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The letters R, Q, N, and Z refers to a set of numbers such that:

R = real numbers includes all real number

Q= rational numbers

N = Natural numbers

z = integers

## What Does V= Mean

In Finite-Dimension Linear Algebra by Mark S. Gockenbach, there is an exercise which I can’t understand the meaning.

It’s chapter is Fields and vector spaces.

“In this exercise, we will define operation on V= in a nonstandard way: …”

I want to know what V= means. Please help!

This means that $V$ is the set of all numbers bigger than zero. In general, the notation $G=$ means that $G$ is the set of numbers between the number $a$ and the number $b$, not including $a$ and $b$ themselves. If we want to include the endpoints $a$ and $b$, we write $G=$ instead.

- $\begingroup$Thanks, but I have another question. Then u, v in V are numbers, not vectors??$\endgroup$Mar 18 ’16 at 3:18
- $\begingroup$Yes .$\endgroup$ Alex SMar 18 ’16 at 3:20
- $\begingroup$@user323870: The elements of $V$ are numbers. They are
*also*vectors, though not the sort with which we may be familiar .$\endgroup$Mar 18 ’16 at 3:29 - $\begingroup$Thank you! I totally understand!$\endgroup$

## Less Than Or Equal To On A Number Line

Inequalities like less than or equal to and greater than or equal to are represented in a different way on a number line. To denote these, we use the closed circle to mark the limit value and point the arrow towards the given condition of inequality. Let us see this on a number line given below:

We can see that when we want to denote ‘x less than or equal to – 5’, we marked a circle at – 5 and pointed an arrow towards the values less than – 5, as suggested in the condition of inequality. Similarly, when we want to denote ‘x greater than or equal to – 2’, we marked a circle at – 2 and pointed an arrow towards the values greater than – 2, as suggested in the condition of inequality.

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## What Is Less Than Or Equal To

‘Less than or equal to’, as the name suggests, means a variable is either less than or equal to another number, variable, or quantity. ‘Less than or equal to’ can also be expressed as at most, no more than, a maximum of, and not exceeding. Observe the following figure to see the symbol that shows a ‘less than’ sign with a sleeping line under it.

## Less Than Or Equal To Examples

**Example 1: **A classroom can occupy 60 study tables at most. Express this statement using the less than or equal to symbol.

**Solution:**

Let’s represent the number of study tables by the variable x.

It is given that the classroom can occupy a maximum of 60 study tables.

So, this can be represented by a simple inequality.

x 60

Therefore, the condition is represented as x 60, where x is the number of tables.

**Example 2:** Charles had 18 chocolates which he was going to distribute to his friends on his birthday. Since the box was open, he lost some chocolates on the way. If x is the number of chocolates that Charles currently has, write an inequality using the ‘less than or equal to’ symbol that represents this situation.

**Solution:**

Number of chocolates Charles had initially = 18

Since he lost some chocolates, the number of chocolates with him currently is less than 18.

x < 18

We know that the number of chocolates should be a non-negative quantity. Hence,

x 0

This inequality can also be written as:

0 x

From the inequalities and , we get,0 x < 18

Therefore, the number of chocolates Charles has is represented as 0 x < 18.

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## What Does The Notation $01$ Mean

I am studying the procedure for bucket sort from Introduction To Algorithms by Cormen et al, which assumes that the input is generated by a random process that distributes the elements uniformly and independently over the interval $” closing bracket for the interval?

- WacDonald’sAug 12 ’12 at 16:44
- 3$\begingroup$The notation is called
*half-closed interval.*A closed interval $$ includes the end points $0, 1.$ An open interval $$ does*not*include the end points $0, 1.$ A half-closed interval is closed on one side, open on the other side. So $[0, 1)$ includes $0$ but does not include $1.$$\endgroup$ user2468Aug 12 ’12 at 16:56 - 1$\begingroup$@Geek: It is the usual notation for
*intervals*. The interval $$. The interval $$ is an*open*interval, while $$ is a*closed*interval.$\endgroup$Aug 12 ’12 at 16:56 - 1$\begingroup$In Section 15.3 they define a closed interval along with notation, they mention open and half-open intervals but do not provide notation.$\endgroup$

## Why Zero Is So Damn Useful In Math

Zeros influence on our mathematics today is twofold. One: Its an important placeholder digit in our number system. Two: Its a useful number in its own right.

The first uses of zero in human history can be traced back to around 5,000 years ago, to ancient Mesopotamia. There, it was used to represent the absence of a digit in a string of numbers.

Heres an example of what I mean: Think of the number 103. The zero in this case stands for theres nothing in the tens column. Its a placeholder, helping us understand that this number is one-hundred and three and not 13.

