Number Systems Overview

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Questions and Answers

What are natural numbers?

  • All rational numbers
  • Imaginary numbers
  • All integers greater than zero (correct)
  • Negative integers

Which of the following describes complex numbers?

  • Only imaginary numbers
  • All integers
  • Numbers of the form a + b, where a and b are real
  • Numbers of the form a + bi, where a and b are real and i is imaginary (correct)

What type of numbers can be expressed as fractions?

  • Only whole numbers
  • Only proper fractions
  • Only negative integers
  • Rational numbers (correct)

What does the Fundamental Theorem of Algebra state?

<p>Every polynomial of degree n has exactly n complex roots (A)</p> Signup and view all the answers

Which option correctly identifies proper and improper fractions?

<p>Proper fractions are less than one, while improper fractions are more than one (D)</p> Signup and view all the answers

How are all real numbers classified in relation to complex numbers?

<p>All real numbers are complex with b = 0 (A)</p> Signup and view all the answers

Which of the following represents the set of whole numbers?

<p>0, 1, 2, 3, 4, 5 (B)</p> Signup and view all the answers

Which of the following is an example of an imaginary number?

<p>sqrt(-49) (A)</p> Signup and view all the answers

What distinguishes irrational numbers from rational numbers?

<p>They cannot be expressed as a fraction. (D)</p> Signup and view all the answers

What includes all possible sums of real and imaginary numbers?

<p>Complex numbers (A)</p> Signup and view all the answers

Which set of numbers includes positive integers?

<p>Natural numbers (C)</p> Signup and view all the answers

Which of the following sets contains negative integers?

<p>Integers (D)</p> Signup and view all the answers

Which statement is true about rational numbers?

<p>They can be expressed as a ratio of two integers. (B)</p> Signup and view all the answers

Which of the following is not a characteristic of real numbers?

<p>They include square roots of negative numbers. (B)</p> Signup and view all the answers

What is a necessary step to check if a solution is valid for an equation?

<p>Plug the solution back into the original equation (C)</p> Signup and view all the answers

What does 'infinitely many solutions' imply about an equation?

<p>The equation is true for all possible values of the variable (B)</p> Signup and view all the answers

What must be done to eliminate fractions in an equation?

<p>Multiply through by the LCM of the denominators (B)</p> Signup and view all the answers

What does the statement 'If b < 0, then ...' indicate in solving inequalities?

<p>Boundary values will be affected by direction of inequality (A)</p> Signup and view all the answers

Which of the following represents a typical boundary point in equations?

<p>A value that makes the equation zero (C)</p> Signup and view all the answers

In the expression '7x + 3 < 0', what does this signify?

<p>Negative x values may satisfy the inequality (C)</p> Signup and view all the answers

In the context of inequalities, what does 'a > b' generally indicate?

<p>A could be any value greater than b (D)</p> Signup and view all the answers

What is the significance of checking large values in the context of solving equations?

<p>To confirm boundary conditions at extreme values (A)</p> Signup and view all the answers

What shape represents intervals where endpoints are not included?

<p>Open interval (C)</p> Signup and view all the answers

In the inequality $x > 5$, what does the endpoint represent?

<p>An excluded value (D)</p> Signup and view all the answers

How is a closed interval denoted on a number line?

<p>With filled circles (A), With brackets (B)</p> Signup and view all the answers

What does the notation $a - b < x < a + b$ imply?

<p>x excludes the values at a and b (A)</p> Signup and view all the answers

What happens to endpoints when inequalities are represented as rays?

<p>Endpoints may disappear (D)</p> Signup and view all the answers

Which of the following represents a closed interval?

<p>([a, b]) or [a, b] (D)</p> Signup and view all the answers

What does an open circle around 'a' signify in graphical representation?

<p>The point 'a' is not included in the solution set. (A)</p> Signup and view all the answers

In the inequality $|x - a| < b$, what does 'b' represent?

<p>The distance from 'a' to 'x'. (A)</p> Signup and view all the answers

Which of the following correctly describes the union of inequalities?

<p>At least one of the inequalities must be true. (D)</p> Signup and view all the answers

What does the shaded area to the left of 'a' in $x < a$ represent?

Signup and view all the answers

What does an open circle around 'a' indicate in graphical representations of inequalities?

<p>Exclusion of the value 'a' (B)</p> Signup and view all the answers

In the inequality $x < a$, what does the shaded area to the left of 'a' represent?

<p>All points less than 'a' (A)</p> Signup and view all the answers

What is represented by the inequality $|x - a| < b$?

<p>x is closer than b units to a (C)</p> Signup and view all the answers

When two inequalities are joined by 'OR', what does this mean in terms of their solutions?

<p>At least one of the inequalities must be true (C)</p> Signup and view all the answers

What does the notation $x < a$ signify on a number line?

<p>All points to the left of 'a' are included (B)</p> Signup and view all the answers

If $|x - a| rac{1}{2} < b$, what does it imply about 'x'?

<p>x is closer than b units to a (D)</p> Signup and view all the answers

What type of graph would represent the inequality $x < a$?

<p>An open ray extending left of 'a' (A)</p> Signup and view all the answers

In the expression $x - 1 > 4$, what are the corresponding values of x?

<p>x &gt; 5 (B)</p> Signup and view all the answers

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Study Notes

Number Systems Overview

  • Natural Numbers: Counting numbers starting from 1; represented as 1, 2, 3, …
  • Whole Numbers: All natural numbers including zero; represented as 0, 1, 2, 3, …
  • Integers: Whole numbers and their negatives; represented as ..., -3, -2, -1, 0, 1, 2, 3, ...
  • Positive Integers: Natural numbers; 1, 2, 3, …
  • Negative Integers: Negative natural numbers; -1, -2, -3, …
  • Rational Numbers: Numbers expressed as fractions; can be positive, negative, proper (less than 1), or improper (greater than 1).
  • Irrational Numbers: Numbers that cannot be expressed as fractions; examples include square roots of non-perfect squares.
  • Real Numbers: Include all rational and irrational numbers; can be represented as points on the number line.
  • Imaginary Numbers: Square roots of negative numbers; written in terms of 'i', where i = √-1.
  • Complex Numbers: Combination of real and imaginary numbers, expressed as a + bi, where a and b are real numbers.
  • All reals are complex (b=0) and all imaginaries are complex (a=0).

The Fundamental Theorem of Algebra

  • States that every polynomial of degree n has exactly n complex roots, including multiple roots.

Inequalities and Solutions

  • To solve inequalities, eliminate fractions by multiplying through by the least common multiple of the denominators.
  • Check for unique solutions, infinitely many solutions, or no solutions by evaluating boundary points.
  • Shading on Graphs: Use open circles for endpoints not included and closed circles for included endpoints.

Interval Notation

  • Represents pieces of the number line; expressions like a - b < x < a + b indicate the range of x values.
  • Open Interval: a < x < b; endpoints are not included.
  • Closed Interval: a ≤ x ≤ b; endpoints are included.
  • Union of inequalities is represented as OR; at least one inequality must hold true.

Additional Points on Equations

  • Solutions involving inequalities may require shading on graphs to represent valid ranges.
  • For boundary points, test points from the intervals to ascertain which portions shade appropriately.
  • Careful consideration is needed if dealing with both positive and negative numbers in equations.

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