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# relationships between sets of real numbers with visual representation

Depict relationship between sets of real numbers with a visual representation.

## Research, Knowledge and Information :

### 8th Grade Texas Mathematics: Unpacked Content

8.2A) Extending previous knowledge of sets and subsets using a visual representation to describe relationships between sets of real numbers. (8.2B) All square roots ...

### IXL - Texas eighth-grade math standards

IXL's dynamic math practice skills offer comprehensive coverage of Texas ... using a visual representation to describe relationships between sets of real numbers;

### 8.2.A [2012] 8.2.B [2012] - Learning Farm

extend previous knowledge of sets and subsets using a visual representation to describe relationships between sets of real numbers;; approximate the value of an ...

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Generate Texas TEKS Math 8 ... Extend previous knowledge of sets and subsets using a visual representation to describe relationships between sets of real numbers ...

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Describing the relationships between sets and subsets of real numbers using a visual representation and ... describe relationships between sets of real numbers.

### Extend previous knowledge of sets and subsets using a visual ...

Extend previous knowledge of sets and subsets using a visual representation to describe relationships between sets of real numbers 8.2A

### The Real Number System - Maneuvering the Middle

6.2A classify whole numbers, integers, and rational numbers using a visual representation such as a Venn diagram to describe relationships between sets of numbers

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using a visual representation to describe relationships between sets of real numbers *8.4A SS Use similar right triangles to develop an

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## Suggested Questions And Answer :

### relationships between sets of real numbers with visual representation

We can define some sets within the superset of all real numbers: A. Integers  A1. Positive integers or natural numbers, including zero   A1.1. Positive integers with integer exponents greater than zero.   A1.2. Negative integers with even integer exponents greater than zero  A2. Negative integers   A2.1. Negative integers with odd integer exponents greater than zero B. Fractions  B1. Proper fractions   B1.1. Positive proper fractions    B1.1.1. Positive integers with integer exponents less than zero   B1.2. Negative proper fractions    B1.2.1. Negative integers with integer exponents less than zero B2. Improper fractions (including mixed numbers)   B2.1. Positive improper fractions   B2.2. Negative improper fractions C. Irrational numbers  C1. Transcendental numbers (cannot be defined as the root of a fraction or integer)  C2. Integer root of a positive integer for integers>1  C3. Integer root of a positive fraction for integers>1  C4. Fractional root of a positive integer  C5. Fractional root of a positive fraction These are arbitrary sets that can be represented visually as circles. Some circles may be completely isolated from other circles; some may completely contain other circles; some may intersect other circles. Indentation above implies that the circle associated with the lesser indentation contains the whole of the circle with the greater indentation. For example, B1 is completely contained by B, B2.2 is contained in B2, which in turn is contained in B. The separate sets of odd and even numbers could be included. We could add the set of prime numbers, positive integers>1 with no other factors than 1 and the number itself. We could add the set of perfect numbers, integers>1 in which the factors, including 1 but excluding the number itself, add up to the number. We could add factorials (the product of consecutive integers up to the factorial integer itself). These would be included in the superset of positive integers. The visual representation is that the subset is totally enclosed by the superset as a circle inside another circle. An example of interlocking circles is the set of factorials with the set of perfect numbers, where 6 is contained in the overlap. 6 belongs to both sets.

### If the operation *is defined on the set of real numbers by x*y=x^2-y

4*5=16-5=11; 5*4=25-4=21. Because x*y is not equal to y*x, the associative property does not apply. Assume that a*b=b*a where a and b are real, but a is not equal to b, then a^2-b=b^2-a. a^2-b^2=b-a; (a-b)(a+b)=b-a=-(a-b); a+b=-1 (dividing through by a-b). But a+b=-1 forces a relationship between a and b, and that means the associative property does not apply generally. Therefore, a*b<>b*a. x*4=0 implies x^2-4=0=(x-2)(x+2), so x=-2 or 2.

### Which produces a counter example to her statement

The range of f(x) = ax + b is the set of all real numbers given that a and b are real numbers. The set of real numbers, R, is a subset of the complex numbers, C. If we let x = i, then f(x)  is a complex number and is not within the real number set.

