Definitions:
The absolute value (or modulus) | x | of a real number x is the non-negative value of x without regard to its sign.
For example, the absolute value of 5 is 5, and the absolute value of −5 is also 5. The absolute value of a number may be thought of as its distance from zero along real number line. Furthermore, the absolute value of the difference of two real numbers is the distance between them.
The absolute value has the following four fundamental properties:
Non-negativity $$|a| ≥ 0$$ Positive-definiteness $$|a| = 0 ⇔ a = 0$$ Multiplicativeness $$|ab| = |a||b|$$ Subadditivity $$|a + b| ≤ |a| + |b|$$
Other important properties of the absolute value include:
Idempotence (the absolute value of the absolute value is the absolute value) $$||a|| = |a|$$ Symmetry $$|-a| = |a|$$ Identity of indiscernibles (equivalent to positive-definiteness) $$|a - b| = 0 ⇔ a = b$$ Triangle inequality (equivalent to subadditivity) $$|a - b| ≤ |a - c| + |c - b|$$ Preservation of division (equivalent to multiplicativeness) $$|a / b| = |a| / |b| \space\space if \space\space b ≠ 0$$ (equivalent to subadditivity) $$|a - b| ≥ ||a| - |b||$$
Two other useful properties concerning inequalities are: $$|a| ≤ b ⇔ -b ≤ a ≤ b$$ $$|a| ≥ b ⇔ a ≤ -b \space or \space b ≤ a$$
These relations may be used to solve inequalities involving absolute values. For example: $$|x - 3| ≤ 9 ⇔ -9 < x - 3 < 9$$ $$⇔ -6 < x < 12$$
FAQs
What is absolute value in math library? ›
The absolute value (or modulus) | x | of a real number x is the non-negative value of x without regard to its sign. For example, the absolute value of 5 is 5, and the absolute value of −5 is also 5. The absolute value of a number may be thought of as its distance from zero along real number line.
What grade do you learn absolute value? ›If you're teaching math to students who are ready to learn about absolute value, typically around Grade 6, here's an overview of the topic, along with two lessons to introduce and develop the concept with your students.
How to do math with absolute values? ›To solve an equation containing absolute value, isolate the absolute value on one side of the equation. Then set its contents equal to both the positive and negative value of the number on the other side of the equation and solve both equations.
What age do you learn absolute value? ›Absolute Value - Chapter Summary
The video lessons in this Absolute Value chapter are designed to help your 6th, 7th or 8th graders understand how to work with and graph absolute value equations.
| Step 1: Isolate the absolute value | |3x - 6| - 9 = -3 |3x - 6| = 6 |
|---|---|
| Step 2: Is the number on the other side of the equation negative? | No, it's a positive number, 6, so continue on to step 3 |
| Step 3: Write two equations without absolute value bars | 3x - 6 = 6 |
| Step 4: Solve both equations | 3x - 6 = 6 3x = 12 x = 4 |
The absolute value of the number is defined as its distance from the origin. For example, to find the absolute value of 7, locate 7 on the real line and then find its distance from the origin. To find the absolute value of −7, locate −7 on the real line and then find its distance from the origin.
What is a in absolute value function? ›Generally, we can represent the absolute value function as, f(x) = a |x - h| + k, where a represents how far the graph stretches vertically, h represents the horizontal shift and k represents the vertical shift from the graph of f(x) = |x|.
What is the formula for graphing absolute value? ›We can graph any absolute value equation of the form y=k|x-a|+h by thinking about function transformations (horizontal shifts, vertical shifts, reflections, and scalings).
What is the absolute value of a number grade 8? ›What is the Absolute Value of a Number? The absolute value of a number or integer is the actual distance of the integer from zero, in a number line. Therefore, the absolute value is always a positive value and not a negative number.
What is the absolute value of 6 and why? ›The absolute value of a number is its distance from 0 on a number line. Let's use the number 6 as an example. Using a number line, we can see that 6 is six units away from 0. Therefore, the absolute value of 6 is 6.
Can absolute value be a decimal? ›
Remarks. The absolute value of a Decimal is its numeric value without its sign. For example, the absolute value of both 1.2 and -1.2 is 1.2.
Why do we use absolute value? ›We often use absolute values when talking about distance because there is no such thing as negative distance. For example, just because the store you're going to is a block behind you doesn't mean it's negative one blocks away.
What is absolute value middle school? ›Remember, absolute value is the distance from zero on the number line. This tutorial shows you how to use a number line to find absolute value.
What is an absolute value function in math example? ›An absolute value equation is an equation in which the unknown variable appears in absolute value bars. For example, |x|=4,|2x−1|=3,|5x+2|−4=9.
What is a fact about absolute value? ›absolute value, Measure of the magnitude of a real number, complex number, or vector. Geometrically, the absolute value represents (absolute) displacement from the origin (or zero) and is therefore always nonnegative. If a real number a is positive or zero, its absolute value is itself.
What is the absolute value of 4? ›The absolute value of 4 is 4 and –3 is 3. Subtract the smaller number from the larger and you get 4 – 3 = 1. The larger absolute value in the equation was 4 or a positive number so you give the result a positive result.
