*absolute value.*Recall that the absolute value |

*x*| of a real number

*x*is itself, if it's positive or zero, but if

*x*is negative, then its absolute value |

*x*| is its negation

*x,*that is, the corresponding positive value. For example, |3|=3, but |4|=4. The absolute value function strips a real number of its sign.

For a complex number *z*=*x*+*yi,* we define the absolute value |*z*| as being the distance from *z* to 0 in the complex plane **C**. This will extend the definition of absolute value for real numbers, since the absolute value |*x*| of a real number *x* can be interpreted as the distance from *x* to 0 on the real number line.We can find the distance |*z*| by using the Pythagorean theorem. Consider the right triangle with one vertex at 0, another at *z* and the third at *x* on the real axis directly below *z* (or above *z* if *z* happens to be below the real axis). The horizontal side of the triangle has length |*x*|, the vertical side has length |*y*|, and the diagonal side has length |*z*|. Therefore,

*z*|

^{2}=

*x*

^{2}+

*y*

^{2}.

(Note that for real numbers like *x,* we can drop absolute value when squaring, since |*x*|^{2}=*x*^{2}.) That gives us a formula for |*z*|, namely,

### The unit circle.

Some complex numbers have absolute value 1. Of course, 1 is the absolute value of both 1 and 1, but it's also the absolute value of both*i*and

*i*since they're both one unit away from 0 on the imaginary axis. The

*unit circle*is the circle of radius 1 centered at 0. It include all complex numbers of absolute value 1, so it has the equation |

*z*|=1.

A complex number *z*=*x*+*yi* will lie on the unit circle when *x*^{2}+*y*^{2}=1. Some examples, besides 1, 1, *i,* and *1* are ±√2/2±*i*√2/2, where the pluses and minuses can be taken in any order. They are the four points at the intersections of the diagonal lines *y*=*x* and *y*=*x* with the unit circle. We'll see them later as square roots of *i* and *i.*

You can find other complex numbers on the unit circle from Pythagorean triples. A *Pythagorean triple* consists of three whole numbers *a, b,* and *c* such that *a*^{2}+*b*^{2}=*c*^{2} If you divide this equation by *c*^{2}, then you find that(*a/c*)^{2}+(*b/c*)^{2}=1. That means that *a/c*+*i**b/c* is a complex number that lies on the unit circle. The best known Pythagorean triple is 3:4:5. That triple gives us the complex number 3/5+*i*4/5 on the unit circle. Some other Pythagorean triples are 5:12:13, 15:8:17, 7:24:25, 21:20:29, 9:40:41, 35:12:27, and 11:60:61. As you might expect, there are infinitely many of them. (For alittle more on Pythagorean triples, see the end of the page at http://www.clarku.edu/~djoyce/trig/right.html.)

### The triangle inequality.

There's an important property of complex numbers relating addition to absolute value called the triangle inequality. If *z* and *w* are any two complex numbers, then

You can see this from the parallelogram rule for addition. Consider the triangle whose vertices are 0, *z,* and *z*+*w.*One side of the triangle, the one from 0 to *z*+*w* has length |*z*+*w*|. A second side of the triangle, the one from 0 to *z,* has length |*z*|. And the third side of the triangle, the one from *z* to *z*+*w,* is parallel and equal to the line from 0 to *w,* and therefore has length |*w*|. Now, in any triangle, any one side is less than or equal to the sum of the other two sides, and, therefore, we have the triangle inequality displayed above.

## FAQs

### Can complex numbers have an absolute value? ›

**Some complex numbers have absolute value 1**. Of course, 1 is the absolute value of both 1 and –1, but it's also the absolute value of both i and –i since they're both one unit away from 0 on the imaginary axis.

**Is the absolute value of a complex number always real? ›**

The absolute square of a complex number is calculated by multiplying it by its complex conjugate. This give the magnitude squared of the complex number. **The absolute square is always real**.

**What is the absolute value of complex value? ›**

The absolute value of a complex number, say x + iy is **the distance from the origin**. The same as the absolute value of a normal number on the number line. x + iy Is simply graphed as the point x on the x axis and y on the y axis.

**What is the easiest way to find absolute value? ›**

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.

**When can you not solve absolute value? ›**

An absolute value equation has no solution **if the absolute value expression equals a negative number** since an absolute value can never be negative.

**What is the rule for absolute value? ›**

**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 can absolute value never be? ›**

Absolute value describes the distance from zero that a number is on the number line, without considering direction. The absolute value of a number is **never negative**.