Okay, you might be thinking, this is basic. But the ancient Romans didnt know this. Do you recall how Romans wrote out their numbers? 103 in Roman numerals is CIII. The number 99 is XCIX. You try adding CIII + XCIX. Its absurd. Placeholder notation is what allows us to easily add, subtract, and otherwise manipulate numbers. Placeholder notation is what allows us to work out complicated math problems on a sheet of paper.

If zero had remained simply a placeholder digit, it would have been a profound tool on its own. But around 1,500 years ago , in India, zero became its own number, signifying nothing. The ancient Mayans, in Central America, also independently developed zero in their number system around the dawn of the common era.

In the seventh century, the Indian mathematician Brahmagupta wrote down whats recognized as the first description of the arithmetic of zero:

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## Common Mathematical Symbols And Terminology: Maths Glossary

Mathematical symbols and terminology can be confusing and can be a barrier to learning and understanding basic numeracy.

This page complements our numeracy skills pages and provides a quick glossary of common mathematical symbols and terminology with concise definitions.

Are we missing something? Get it touch to let us know.

## Why Does Zero Factorial Equal One

- M.S., Mathematics, Purdue University
- B.A., Mathematics, Physics, and Chemistry, Anderson University

A zero factorial is a mathematical expression for the number of ways to arrange a data set with no values in it, which equals one. In general, the factorial of a number is a shorthand way to write a multiplication expression wherein the number is multiplied by each number less than it but greater than zero. 4! = 24, for example, is the same as writing 4 x 3 x 2 x 1 = 24, but one uses an exclamation mark to the right of the factorial number to express the same equation.

It is pretty clear from these examples how to calculate the factorial of any whole number greater than or equal to one, but why is the value of zero factorial one despite the mathematical rule that anything multiplied by zero is equal to zero?

The definition of the factorial states that 0! = 1. This typically confuses people the first time that they see this equation, but we will see in the below examples why this makes sense when you look at the definition, permutations of, and formulas for the zero factorial.

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## Necessary And Sufficient Conditions

Biconditional statements are related to conditions that are both necessary and sufficient. Consider the statement if today is Easter, then tomorrow is Monday. Today being Easter is sufficient for tomorrow to be Monday, however, it is not necessary. Today could be any Sunday other than Easter, and tomorrow would still be Monday.

## E In Scientific Notation And The Meaning Of 1e6

You don’t need a calculator to use E to express a number in scientific notation. You can simply let E stand for the base root of an exponent, but only when the base is 10. You wouldn’t use E to stand for base 8, 4 or any other base, especially if the base is Euler’s number, e.

When you use E in this way, you write the number *x*E*y*, where *x* is the first set of integers in the number and *y* is the exponent. For example, you would write the number 1 million as 1E6. In regular scientific notation, this is 1 × 106, or 1 followed by 6 zeros. Similarly 5 million would be 5E6, and 42,732 would be 4.27E4. When writing a number in scientific notation, whether you use E or not, you usually round to two decimal places.

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## What Is The Origin Of Zero How Did We Indicate Nothingness Before Zero

**Robert Kaplan, author of The Nothing That Is: A Natural History of Zero and former professor of mathematics at Harvard University, provides this answer:**

The first evidence we have of zero is from the Sumerian culture in Mesopotamia, some 5,000 years ago. There, a slanted double wedge was inserted between cuneiform symbols for numbers, written positionally, to indicate the absence of a number in a place .

Image: KRISTEN MCQUILLINTIMELINE shows the development of zero throughout the world. The first recorded zero appeared in Mesopotamia around 3 B.C. The Mayans invented it independently circa 4 A.D. It was later devised in India in the mid-fifth century, spread to Cambodia near the end of the seventh century, and into China and the Islamic countries at the end of the eighth. Zero reached western Europe in the 12th century. |

Writing Numbers |

**The Babylonians****The Mayans****The Chinese****The Hindus**

The symbol changed over time as positional notation , made its way to the Babylonian empire and from there to India, via the Greeks . Arab merchants brought the zero they found in India to the West. After many adventures and much opposition, the symbol we use was accepted and the concept flourished, as zero took on much more than a positional meaning. Since then, it has played avital role in mathematizing the world.

*Answer originally posted Feb. 28, 2000.*

## Why Can’t We Divide By Zero

The reason that the result of a division by zero is undefined is the fact that **any attempt at a definition leads to a contradiction.**

To begin with, how do we define *division*? The ratio *r* of two numbers *a* and *b*:

*r=a/b*

is that number *r* that satisfies

Well, if *b=0*, i.e., we are trying to divide by zero, we have to find a number *r* such that

*r*0=a**r*0=0**r**a=0*

Now you could say that *r=infinity* satisfies . That’s a common way of putting things, but what’s infinity? It is not a number! Why not? Because if we treated it like a number we’d run into contradictions. Ask for example what we obtain when adding a number to infinity. The common perception is that infinity plus any number is still infinity. If that’s so, then

*infinity = infinity+1 = infinity + 2*

which would imply that 1 equals 2 if infinity was a number. That in turn would imply that all integers are equal, for example, and our whole number system would collapse.