What words describe absolute value? ›In math, absolute value is also called a modulus. Picture a line of numbers with zero in the middle, positive values extending to the right and negative numbers to the left.
What is the absolute value of 10? ›Answer and Explanation: The absolute value of 10 is 10.
What is absolute value lesson? ›The absolute value of a number is its distance from 0 on the number line. The absolute value of -7 is 7, because it is 7 units away from 0. The absolute value of 5 is 5, because it is 5 units away from 0.
Is absolute value always positive yes or no? ›The absolute value of a number is always positive or zero. If the original number is negative, its absolute value is that number without the negative sign.
What are the 4 main things in math? ›
Addition, subtraction, multiplication and division
Knowing these basics is extremely helpful, especially when dealing with money.
A geophysicist uses absolute value to look at the total amount of energy used. In an energy wave, there are both negative and positive directions of movement. Another example is when scuba divers discuss their location in regards to sea level. “50 feet below sea level” doesn't have to be represented as -50 feet.
Can absolute value equal zero? ›Hence, the absolute value is always positive (or zero if you are taking the absolute value of 0).
What is the absolute value in algebra 2? ›Every real number is either positive or negative. The two numerals +7 and -7 have different signs but the numerical part of each is 7. This part of the numeral designates the absolute value of the number. The absolute value of a is denoted by | a | (a vertical bar on each side of the quantity).
Why do absolute values have two solutions? ›Absolute values must be solved for values greater than or equal to zero, and less than zero. Two equations must be performed, one for positive values and the other for negative values. Absolute values will have two solutions when they are equations, functions, in the inequalities that will give a set of results.
What are the special cases of absolute value equations? ›When solving equations that involve absolute values, there are two cases to consider. Case 1: The expression inside the absolute value bars is positive. Case 2: The expression inside the absolute value bars is negative. For example, consider the expression | 4x+2 |=8 .
What is the absolute value of 65? ›The absolute value of 65 is65.
What is the symbol for absolute value in calculator? ›Absolute value. ABS(x) returns the absolute value of the number x.
What is the absolute value of negative 80? ›The absolute value of −80 (usually written |−80| ) is 80.
What is the absolute value of |- 9? ›The absolute value of −9 is 9. The absolute value of 3 is 3. The absolute value of 0 is 0. The absolute value of −156 is 156.
Can you expand an absolute value? ›
Absolute value can be expressed as a function: f(x) = |x| = x if x is positive, = -x if x is negative. The domain includes all real numbers (absolute value can be expanded to more numbers once we introduce them), and the range is all non-negative numbers (all positive numbers and 0).
What is the absolute value of 0? ›The absolute value of 0 is 0. (0 is 0 places from 0.)
Which Math library function returns the absolute value? ›The C library function int abs(int x) returns the absolute value of int x.
What is absolute value in Python? ›Python Absolute Value – abs() for real and complex numbers
The absolute value of a number refers to that value's magnitude, regardless of whether it is positive or negative. For example, the absolute value of -5 is 5.
The abs() function in C++ returns the absolute value of an integer number.
What is an absolute value function in Math example? ›An absolute value equation is an equation in which the unknown variable appears in absolute value bars. For example, |x|=4,|2x−1|=3,|5x+2|−4=9.
What is the rule for absolute value function? ›An absolute value function is an important function in algebra that consists of the variable in the absolute value bars. The general form of the absolute value function is f(x) = a |x - h| + k and the most commonly used form of this function is f(x) = |x|, where a = 1 and h = k = 0.
What function is used for absolute value? ›Returns the absolute value of a number. The absolute value of a number is the number without its sign.
What are the types of absolute value function? ›- absolute value and a static value.
- absolute value and an expression involving unknown pronumerals.
- two absolute values in both sides.
- two absolute values and a value.
The absolute value of −9 is 9.
What is the absolute value of and 12? ›
Picture a line of numbers with zero in the middle, positive values extending to the right and negative numbers to the left. It's easy to understand that the absolute value of 12 and -12 are identical: they're both 12, since each number is 12 units away from zero.
What is absolute value with variables? ›Absolute value changes the value of the expression within them. The absolute value of any expression is either positive or zero. When we are given the absolute value of an expression containing a variable, we don't know whether the variable is positive or negative.
What library is absolute value in python? ›Python abs() is a built-in function available with the standard library of python. It returns the absolute value for the given number. Absolute value of a number is the value without considering its sign. The number can be integer, floating point number or complex number.
What key is absolute function? ›There is a shortcut for placing absolute cell references in your formulas! When you are typing your formula, after you type a cell reference - press the F4 key. Excel automatically makes the cell reference absolute! By continuing to press F4, Excel will cycle through all of the absolute reference possibilities.
What's the absolute value of 4? ›The absolute value of 4 is 4 and –3 is 3. Subtract the smaller number from the larger and you get 4 – 3 = 1. The larger absolute value in the equation was 4 or a positive number so you give the result a positive result.
Why is absolute value always positive? ›Absolute Value Symbol
The distance of any number from the origin on the number line is the absolute value of that number. It also shows the polarity of the number whether it is positive or negative. It can be negative ever a s it shows the distance and the distance can't be negative. So, it is always positive.