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Chapter 1: Problem 188

1 / 2

Short Answer

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0.5

Step by step solution

01

Understand the Problem

The problem requires finding the division of 1 by 2.
02

Set Up the Equation

Express the problem as the fraction \( \frac{1}{2} \).
03

Perform the Division

To divide 1 by 2, you can calculate it directly: \( 1 \div 2 = 0.5 \).
04

Write the Answer

The result of dividing 1 by 2 is \( 0.5 \).

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Key Concepts

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

Fraction Operations
Fraction operations are critical skills in mathematics. They include addition, subtraction, multiplication, and division of fractions. These operations are essential for problem-solving in both simple and complex math problems. For example, when you see the fraction \( \frac{1}{2} \), it represents '1' divided by '2'. Here are a few things to remember about fraction operations:

  • Addition and subtraction of fractions require a common denominator. This makes the numerators compatible so you can add or subtract them directly.
  • When multiplying fractions, multiply the numerators together and the denominators together.
  • Dividing fractions involves multiplying by the reciprocal of the divisor. The reciprocal of a fraction is simply swapping its numerator and denominator. For instance, the reciprocal of \( \frac{2}{3} \) is \( \frac{3}{2} \).
Understanding these basics will help you handle questions involving fraction operations easily.
Division
Division is a fundamental arithmetic operation. It splits a number into equal parts. For example, dividing '1' by '2' means finding how many times '2' fits into '1'. The division can be represented in various forms:

  • As a fraction: \( \frac{1}{2} \)
  • As a division expression: 1 梅 2
  • As a decimal: 0.5
When you see a fraction, it is essentially a division problem. For instance, to solve \( 1 \div 2 \), you understand that '1' divided by '2' equals '0.5'. This conversion is useful in fields such as algebra, where fractions and decimals often interchange.
Understanding Fractions
Fractions represent parts of a whole and are written with two numbers separated by a slash. The top number is called the numerator, representing the portion, and the bottom number is the denominator, representing the total number of parts.

In the fraction \( \frac{1}{2} \):
  • The numerator is '1', showing one part.
  • The denominator is '2', showing a total of two parts.
So, \( \frac{1}{2} \) means one part out of two equal parts. Understanding fractions is vital, as it helps in dividing quantities equally and comparing different quantities. Fractions are everywhere in real life, from measuring ingredients in a recipe to understanding probabilities in games and predictions.

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Most popular questions from this chapter

Prove that $$ \begin{aligned} \operatorname{det}\left(\begin{array}{ccccc} 1 & 4 & 9 & \cdots & n^2 \\ n^2 & 1 & 4 & \cdots & (n-1)^2 \\ \vdots & \vdots & \vdots & \vdots & \vdots \\ 4 & 9 & 16 & \cdots & 1 \end{array}\right) \\ &=(-1)^{n-1} \frac{n^{n-2}(n+1)(2 n+1)\left((n+2)^n-n^n\right)}{12} . \end{aligned} $$

Let \(f\) be a continuous function on \([0,1]\), which is bounded below by 1 , but is not identically 1 . Let \(R\) be the region in the plane given by \(0 \leq x \leq 1\), \(1 \leq y \leq f(x)\). Let $$ R_1=\\{(x, y) \in R \mid y \leq \bar{y}\\} \quad \text { and } \quad R_2=\\{(x, y) \in R \mid y \geq \bar{y}\\} $$ where \(\bar{y}\) is the \(y\)-coordinate of the centroid of \(R\). Can the volume obtained by rotating \(R_1\) about the \(x\)-axis equal that obtained by rotating \(R_2\) about the \(x\)-axis?

Find all twice continuously differentiable functions \(f\) for which there exists a constant \(c\) such that, for all real numbers \(a\) and \(b\), $$ \left|\int_a^b f(x) d x-\frac{b-a}{2}(f(b)+f(a))\right| \leq c(b-a)^4 $$

Every year, the first warm days of summer tempt Lake Wohascum's citizens to venture out into the local parks; in fact, one day last May, the MAA Student Chapter held an impromptu picnic. A few insects were out as well, and at one point an insect dropped from a tree onto a paper plate (fortunately an empty one) and crawled off. Although this did not rank with Newton's apple as a source of inspiration, it did lead the club to wonder: If an insect starts at a random point inside a circle of radius \(R\) and crawls in a straight line in a random direction until it reaches the edge of the circle, what will be the average distance it travels to the perimeter of the circle? ("Random point" means that given two equal areas within the circle, the insect is equally likely to start in one as in the other; "random direction" means that given two equal angles with vertex at the point, the insect is equally likely to crawl off inside one as the other.)

Suppose you draw \(n\) parabolas in the plane. What is the largest number of (connected) regions that the plane may be divided into by those parabolas? (The parabolas can be positioned in any way; in particular, their axes need not be parallel to either the \(x\) - or the \(y\)-axis.)
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