Examples of number line in the following topics:
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- 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 ∣a∣.
- Therefore, ∣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.
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- 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.
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- 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≤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.
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- 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.
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- 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.
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- 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 i2=−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.
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- 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.
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- 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 a2+b2 is the square of the length of the line segment from the origin to the number a+bi.
- The number √a2+b2 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 √a2+b2.
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- 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.
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- 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.