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## What Do The Letters R Q N And Z Mean In Math

In math, the letters R, Q, N, and Z refer, respectively, to real numbers, rational numbers, natural numbers, and integers.

**Who are the experts?**Our certified Educators are real professors, teachers, and scholars who use their academic expertise to tackle your toughest questions. Educators go through a rigorous application process, and every answer they submit is reviewed by our in-house editorial team.

The letters R, Q, N, and Z refers to a set of numbers such that:

R = real numbers includes all real number

Q= rational numbers

N = Natural numbers

z = integers ( all integers…

## Introduction: Connecting Your Learning

Mathematics is commonly referred to as the universal language. With the overwhelming popularity of social media, this is becoming increasingly clear. You cannot communicate with others from various walks of life without the common language of mathematics driving your Internet connection, moving satellites, or translating words.

Furthermore, people cannot map the earth from smartphones without computer programmers to generate mathematical algorithms to provide this technology. If you choose to pursue a career in information technology, whether it is in security, networking, mobile applications, or geospatial technology, you will need a certain level of mathematical knowledge.

In this lesson, you will learn how real numbers are ordered, how many categories of numbers exist, and mathematical symbolism that allows you to quickly compare or categorize numbers.

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## What Does If And Only If Mean In Mathematics

To understand if and only if, we must first know what is meant by a conditional statement. A conditional statement is one that is formed from two other statements, which we will denote by P and Q. To form a conditional statement, we could say if P then Q.

The following are examples of this kind of statement:

- If it is raining outside, then I take my umbrella with me on my walk.
- If you study hard, then you will earn an A.
- If
*n*is divisible by 4, then*n*is divisible by 2.

## What Does Less Than Or Equal To Mean

Less than or equal to in math means that you can’t have more than something, you must have either less than or equal to the given limit. ‘Less than or equal to’, as the name suggests, means a number is either less than or equal to another number. It can also be expressed as at most, no more than, a maximum of, and not exceeding.

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## How Do I Construct A Real Number Line

Here are three steps to follow to create a real number line.

Choose any point on the line and label it 0. This point is called the origin.

Now that you have created a number line, it is time see how points on a number line are defined.

**Real Numbers**

A real number is any number that is the coordinate of a point on the real number line.

## What Else Can Understand Nothing

We may not be born with the ability to understand zero. But our capacity to learn it may have deep evolutionary roots, as some new science shows us.

The fourth step in thinking of zero that is thinking of zero as a symbol may be unique to humans. But a surprising number of animals can get to step three: recognizing that zero is less than one.

Even bees can do it.

Scarlett Howard, a PhD student at Royal Melbourne Institute of Technology, recently in *Science *thats almost identical to the one Brannon did with kids. The bees chose the blank page 60 to 70 percent of the time. And they were significantly better at discriminating a large number, like six, from zero, than they were in discriminating one from zero. Just like the kids.

This is impressive, considering that weve got this big mammalian brain but bees have a brain thats so small weighs less than a milligram, Howard says. Her research group is hoping to understand how bees do these calculations in their minds, with the goal of one day using those insights to build more efficient computers.

In similar experiments, researchers have shown that monkeys can recognize the empty set . But the fact that bees can do it is kind of amazing, considering how far they are away from us on the evolutionary trees of life. The last common ancestor between us and the bees lived about 600 million years ago, which is an eternity in evolutionary times, Nieder says.

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## + Addition Plus Positive

**The addition symbol + is usually used to indicate that two or more numbers should be added together, for example, 2 + 2.**

The + symbol can also be used to indicate a positive number although this is less common, for example, +2. Our page on **Positive and Negative Numbers** explains that a number without a sign is considered to be positive, so the plus is not usually necessary.

See our page onAdditionfor more.

## Euler’s Number In Nature

Exponents with e as a base are known as natural exponents, and here’s the reason. If you plot a graph of

you’ll get a curve that increases exponentially, just as you would if you plotted the curve with base 10 or any other number. However, the curve *y* = e*x* has two special properties. For any value of *x*, the value of *y* equals the value of the slope of the graph at that point, and it also equals the area under the curve up to that point. This makes e an especially important number in calculus and in all the areas of science that use calculus.

The logarithmic spiral, which is represented by the equation

is found throughout nature, in seashells, fossils and and flowers. Moreover, e turns up in numerous scientific contexts, including the studies of electric circuits, the laws of heating and cooling, and spring damping. Even though it was discovered 350 years ago, scientists continue to find new examples of Euler’s number in nature.

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