the clinic to be weighed. But the baby would not stay still and caused the needle on the scales to wobble. So Dad held the baby still and stood on the scales, while nurse read off their combined weight: \(78 \mathrm{~kg}\). Then nurse held the baby, while Dad read off their combined weight: \(69 \mathrm{~kg}\). Finally Dad held the nurse, while the baby read off their combined weight: \(137 \mathrm{~kg}\). How heavy was the baby? The situation described in Problem 92 is representative of a whole class of problems, where the given information incorporates a certain symmetry, which the solver would be wise to respect. Hence one should hesitate before applying systematic brute force (as when using the information from one weighing to substitute for one of the three unknown weights - a move which effectively reduces the number of unknowns, but which fails to respect the symmetry in the data). A similar situation arises in certain puzzles like the following.

(a) Prove that the number \(\sqrt{2}+\sqrt{3}\) is irrational. (b) Prove that the number \(\sqrt{2}+\sqrt{3}+\sqrt{5}\) is irrational

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