Irrational Numbers - Algebra II
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Which of the following is an irrational number?
Which of the following is an irrational number?
A rational number can be expressed as a fraction of integers, while an irrational number cannot.
can be written as 
.
is simply 
, which is a rational number.
The number 
can be rewritten as a fraction of whole numbers, 
, which makes it a rational number.
is also a rational number because it is a ratio of whole numbers.
The number, 
, on the other hand, is irrational, since it has an irregular sequence of numbers (
...) that cannot be written as a fraction.
A rational number can be expressed as a fraction of integers, while an irrational number cannot.
The number
The number,
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Which of the following is an irrational number?
Which of the following is an irrational number?
An irrational number is any number that cannot be written as a fraction of whole numbers. The number pi and square roots of non-perfect squares are examples of irrational numbers.
can be written as the fraction 
. The term 
is a whole number. The square root of 
is 
, also a rational number. 
, however, is not a perfect square, and its square root, therefore, is irrational.
An irrational number is any number that cannot be written as a fraction of whole numbers. The number pi and square roots of non-perfect squares are examples of irrational numbers.
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Of the following, which is a rational number?
Of the following, which is a rational number?
A rational number is any number that can be expressed as a fraction/ratio, with both the numerator and denominator being integers. The one limitation to this definition is that the denominator cannot be equal to 
.
Using the above definition, we see 
, 
and 
(which is 
) cannot be expressed as fractions. These are non-terminating numbers that are not repeating, meaning the decimal has no pattern and constantly changes. When a decimal is non-terminating and constantly changes, it cannot be expressed as a fraction.
is the correct answer because 
, which can be expressed as 
, fullfilling our above defintion of a rational number.
A rational number is any number that can be expressed as a fraction/ratio, with both the numerator and denominator being integers. The one limitation to this definition is that the denominator cannot be equal to
Using the above definition, we see
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Of the following, which is an irrational number?
Of the following, which is an irrational number?
The definition of an irrational number is a number which cannot be expressed in a simple fraction, or a number that is not rational.
Using the above definition, we see that 
is already expressed as a simple fraction.
any number 
and
. All of these options can be expressed as simple fractions, making them all rational numbers, and the incorrect answers.
cannot be expressed as a simple fraction and is equal to a non-terminating, non-repeating (ever-changing) decimal, begining with 
This is an irrational number and our correct answer.
The definition of an irrational number is a number which cannot be expressed in a simple fraction, or a number that is not rational.
Using the above definition, we see that
This is an irrational number and our correct answer.
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Which of the following numbers is an irrational number?
, 
Which of the following numbers is an irrational number?
An irrational number is one that cannot be written as a fraction. All integers are rational numberes.
Repeating decimals are never irrational, 
can be eliminated because
.
and 
are perfect squares making them both integers.
Therefore, the only remaining answer is 
.
An irrational number is one that cannot be written as a fraction. All integers are rational numberes.
Repeating decimals are never irrational,
Therefore, the only remaining answer is
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Which of the following is/are an irrational number(s)?
I. 
II. 
III. 
IV. 
Which of the following is/are an irrational number(s)?
I.
II.
III.
IV.
Irrational numbers are numbers that can't be expressed as a fracton. This elminates statement III automatically as it's a fraction.
Statement I's fraction is 
so this statement is false.
Statement IV. may not be easy to spot but if you let that decimal be 
and multiply that by 
you will get 
. This becomes 
. Subtract it from 
and you get an equation of 
.
becomes 
which is a fraction.
Statement II can't be expressed as a fraction which makes this the correct answer.
Irrational numbers are numbers that can't be expressed as a fracton. This elminates statement III automatically as it's a fraction.
Statement I's fraction is
Statement IV. may not be easy to spot but if you let that decimal be
Statement II can't be expressed as a fraction which makes this the correct answer.
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Is 
rational or irrational?
Is
Irrational numbers can't be expressed as a fraction with integer values in the numerator and denominator of the fraction.
Irrational numbers don't have repeating decimals.
Because of that, there is no definite value of irrational numbers.
Therefore, 
is irrational because it can't be expressed as a fraction.
Irrational numbers can't be expressed as a fraction with integer values in the numerator and denominator of the fraction.
Irrational numbers don't have repeating decimals.
Because of that, there is no definite value of irrational numbers.
Therefore,
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What do you get when you multiply two irrational numbers?
What do you get when you multiply two irrational numbers?
Let's take two irrationals like 
and multiply them. The answer is 
which is rational.
But what if we took the product of 
and 
. We would get 
which doesn't have a definite value and can't be expressed as a fraction.
This makes it irrational and therefore, the answer is sometimes irrational, sometimes rational.
Let's take two irrationals like
But what if we took the product of
This makes it irrational and therefore, the answer is sometimes irrational, sometimes rational.
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Which of the following is not irrational?
Which of the following is not irrational?
Some answers can be solved. Let's look at some obvious irrational numbers.
is surely irrational as we can't get an exact value.
The same goes for 
and 
.
is not a perfect cube so that answer choice is wrong.
Although 
is a square root, the sum inside however, makes it a perfect square so that means 
is rational.
Some answers can be solved. Let's look at some obvious irrational numbers.
The same goes for
Although
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Which concept of mathematics will always generate irrational answers?
Which concept of mathematics will always generate irrational answers?
Let's look at all the answer choices.