**How do you know if a complex number is real or imaginary? ›**

In a complex number z=a+bi , a is called the "real part" of z and b is called the "imaginary part." **If b=0 , the complex number is a real number; if a=0 , then the complex number is "purely imaginary."**

**Is every numbers absolute value always the number itself? ›**

The term “absolute value” refers to the magnitude of a quantity without regard to its sign. In other words, absolute value is the distance of a number from zero on a number line. **The absolute value of a positive number is the number itself**, and the absolute value of a negative number is its opposite.

**What makes a complex number real? ›**

All real numbers are complex numbers (**by having b = 0**), but there are complex numbers that are not real, e.g. 2 - 3i. We add or subtract two complex numbers by adding and substracting the corre- sponding real and complex part of the number.

### Is absolute value of a complex number differentiable? ›

Absolute value function

**It is differentiable everywhere except for x = 0**. It is monotonically decreasing on the interval (−∞, 0] and monotonically increasing on the interval [0, +∞).

**What is the absolute value of the complex number 4 -/ 2? ›**

The absolute value of the complex number -4 - √2i is **√18**.

**What are the 4 properties of absolute value? ›**

Absolute value has the following fundamental properties: **Non-negativity |a| ≥ 0**. **Positive-definiteness |a| = 0a = 0**. **Multiplicativity |ab| = |a| |b|**

**Can there ever be one solution to an absolute value equation? ›**

**When the right hand side of the equation is zero, then the absolute value equation has only one solution**. Therefore, the absolute value equation has only one solution.

**How many solutions can you get when you solve absolute value equations? ›**

An absolute value equation may have **one solution, two solutions, or no solutions**.

**What are the 4 steps to solving an absolute value equation? ›**

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 |

**What is the absolute value of 65? ›**

The absolute value of 65 **is65**.

**What makes an absolute value equation false? ›**

There's one MAJOR red flag of an equation with an absolute value that has no solution. Recall that **an absolute value expression can never be less than zero**. That is, a fully reduced absolute value expression must be greater than or equal to zero.

**Do all absolute value equations have at least two solutions? ›**

In conclusion, **an absolute-value problem will not always have two solutions**, because absolute-value inequalities result in one set of solutions. If the problem contains only one absolute number, it will have only one solution and that will be the positive number.

**What is the absolute value of 11? ›**

The absolute value of any positive number is **the number itself**, so 11 has 11 as an absolute value.

### 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.

**Can absolute value have a negative answer? ›**

Since the absolute value of any number other than zero is positive, **it is not permissible to set an absolute value expression equal to a negative number**.

**Can an absolute value be infinity? ›**

**The Absolute Infinite (symbol: Ω) is an extension of the idea of infinity proposed by mathematician Georg Cantor**. It can be thought of as a number that is bigger than any other conceivable or inconceivable quantity, either finite or transfinite.

**What are true facts 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 if a complex number is purely real? ›**

A complex number is purely real **iff the imaginary part is 0**.

**What are non real complex numbers? ›**

**A pure imaginary number or a number, like 7 + 2i with a ≠ 0 and b ≠ 0**, is a nonreal complex number. The form a + bi (or a + ib) is called standard. form.

**What is an example of real complex number? ›**

Complex Numbers in Maths. Complex numbers are the numbers that are expressed in the form of a+ib where, a,b are real numbers and 'i' is an imaginary number called “iota”. The value of i = (√-1). For example, **2+3i** is a complex number, where 2 is a real number (Re) and 3i is an imaginary number (Im).

**Does 0 have an absolute value? ›**

Since 0 is zero units away from itself, the absolute value of 0 is just 0. The absolute value of 0 is written as |0| and is equal to 0.

**What is the absolute value of 24? ›**

The number line also shows that |24| is the distance between 24 and 0 is 4. Thus, **|24| 5 4**. The absolute value of a positive number or zero is the number itself. The absolute value of a negative number is its opposite.

**Can two numbers not have the same absolute value? ›**

Answer: **Two different integers can have the same absolute value**.

### Do complex numbers exist in real life? ›

Complex numbers (the sum of real and imaginary numbers) **occur quite naturally in the study of quantum physics**. They're useful for modelling periodic motions (such as water or light waves) as well as alternating currents.

**Can a complex number have no real part? ›**

**Either Part Can Be Zero**

So, a Complex Number has a real part and an imaginary part. But either part can be 0, so all Real Numbers and Imaginary Numbers are also Complex Numbers.

