GED Math - Types of Numbers and Number Theory

Express $5432_{\textrm{six}}$ as a number in base ten.

Explanation

Each digit of a base six number has a place value that is a power of six. The lowest four powers of 6, beginning with 1, are $6^{0} = 1,6^{1} = 6, 6^{2} = 36, 6^{3} = 216$ , so

$5432_{\textrm{six}} = 5 \times 216 + 4 \times 36 + 3 \times 6 + 2$

To how many of the following sets does the number belong?

I) The set of whole numbers

II) The set of integers

III) The set of rational numbers

Two

Three

None

One

Explanation

The whole numbers comprise 0 and the positive integers. is a negative integer, so it belongs to the set of integers, but not the set of whole numbers. Also, all integers are rational by definition, since every integer can be expressed as the quotient of two integers (the integer divided by 1, for example). Therefore, is an integer and a rational number, but not a whole number, and the correct response is two.

How many elements are in a set that has exactly 64 subsets?

Explanation

A set with elements has $2 ^{N}$ subsets. Since $64 = 2 ^{6}$ , a set with 64 subsets has 6 elements.

The set of real numbers is divided into several subsets including positive numbers and negative numbers, prime numbers and composite number, rational numbers and irrational numbers, etc. For the following questions, select the answer that is a member of the stated subset.

Which number is both rational and real?

$\sqrt{9}$

$\sqrt{8}$

$\sqrt{-8}$

$\sqrt{-9}$

Explanation

A rational number is a number that can be expressed in the form $\frac{p}{q}$ where both and are integers.

$\sqrt{9}=\sqrt{3*3}=\sqrt{3^2}=3$ which is not only rational but an integer.

$\sqrt{8}=\sqrt{2*2*2}=\sqrt{2^3}=2\sqrt{2}$ . Any square root of a prime number, the number 2 here, is irrational.

Finally, any square root of a negative number is complex and not real.

How many subsets does the set $\left { 1, 2, 3, 4 ,a,b , c, d\right }$ have?

Explanation

The number of subsets of a set with elements is $2^{N}$ , so this eight-element set has $2^{8} = 256$ subsets.

; , , and are distinct integers.

Which of the following could be equal to ?

Explanation

We need to find ways to factor 32 such that the three factors are different, and then find the sum of those factors in each case.

32 can be factored as the product of three integers in five differrent ways:

I) $1 \times 1 \times 32$

II) $1 \times 2 \times 16$

III) $1 \times 4 \times 8$

IV) $2 \times 2 \times 8$

V) $2 \times 4 \times 4$

Of the five ways, only the second and third involve three distinct factors.

In Case II, the sum of the factors is

In Case III, the sum of the factors is

The only possible correct response is 19.

Which expression is complex? Specifically, which number cannot be written as a real number?

$\sqrt{19}$

$\frac{19}{91}$

$\sqrt{91 - 19}$

Explanation

Remember that a complex number is any number that includes or $\sqrt{-1}$ .

$\frac{19}{91} \approx 0.208791$

$\sqrt{19}$ is irrational but real.

Similarly, $\sqrt{91 - 19} = \sqrt{72} = \sqrt{9 * 8} = \sqrt{3^{2} * 2^{}3} = 3 * 2 *\sqrt{2} = 6\sqrt{2}$ which is irrational but real.

Which of the following sets does belong to?

(a) Whole numbers

(b) Integers

(b) only

(a) only

Both (a) and (b)

Neither (a) nor (b)

Explanation

The set of whole numbers comprises 0 and the so-called natural, or counting, numbers; that is, it is the set $\left { 0, 1, 2, 3, 4,...\right }$ . is not one of these numbers.

The set of integers comprises these numbers as well as their (negative) opposites; that is, it is the set $\left { ...,-4, -3, -2, -3, 0, 1, 2, 3, 4,...\right }$ . is one of these numbers.

To how many of the following sets does the number belong?

I) The set of whole numbers

II) The set of integers

III) The set of rational numbers

Two

Three

None

One

Explanation

Which number is prime?

Explanation

A prime number is a positive integer which has exactly two factors - 1 and the number itself. 36, 38, and 39 can each be shown to have at least one other factor:

$36 \div 2 = 18$ , so 2 and 18 are factors of 36, making 36 not prime.