The area of a triangle is base times height divided by two. Since base and height can be any value, this statement is wrong. We can have irrational values or rational values, thus generating both irrational or rational answers.
The diagonal of a right triangle will generate sometimes rational answers or irrational values. If you have a perfect Pythagorean Triple 
or 
etc..., then the diagonal is a rational number. A Pythagorean Triple is having all the lengths of a right triangle being rational values. One way the right triangle creates an irrational value is when it's an isosceles right triangle. If both the legs of the triangle are 
, the hypotenuse is 
, 
, 
, 
can't be negative since lengths of triangle aren't negative.
The same idea goes for volume of cube and area of square. It will generate both irrational and rational values.
The only answer is finding value of 
. 
is irrational and raised to any power except 0 is always irrational.
Let's look at all the answer choices.
The area of a triangle is base times height divided by two. Since base and height can be any value, this statement is wrong. We can have irrational values or rational values, thus generating both irrational or rational answers.
The diagonal of a right triangle will generate sometimes rational answers or irrational values. If you have a perfect Pythagorean Triple
The same idea goes for volume of cube and area of square. It will generate both irrational and rational values.
The only answer is finding value of
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Which of the following numbers are irrational?
Which of the following numbers are irrational?
The definition of irrational numbers is that they are real numbers that cannot be expressed in a common ratio or fraction.
The term 
is imaginary which equals to 
.
The other answers can either be simplified or can be written in fractions.
The only possible answer shown is 
.
The definition of irrational numbers is that they are real numbers that cannot be expressed in a common ratio or fraction.
The term
The other answers can either be simplified or can be written in fractions.
The only possible answer shown is
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Which irrational number is between 9 and 10?
Which irrational number is between 9 and 10?
Since all the possible answers are square roots, we can square both the limits of our problem and all the possible answers. This allows us to see which number is correct. After squaring everything, we notice that we need a number between 81 and 100. The only possible answer given is 90. Now we can square root everything back, giving us our final answer of 
.
Since all the possible answers are square roots, we can square both the limits of our problem and all the possible answers. This allows us to see which number is correct. After squaring everything, we notice that we need a number between 81 and 100. The only possible answer given is 90. Now we can square root everything back, giving us our final answer of
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Rationalize: 
Rationalize:
In order to rationalize, we will need to multiply the top and bottom by the denominator in order to eliminate the square root in the denominator.
Distribute and simplify the numerator. Multiplying unlike numbers inside square roots will not eliminate the square root!
The answer is: 
In order to rationalize, we will need to multiply the top and bottom by the denominator in order to eliminate the square root in the denominator.
Distribute and simplify the numerator. Multiplying unlike numbers inside square roots will not eliminate the square root!
The answer is:
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What is the sum of 
and 
?
What is the sum of
Distribute the negative
Combine like terms
Distribute the negative
Combine like terms
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Write the following expression in the standard form for a complex number
Write the following expression in the standard form for a complex number
Multiply Binomials ( you may use the FOIL method)
We know that 
, so we replace 
with 
combine like terms
Distribute the i
We know that 
, so we replace 
with 
Swich to standard form
Multiply Binomials ( you may use the FOIL method)
We know that
combine like terms
Distribute the i
We know that
Swich to standard form
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Which of the following is equivalent to
Which of the following is equivalent to
Factor the number -96
-1 2 2 2 2 2 3
The -1 come out the radical as an i. We look for pairs in the factors.
-1 ( 2 2 )( 2 2 ) 2 3
We see two pairs of 2, both can come out from under the radical .
Multiply
Factor the number -96
-1 2 2 2 2 2 3
The -1 come out the radical as an i. We look for pairs in the factors.
-1 ( 2 2 )( 2 2 ) 2 3
We see two pairs of 2, both can come out from under the radical .
Multiply
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Find the differance
Find the differance
simplify the radicals
Distribute the negative
Combine like terms
simplify the radicals
Distribute the negative
Combine like terms
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What is the complex conjugate of 
in standard form?
What is the complex conjugate of
To find a complex conjugate we change the sign of the imaginary part of the number.
To be in standard form the real number should come before the imaginary part of the number
To find a complex conjugate we change the sign of the imaginary part of the number.
To be in standard form the real number should come before the imaginary part of the number
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Simplify
Simplify
To simplify we need to remove the complex number from the denominator. To do this our first step is to multiply the expression by the complex conjugate of the denominator.
Multiply the binomials in the numorator and the denominator. You may use the FOIL method.
We know that 
so we replace 
with 
Combine like terms
To simplify we need to remove the complex number from the denominator. To do this our first step is to multiply the expression by the complex conjugate of the denominator.
Multiply the binomials in the numorator and the denominator. You may use the FOIL method.
We know that
Combine like terms
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Which of the following is equal to
Which of the following is equal to
Simplify the radicals
We notice that we have a complex number in the denominator. To get rid of this we multiply the numerator and denominator by the complex conjugate of the denominator.
Distribute across the numerator and multiply the binomials in the denominator. You may use the FOIL method.
We know that 
so we replace 
with -1
Combine like terms
Reduce and put in standard form
or 
Simplify the radicals
We notice that we have a complex number in the denominator. To get rid of this we multiply the numerator and denominator by the complex conjugate of the denominator.
Distribute across the numerator and multiply the binomials in the denominator. You may use the FOIL method.
We know that
Combine like terms
Reduce and put in standard form
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