### how, express the set of all real numbers larger than 2 in set builder notation?

Set builder notation uses symbols to represent words or phrases. The set parentheses (curly brackets { }) enclose the set. A variable, usually x represents an element. R represents all real numbers. : or vertical bar (|) means "for all".  The sign resembling € or e means "in", "belongs to", or "in the set (of)". The intersection of sets is represented by an inverted V or U, for which I will use the caret symbol ^. {x | x € R ^ x > 2} might be a way of expressing the set of all real numbers greater than 2.

### solve for y=3x+4, and show both ordered sets on graph

The equation is linear in two variables, x and y. It can't be solved, as such, for particular values of x and y, but a straight line graph represents the dependency of one variable on the other. The graph is a mapping of the function in which the range of values of x and y is the set of all real numbers, so the graph maps x values onto y and y onto x in a unique way: there is only one value of x for a particular value of y, and only one value of y for a particular value of x. There is a real value of x for every real value of y and vice versa, subject to the dependency function y=3x+4, and no real values are excluded. The ordered pair (x,y) forms the set of real numbers satisfying the equation. Every point on the line (and there are an infinite number of them) is an ordered pair, and forms an infinite set. Each ordered pair is effectively a solution to the equation, so there are an infinite number of solutions.

### problem 6b-3/3b^2-12, which part of the problem are the excluded value, the sets and real numbers

Divide top and bottom by 3: (2b-1)/((b-2)(b+2). Excluded values: b=-2 and 2. The variable b can take any other values in the set of real numbers. As b gets larger, the expression approaches 2b/b^2=2/b, so the expression gets closer and closer to zero from a negative and positive direction, and where b is between -2 and 2. The range of the expression is all real numbers.

### Finding a polynomial of a given degree with given zeros: Complex zeros

If the polynomial has an even degree, meaning that the highest power of the variable (for example: x) is an even number (for example: x^4), then all the zeroes could be complex. If odd, there must be at least one real zero. A complex zero is given by the complex expression: a+ib, where a and b are real, and i=sqrt(-1). If the polynomial has real coefficients (all the numbers in at are, or represent, real numbers), as is usually the case, then the imaginary part of the complex zeroes must combine to make a real number. For example, if one complex zero is identified as a+ib, then there will be another zero a-ib such that (x-a-ib)(x-a+ib)=x^2-2ax+a^2+b^2. (Note that, although a and b are real, they are not necessarily rational numbers.) There are no imaginary components in this expansion. Example: x^4+x^3-x-1 is degree 4 polynomial. It has real zeroes 1 and -1 so that it factorises to (x-1)(x+1)(x^2+x+1). The zeroes of x^2+x+1 are found by setting the quadratic to zero and solving using the formula; so x=(-1+sqrt(1-4))/2=(-1+sqrt(-3))/2=(-1+isqrt(3))/2. The two complex roots are -1/2-isqrt(3)/2 and -1/2+isqrt(3)/2. In this case a=-1/2 and b=sqrt(3)/2. Note that a^2+b^2=1. Example: you're told that a degree 5 polynomial has a real zero 1, and two complex zeroes 1+i and -2-i. Find all zeroes and the polynomial. The two missing zeroes will be 1-i and -2+i. The polynomial is (x-1)(x-1-i)(x-1+i)(x+2+i)(x+2-i)=(x-1)(x^2-2x+2)(x^2+4x+5)=x^5+x^4-3x^3-x^2+12x-10. Strictly speaking, the polynomial is a(x^5+x^4-3x^3-x^2+12x-10), where a is a real number, because multiplying a polynomial by a constant doesn't affect its zeroes.

### what is real in real analysis?

A real number is one of the set of rational and irrational numbers. It excludes imaginary numbers that are based on the imaginary square root of -1. The range of real numbers is infinite. The study of them is real analysis including functions, series, etc.

### In the set of real numbers, the first number less than -3 is -4.

FAWLS if  it sed INTEGERS, it wood be true

### Let X be the set of real numbers with the usual pseudometric

me dont put metriks in me suep