# Complex numbers: absolute value (2023)

An important concept for numbers, either real or complex is that of 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 = x2 + y2.

(Note that for real numbers like x, we can drop absolute value when squaring, since |x|2=x2.) 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 x2+y2=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 a2+b2=c2 If you divide this equation by c2, then you find that(a/c)2+(b/c)2=1. That means that a/c+ib/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+i4/5 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 bars3x - 6 = 6
Step 4: Solve both equations3x - 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. x3+y3+z3=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 x3+y3+z3=k is known as the sum of cubes problem.

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