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Chapter 4: Problem 17

Fill in the blanks. Any nonzero value divided by itself is \(____\)

Short Answer

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Step by step solution

01

Understanding the basic mathematical concept

Firstly, recognize the basic division rule that any number, except zero, when divided by itself gives the quotient as 1. This rule is fundamental to arithmetic.
02

Applying the concept to fill in the blank

According to the rule mentioned above, it's clear that any nonzero value divided by itself will always be 1. Therefore, the blank should be filled with '1'

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Key Concepts

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

Division
Division is a fundamental operation in arithmetic, often paired with multiplication as its opposite. The process involves splitting a number, known as the dividend, into equal parts determined by another number, the divisor. Essentially, division tells us how many times one number is contained within another.
In mathematical terms, if you divide a number 'a' (dividend) by another number 'b' (divisor), you express it as: \[ a \div b \] or \[ \frac{a}{b} \].
  • The dividend is the number being divided.
  • The divisor is the number by which the dividend is divided.
The result of a division operation is called the quotient. Understanding this operation is crucial since it's used in various aspects of everyday life, such as calculating averages, budgeting, or even determining speeds.
Nonzero Values
Nonzero values are numbers that are any integer, real number, or complex number except zero. When dealing with division, particularly in arithmetic, it is important to understand this concept.
  • Zero is unique since dividing by zero is undefined in mathematics.
  • Nonzero values, on the other hand, can readily act as divisors.
The rule that a nonzero value divided by itself equals one highlights the importance of recognizing nonzero conditions. This rule emphasizes a consistent result in division beyond zero, reinforcing the reliability of calculations in both simple and complex mathematical problems. It鈥檚 crucial to remember in arithmetic that the concept and calculations with nonzero values are not only essential but necessary to avoid mathematical errors, especially when programming or handling equations.
Quotients
Quotients are the results we obtain after performing a division operation. Understanding what a quotient represents is key to grasping the fundamentals of division.
When you divide a number by another, such as in \( 10 \div 2 \), the quotient is 5, meaning 2 is contained in 10 exactly 5 times.
  • The quotient demonstrates how many units of the divisor fit into the dividend.
  • It can be an integer or a decimal, depending on whether the division is exact or not.
Another important aspect is recognizing that when any nonzero value is divided by itself, the quotient is always 1. This is a vital rule in solving mathematical expressions and understanding the intrinsic properties of numbers. Mastery of quotients allows deeper insights into more complex arithmetic operations, setting the foundation for advanced mathematical concepts.

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Most popular questions from this chapter

Perform each division. If there is a remainder, leave the answer in quotient \(+\frac{\text { remainder }}{\text { divisor }}\) form. Assume no division by \(0 .\) $$\frac{x^{3}+3 x^{2}+3 x+1}{x+1}$$

Simplify. $$2\left(x^{2}-5 x-4\right)-3\left(x^{2}-5 x-4\right)+6\left(x^{2}-5 x-4\right)$$

Find the difference when \(\left(-3 z^{3}-4 z+7\right)\) is subtracted from the sum of \(\left(2 z^{2}+3 z-7\right)\) and \(\left(-4 z^{3}-2 z-3\right)\)

Perform the operations. $$3\left(x y^{2}+y^{2}\right)-2\left(x y^{2}-4 y^{2}+y^{3}\right)+2\left(y^{3}+y^{2}\right)$$

The number of feet that a car travels before stopping depends on the driver's reaction time and the braking distance. For one driver, the stopping distance \(d\) is given by the function \(d=f(v)=0.04 v^{2}+0.9 v,\) where \(v\) is the velocity of the car. Find the stopping distance when the driver is traveling at \(30 \mathrm{mph}\).
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