Algebra II

Consider the numbers 1/2 and 9. Which of the following statements is completely true regarding the most specific classification of these two numbers?

1/2 is a rational number and 9 is a natural number. The most specific classification for the number 1/2 is a rational number as it is not an integer. The number 9 is an integer, but can be more specifically classified as a natural number.

Which of the numbers in the following set are rational numbers?

500, -15, √2, 1/4, 0.5, -2.50, π

500, -15, 1/4, 0.5, -2.50 Rational numbers include integers, natural numbers and whole numbers. √2 and π are not rational numbers. The rest are.

Which of these statements is FALSE regarding the number 0?

0 is the smallest natural number. 0 is a real, rational number. It is also the smallest whole number. It is not a natural number, though.

Which family of numbers begins with the numbers 0, 1, 2, 3, …?

Whole numbers Whole numbers include the counting numbers 1, 2, 3, and so on plus the number 0.

Which is NOT a rational number?

√2 The √2 is NOT a rational number. All the others are rational numbers. A rational number is a number that is made by dividing two integers, meaning it can be written as a fraction, such as 1/3, 0.25 (25/100), 0.666... (2/3), 1000 (1000/1).

What numbers are graphed on the number line?

63, 85, 96 The graph shows increments of 5. From left to right, it shows 50, 55, 60, 65, 70....and so on. Based on this information, we know the numbers graphed on the line are 63, 85, and 96.

Which number is graphed on the number line?

65 Remember that you can graph rational numbers on a number line by drawing a line with the appropriate scale and then placing a dot at the correct position on the number line. The number on the line is 65.

Graph the following daily high temperatures on a number line.

Monday - 78

Tuesday - 72

Wednesday - 65

Thursday - 80

Which number line would be best for creating a timeline depicting major events of the 1700s?

Because the 1700s is an entire century of events, the image showing a 100 year period starting at 1700 and going to 1800 is correct. The other models show too large of a span of time or incorrect labeling of the time period.

Convert the following fraction to a decimal: 9/10

0.9 Remember that the line in a fraction is the same as a division symbol. 9 over 10 is the same as 9 divided by 10. So, one method is to divide the numerator by the denominator. 9 divided by 10 is 0.9. And that's our decimal.

Convert the following fraction to a decimal: 2/3

0.667 Divide 2 by 3

2 /3 = 0.667

Convert the following fraction to a decimal: 35/70

0.5 35 over 70 is the same as 35 divided by 70.

35 / 70 = 0.5

Convert the following decimal to a fraction: 0.07

7/100 0.07 = 7/100

Convert the following decimal to a fraction: 0.25

¼ We have 25 hundredths and we can write it as 25/100.

So 25/100 can be simplified by dividing the numerator and denominator by 25, which gives us 1/4.

Which statement is INCORRECT about inequalities?

One of the inequality symbols is = An equal sign is not an inequality.

Which of the answer choices is an INCORRECT statement?

-2 < -4 Everything is correct except the -2 one because the -2 is greater than the -4.

Fill in the blank

-3 _____ -√11

> The square root of 11 is 3.316. So -3 is greater than, >, the negative square root of 11.

Fill in the blank

2 + √5 _____ 7 - √10

> 2 + √5 = 4.236 and 7 - √10 = 3.8377 so the first number is greater than, >, the second number.

Which set of numbers make this inequality true?

a < b

a = pi

b = 3.5 The only one where the a number is less than or smaller than the b number is 3.14, pi, for a and 3.5 for b.

Simplify |1 - 3|

2 Remember that absolute value is always positive, and whatever number or variable is inside the absolute value symbols, the result will be positive. We know that 1-3 usually equals -2, but because it is in between the absolute value symbols, we know that the correct answer is 2.

Which of the following is not a real number?

infinity The imaginary numbers are the square root of -1 and infinity.

Solve |17|

17 Keep in mind that the absolute value of a number is its distance from zero on a number line, so the correct answer is 17.

Solve -(-|-5|)

5 Since the absolute value of negative 5 is positive 5, we end up with negative 5 in the parentheses. Because there is a negative sign outside of the parentheses, and two negatives equal a positive, the correct answer is 5.

