number line

(noun)

A line that graphically represents the real numbers as a series of points whose distance from an origin is proportional to their value.

Related Terms

  • Other Inequalities
  • absolute value
  • inequality

(noun)

A visual representation of the set of real numbers as a series of points.

Related Terms

  • Other Inequalities
  • absolute value
  • inequality

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 aaa is denoted ∣a∣\left | a \right |∣a∣.
    • Therefore, ∣a∣>0\left | a \right |>0∣a∣>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:
    • aaa is to the right of bbb on this number line.
    • aaa is to the left of bbb 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 xxx satisfying 0≤x≤10 \leq x \leq 10≤x≤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<ba < x < ba<x<b indicates "betweenness"—the number xxx is between the numbers aaa and bbb.
    • This states that xxx is some number strictly between 4 and 9.
    • For a visualization of this inequality, refer to the number line below.
    • In this case, zzz is some number strictly between -2 and 0.
    • The expression x+6x + 6x+6 represents some number strictly between 1 and 8.
  • Introduction to Complex Numbers

    • A complex number has the form a+bia+bia+bi, where aaa and bbb are real numbers and iii is the imaginary unit.
    • A complex number is a number that can be put in the form a+bia+bia+bi where aaa and bbb are real numbers and iii is called the imaginary unit, where i2=−1i^2=-1i​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 (mmm) of a line, and its yyy-intercept (bbb).
    • In other words, a line with a slope of −9-9−9 is steeper than a line with a slope of 777.
    • 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+bia+bia+bi is a−bia-bia−bi.
    • The complex conjugate (sometimes just called the conjugate) of a complex number a+bia+bia+bi is the complex number a−bia-bia−bi.
    • The number a2+b2a^2+b^2a​2​​+b​2​​ is the square of the length of the line segment from the origin to the number a+bia+bia+bi.
    • The number a2+b2\sqrt{a^2+b^2}√​a​2​​+b​2​​​​​ is called the length or the modulus of the complex number z=a+biz=a+biz=a+bi.
    • The length of the line segment from the origin to the point a+bia+bia+bi is a2+b2\sqrt{a^2+b^2}√​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 ppp is a positive real number, and the eccentricity is a positive real number eee, the conic has a polar equation:
    • For a conic with a focus at the origin, if the directrix is $y=±p$, where ppp is a positive real number, and the eccentricity is a positive real number eee, 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.  
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