**What is the condition for a complex number to become a real number? ›**

So, in general, for the product of two complex numbers to be real, **the ratio of the real to imaginary parts of each complex number must be equal up to a minus sign**. Or an imaginary number?

**What is the chain rule for complex numbers? ›**

The chain rule states that **the derivative of f(g(x)) is f'(g(x))⋅g'(x)**. In other words, it helps us differentiate *composite functions*. For example, sin(x²) is a composite function because it can be constructed as f(g(x)) for f(x)=sin(x) and g(x)=x².

**Why is absolute value always positive? ›**

Why is the Absolute Value Always Positive? Absolute value means the distance of the number from the origin 0. The number represented on a number line can be negative but the absolute value is always positive since the distance is never negative.

**Can an absolute value have a derivative? ›**

**The derivative of an absolute value function and of any function, for that matter, is the slope of the tangent line to the curve at a given point**. Because such functions are piecewise ones, one can differentiate each piece separately.

**What is the absolute value of the complex number 3 − 4i? ›**

The absolute value of z = 3 - 4i, denoted |3 - 4i|, is **5**.

**What is the opposite of 4 absolute value? ›**

The opposite of 4 is **-4** (the sign changes from + to -). The opposite of -8 is 8 (the sign changes from - to +).

**What is the absolute value of 2 and of − 2? ›**

The absolute value is just the distance from the origin

we can say that 2 is a distance of two units away from 0, but also that −2, the opposite of 2 (and on the other side of 0), is two units away from 0. So both 2 and −2 are **two units away from 0**.

**Why do 4 and 4 have the same absolute value? ›**

The absolute value of a number is its distance from 0. 4 **4 4 and −4 are the same distance from 0**, so they have the same absolute value of 4start color #11accd, 4, end color #11accd.

### What is the absolute value of a 3? ›

The absolute value of 3 is **3**. The absolute value of 0 is 0.

**What are all the absolute values of 6? ›**

Since 6 is six units away towards right from 0, the absolute value of 6 is just 6. The absolute value of 6 is written as |6| and is equal to 6.

**What is the absolute value of the complex number − 4 − i 2 √? ›**

The absolute value of the complex number -4 - √2i is **√18**.

**What is the formula for solving complex numbers? ›**

The complex number is in the form of **a+ib**, where a = real number and ib = imaginary number. Also, a,b belongs to real numbers and i = √-1. Hence, a complex number is a simple representation of addition of two numbers, i.e., real number and an imaginary number.

**What is the absolute value of − 4? ›**

Therefore, the absolute value of **-4** is 4.

**What is the absolute value of 4 and 4? ›**

The absolute value of a number is its distance from zero on a number line . For instance, 4 and −4 have the same absolute value **( 4 )**.

**What is the complex conjugate of the complex number − 3 − 2i? ›**

To find a complex conjugate, simply change the sign of the imaginary part (the part with the i ). This means that it either goes from positive to negative or from negative to positive. As a general rule, the complex conjugate of a+bi is a−bi . Therefore, the complex conjugate of 3−2i is **3+2i** .

**What is the absolute value of the given complex number − 2 i? ›**

Explanation: Absolute value of a complex number a+ib is written as |a+ib| and its value as √a2+b2 . Hence, absolute value of −2−i is √(−2)2+(−1)2=√4+1=**√5** .

**What is the absolute value of − 3? ›**

For example, the absolute value of 3 is **3**, and the absolute value of −3 is also 3. The absolute value of a number may be thought of as its distance from zero.

**Does absolute value have 2 answers? ›**

**Absolute values will have two solutions when they are equations, functions, in the inequalities that will give a set of results**. For a specific number with an absolute value, it will only have one result which will always be positive.

### How many solutions can an absolute value equation have? ›

An absolute value equation may have **one solution, two solutions, or no solutions**.

**How many answers do you usually have when solving an absolute value equation? ›**

Summary. Absolute value equations are always solved with the same steps: isolate the absolute value term and then write equations based on the definition of the absolute value. There may end up being **two solutions, one solution, or no solutions**.

**What's the hardest math equation? ›**

For decades, a math puzzle has stumped the smartest mathematicians in the world. **x ^{3}+y^{3}+z^{3}=k**, with k being all the numbers from one to 100, is a Diophantine equation that's sometimes known as "summing of three cubes."

**What's the answer to x3 y3 z3 K? ›**

In mathematics, entirely by coincidence, there exists a polynomial equation for which the answer, **42**, had similarly eluded mathematicians for decades. The equation x^{3}+y^{3}+z^{3}=k is known as the sum of cubes problem.