Which statement is true about the absolute value?

It is notated by |x| It is TRUE that absolute value is designated by a set of straight vertical lines - |x|.

What do you multiply by to rationalize this denominator?

sqrt(6)/sqrt(6) To rationalize the denominator, you multiply both numerator and denominator by the square root of 6.

Rationalize the denominator.

sqrt(5) Multiplying both numerator and denominator by the square root of 5 and simplifying gives us the square root of 5.

What do you multiply the numerator and denominator with to rationalize the denominator?

5 + sqrt(2) To rationalize this denominator, we multiply by the conjugate, 5 + sqrt(2) where the only difference is the sign between the two terms in the denominator.

Rationalize the denominator.

sqrt(14)/2 Multiply both the numerator and denominator by the square root of 14 and then simplify to get the square root of 14 divided by 2.

Rationalize the denominator.

(2 - sqrt(2))/2 Multiplying both the numerator and denominator by 2 - sqrt(2) gives us (2 - sqrt(2))/(4 - 2) which simplifies to (2 - sqrt(2))/2.

An algebraic number is what to a polynomial?

a solution An algebraic number is any number that is the solution to a polynomial with rational coefficients.

Algebraic numbers are _.

countable Algebraic numbers can be radicals, irrational numbers and even the imaginary number i. And even though they're infinite, they are countable and definable.

Which of these numbers is transcendental?

The Liouville Constant The first transcendental number that was proved was the Liouville constant.

Transcendental numbers are _ .

uncountable Transcendental numbers are infinite and uncountable because there are far more transcendentals than there are algebraics.

Which of these numbers is a transcendental number?

pi pi is a proven transcendental number.

Which of the following is a Pythagorean Triple?

(3, 4, 5)

What formula must all Pythagorean Triple obey?

a^2 + b^2 = c^2

Which of the following is a Pythagorean Triple?

(9, 40, 41) (9, 40, 41) follows the formula a^2 + b^2 = c^2.

9^2 + 40^2 = 41^2

81 + 1600 = 1681

Which of the following is a scaled up version of (5, 12, 13)?

(10, 24, 26) (10, 24, 26) are all the numbers in the original multiplied by 2. It is scaled by a factor of 2.

Which of the following is NOT a Pythagorean Triple?

(5, 10, 11)

Which of the following is an imaginary number?

2i 2i is an imaginary number. Infinity is not an imaginary number or really a number at all. It is a concept or idea of something without end.

What is the simplified value of the expression below?

√-110 equals -1

Which of the following expressions is written as a complex number?

-1 + i When you combine an imaginary number with a real one you get a complex number: -1 + i.

When solving a quadratic equation, under what circumstances would the result be two answers that are imaginary numbers?

When the discriminant is negative When the discriminant is negative, a quadratic equation will result in two answers that are imaginary.

In math, i equals _____.

In math, i equals √-1.

Subtract and simplify the following expression:

Remember (a + bi) - (c + di)=(a - c) + (b - d)i.

Therefore, (-1 + 4i) - (-2 + i) = 1 + 3i.

Add and simplify the following expression:

Remember a + bi and c + di gives us an answer of (a + c) + (b + d) i.

So (2 - i) + (-5 -3i) = -3 - 4i.

Multiply and simplify the following expression:

The FOIL method can be used to multiply. This will give:

8 - 4i + 12i - 6i^2

= 8 + 8i - (-6) (since i^2 = -1)

= 8 + 8i + 6

= 14 + 8i

Which of the following is a complex number with a non-zero real part and a non-zero imaginary part?

8 + 4i When we take an imaginary number and add a real number to it, we end up with a complex number, often denoted by a+bi, where a represents the real and b the imaginary portion of the number.

Simplify the following expression:

First simplify (1 + i)(1 - i) by using FOIL:

This gives (1 - i^2) but since i^2 = -1 this can be simplified

(1 - i^2) = (1 - (-1)) = 1 + 1 = 2

Now put this back into the original problem:

3 + 2i + 2 = 5 + 2i.