Number Theory - Algebra II
Card 0 of 376
Which of these numbers is prime?
Which of these numbers is prime?
For a number to be prime it must only have factors of one and itself.
10 has factors 1, 2, 5, 10.
15 has factors 1, 3, 5, 15.
18 has factors 1, 2, 3, 6, 9, 18.
The only factors of 13 are 1 and 13. As such it is prime.
For a number to be prime it must only have factors of one and itself.
10 has factors 1, 2, 5, 10.
15 has factors 1, 3, 5, 15.
18 has factors 1, 2, 3, 6, 9, 18.
The only factors of 13 are 1 and 13. As such it is prime.
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True or false: The set 
comprises only whole numbers.
True or false: The set
The whole numbers are defined to be 0 and the so-called counting numbers, or natural numbers 1, 2, 3, and so forth. Negative integers 
and 
are not included in this set.
The whole numbers are defined to be 0 and the so-called counting numbers, or natural numbers 1, 2, 3, and so forth. Negative integers
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True or false:
The following set comprises only imaginary numbers:
True or false:
The following set comprises only imaginary numbers:
To raise 
to the power of any positive integer, divide the integer by 4 and note the remainder. The correct power is given according to the table below.
Every element in the set is equal to 
raised to an even-numbered power, so when each exponent is divided by 4, the remainder will be either 0 or 2. Therefore, each element is equal to either 1 or 
. Consequently, the set includes no imaginary numbers.
To raise
Every element in the set is equal to
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Which set does NOT contain an irrational number?
Which set does NOT contain an irrational number?
Irrational numbers are nonrepeating decimals-- they cannot be written as fractions.
has only real numbers because the square root of 4 is 2, a rational number.
Irrational numbers are nonrepeating decimals-- they cannot be written as fractions.
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Try without a calculator.
True or false: the set
comprises only irrational numbers.
Try without a calculator.
True or false: the set
comprises only irrational numbers.
is an irrational number, as is any integer multiple of 
. All of the elements are integer multiples of 
, so all of them are irrational.
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Find the intersection of the two sets:
Find the intersection of the two sets:
To find the intersection of the two sets, 
, we must find the elements that are shared by both sets:
To find the intersection of the two sets,
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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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Using the properties of negative numbers, determine which answer is equivalent to
Using the properties of negative numbers, determine which answer is equivalent to
Recall the following property of negative numbers:
It follows that:
Recall the following property of negative numbers:
It follows that:
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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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Using the properties of negative numbers, determine which answer is equivalent to
Using the properties of negative numbers, determine which answer is equivalent to
Recall the following property of negative numbers:
It follows that:
Recall the following property of negative numbers:
It follows that:
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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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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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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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