In terms of mathematics, the rise or run between any two points on a particular line is termed as a Slope. A slope is basically used to measure the steepness of a particular line. It comprises of two points or coordinates. These points are often shown by variables, “X” and “Y” letters. A change in either one of the variables will affect the other and vice versa.

The letters “X” and “Y” have two different axes. Lines and points are placed with the help on integers on these axes. These integers could either be positive or negative, with zero always being at the centre of the graph. Zero always lies at the intersection of these two axes.

The concept of slopes is very commonly used. Different areas make use of this concept. Fields like economics, construction, architecture and so on make use of this concept. Fields related to health and trend analysis also use the concept of slope in their day to day activities. Basically, anything which makes use of an angle or steepness can be measured through the formula for the slope.

In most cases, a slope is expressed in positive or negative integers. Though in a few cases, the value of both the “X” and the “Y” can be equal to zero. In such cases, an undefined and zero slope comes into existence, wherein the numerator or the denominator is zero.

**Undefined vs Zero Slope**

The difference between Undefined Slope and Zero Slope is that an undefined slope means that it has a vertical line whereas, on the other hand, a horizontal line has zero slope. Zero is the denominator of an Undefined Slope whereas zero is the numerator of a Zero Slope.

**Main Differences Between Undefined and Zero Slope**

Parameter of Comparison | Undefined Slope | Zero Slope |

Characteristics | The characteristics of an Undefined Slope is of a vertical line. | The characteristics of a Zero Slope is of a horizontal line. |

Value | An Undefined Slope has a non-existent value since it can not have any concrete value. | A Zero Slope has a value of zero, which is determined. |

Determinants | An Undefined Slope is determined by the variable “X”. | A Zero Slope is determined by the variable “Y”. |

Zero | An Undefined Slope has zero as its denominator. | A Zero Slope has zero as a difference between its numerators. |

Change | In an Undefined Slope, the “X” does not change, while the “Y” changes. | In a Zero Slope, the “Y” does not change, whereas the “X” changes. |

**What is Undefined Slope?**

In simple terms, an Undefined Slope can be defined as a straight line on any graph. It is basically the slope of a vertical line. In an Undefined Slope, the variable “X” does not have an existing value. It is undetermined. The denominator of the Undefined Slope is zero. It is because of this reason that the value of this slope is non-existent, irrespective of the numerator. Since any numerator cannot be divided by zero, the value is always non-existent.

An Undefined Slope is represented by “X” variable. The difference between the two “X” points is zero. Any line in this slope moves neither to the left nor to the right, along the “Y” variable., since there is no change horizontally. The variable “Y” doesn’t change in the case of an Undefined Slope, whereas the variable “X” changes.

**What is Zero Slope?**

In simple words, a Zero Slope is a slope of a horizontal line. A line on a graph which is horizontal is characterized as a Zero Slope. It is represented by “Y” variable. The variable “Y” does not change, whereas the variable “X” keeps changing in case of a Zero Slope.

The numerator of a Zero Slope is always zero. Thus, the difference between two points on the “Y” variable is zero. Irrespective of the denominator, the value of the Zero Slope is zero. This makes the slope a determined number. This is because the numerator is zero and when zero is divided by any number, the result is zero. The Zero Slope is basically a straight line which does not move upwards or downwards towards the “X” variable. This line runs parallel to the variable “X”.

**Main Differences Between Undefined and Zero Slope**

- In an Undefined Slope, the graph of the line is vertical, whereas on the other hand in a Zero Slope the graph of the line is horizontal.
- In an Undefined Slope, the denominator is zero whereas on the other hand in a Zero Slope the difference between the numerators is zero.
- The value of an Undefined Slope is not determined and is non-existent. On the other hand, in the case of a Zero Slope, the value of the slope is determined and is zero.
- The Undefined Slope is represented by the “X” variable whereas on the other hand the Zero Slope is represented by the variable “Y”.
- An Undefined Slope runs parallel to the “Y” variable, whereas, on the other hand, a Zero Slope runs parallel to the “X” variable.
- In case of an Undefined Slope, the variable “X” remains constant whereas the variable “Y” changes. On the other hand in case of a Zero Slope, the variable “Y” remains constant whereas the variable “X” changes.

**Conclusion**

The concept of Slopes is very popularly used in day-to-day activities. Slopes are represented by the “X” and “Y” variable. The slopes are represented as integers which could be positive or negative. In certain cases though, the variables “X” and “Y” can be equal to zero or they could be non-existent. In such cases, wither the numerator or the denominator is zero. These cases are termed as Undefined Slope or Zero Slope.

Undefined Slope and Zero Slope differ from each other as an Undefined Slope is represented by the variable “X”, whereas on the other hand the Zero Slope is represented by the variable “Y”. the Undefined Slope is a vertical line whereas the Zero Slope os a horizontal line.

**References**

- https://link.springer.com/content/pdf/10.1007/s11053-005-6951-3.pdf
- https://agupubs.onlinelibrary.wiley.com/doi/abs/10.1029/JB076i008p01905

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