number line

Examples of number line in the following topics:

• Absolute Value

  • Absolute value can be thought of as the distance of a real number from zero.
  • In mathematics, the absolute value (sometimes called the modulus) of a real number $a$ is denoted $\left | a \right |$.
  • Therefore, $\left | a \right |>0$ for all numbers.
  • When applied to the difference between real numbers, the absolute value represents the distance between the numbers on a number line.
  • The absolute values of 5 and -5 shown on a number line.

• Introduction to Inequalities

  • The above relations can be demonstrated on a number line.
  • For a visualization of this, see the number line below:
  • For a visualization of this, see the number line below:
  • $a$ is to the right of $b$ on this number line.
  • $a$ is to the left of $b$ on this number line.

• Interval Notation

  • A "real interval" is a set of real numbers such that any number that lies between two numbers in the set is also included in the set.
  • For example, the set of all numbers $x$ satisfying $0 \leq x \leq 1$ is an interval that contains 0 and 1, as well as all the numbers between them.
  • Other examples of intervals include the set of all real numbers and the set of all negative real numbers.
  • The image below illustrates open and closed intervals on a number line.
  • Representations of open and closed intervals on the real number line.

• Inequalities with Absolute Value

  • Inequalities with absolute values can be solved by thinking about absolute value as a number's distance from 0 on the number line.
  • What numbers work?
  • This answer can be visualized on the number line as shown below, in which all numbers whose absolute value is less than 10 are highlighted.
  • Now think about the number line.
  • All numbers therefore work.

• Compound Inequalities

  • The compound inequality $a < x < b$ indicates "betweenness"—the number $x$ is between the numbers $a$ and $b$.
  • This states that $x$ is some number strictly between 4 and 9.
  • For a visualization of this inequality, refer to the number line below.
  • In this case, $z$ is some number strictly between -2 and 0.
  • The expression $x + 6$ represents some number strictly between 1 and 8.

• Introduction to Complex Numbers

  • A complex number has the form $a+bi$, where $a$ and $b$ are real numbers and $i$ is the imaginary unit.
  • A complex number is a number that can be put in the form $a+bi$ where $a$ and $b$ are real numbers and $i$ is called the imaginary unit, where $i^2=-1$.
  • Complex numbers extend the idea of the one-dimensional number line to the two-dimensional complex plane by using the horizontal axis for the real part and the vertical axis for the imaginary part.
  • A complex number whose real part is zero is said to be purely imaginary, whereas a complex number whose imaginary part is zero is a real number.
  • Complex numbers allow for solutions to certain equations that have no real number solutions.

• Slope

  • Slope describes the direction and steepness of a line, and can be calculated given two points on the line.
  • In mathematics, the slope of a line is a number that describes both the direction and the steepness of the line.
  • Putting the equation of a line into this form gives you the slope ($m$) of a line, and its $y$-intercept ($b$).
  • In other words, a line with a slope of $-9$ is steeper than a line with a slope of $7$.
  • This ratio is represented by a quotient ("rise over run"), and gives the same number for any two distinct points on the same line.

• Complex Conjugates

  • The complex conjugate of the number $a+bi$ is $a-bi$.
  • The complex conjugate (sometimes just called the conjugate) of a complex number $a+bi$ is the complex number $a-bi$.
  • The number $a^2+b^2$ is the square of the length of the line segment from the origin to the number $a+bi$.
  • The number $\sqrt{a^2+b^2}$ is called the length or the modulus of the complex number $z=a+bi$.
  • The length of the line segment from the origin to the point $a+bi$ is $\sqrt{a^2+b^2}$.  

• Fitting a Curve

  • Curve fitting with a line attempts to draw a line so that it "best fits" all of the data.
  • In this section, we will only be fitting lines to data points, but it should be noted that one can fit polynomial functions, circles, piece-wise functions, and any number of functions to data and it is a heavily used topic in statistics.
  • Example:  Write the least squares fit line and then graph the line that best fits the data 
  • If we have a point that is far away from the approximating line, then it will skew the results and make the line much worse.  
  • Notice 4 points are above the line, and 4 points are below the line.

• Conics in Polar Coordinates

  • Previously, we learned how a parabola is defined by the focus (a fixed point) and the directrix (a fixed line).
  • We can define any conic in the polar coordinate system in terms of a fixed point, the focus $P(r,θ)$ at the pole, and a line, the directrix, which is perpendicular to the polar axis.
  • For a conic with a focus at the origin, if the directrix is $x=±p$, where $p$ is a positive real number, and the eccentricity is a positive real number $e$, the conic has a polar equation:
  • For a conic with a focus at the origin, if the directrix is $y=±p$, where $p$ is a positive real number, and the eccentricity is a positive real number $e$, the conic has a polar equation:
  • Any conic may be determined by three characteristics: a single focus, a fixed line called the directrix, and the ratio of the distances of each to a point on the graph.