Standardized tests like the ACT and SAT will directly and indirectly test your knowledge of number types. For example, a word problem may include rules where its values may only be integers or whole numbers.
Real numbers are numbers that exist. They are numbers that can be plotted on a number line. Real numbers can be broken down into two main categories - rational and irrational numbers. Rational numbers are numbers that can be written as a fraction, while irrational numbers cannot. Rational numbers can be broken down into increasingly specific categories. Some rational numbers are also integers, some integers are also whole numbers, and some whole numbers are also natural numbers. However, not all whole numbers are natural numbers, not all integers are whole numbers, and not all rational numbers are integers.
Imaginary numbers are numbers that don’t exist on a normal number line. They are created by taking an even root of a negative number. The square root of -1 ( -1 ) is known as i.
Complex numbers contain both a real and an imaginary component, and is written in the form of a + bi. Since either component of a complex number could equal 0, all real and imaginary numbers can also be complex numbers.
Often we think of taking a square root as the inverse or opposite of that number being squared. While this may be true when taking the square root of a positive number, it is not true of taking the square root of a negative number. Since multiplying two negative numbers always results in a positive product, squaring a negative number results in a positive number. Thus, if square roots are meant to undo the product resulting from a number raised to the second power, but it is impossible for a squared number to have a negative product, the square root of a negative number does not exist. After all, no number raised to the second power will have a negative product, so there is no real number solution to taking the square root of a negative number.
Remember, this is not only true for square roots, but also for all even roots. However, it is not necessarily true for odd roots. This is because odd roots, like imply finding a number that has been raised to an odd-numbered power, as in x3, x5, x7, etc. Since odd powers mean that that number is being multiplied by itself an odd amount of times, it means the end product can be negative. For instance, (-2)3 = - 2 × -2 × -2 = -8. The fact that there is an odd amount of negative numbers being multiplied means that the product will be negative. Therefore, if you are taking an odd root of a negative radicand, do not immediately think that the simplified version will be an imaginary number.
Complex numbers look like binomials (a + bi), so you can use properties such as the distributive property and FOIL.
(a + bi) × 3 = 3a + 3bi
Prime numbers are natural numbers that can only be divided by 1 and itself. Composite numbers have more than two factors, and are the opposite of prime numbers. The number “1” is the only number that is neither a prime nor a composite number, since it can only be divided by itself, which is 1. The number 2 is the only even prime number, since every other even number can be divided by 2 as well as itself.
Here is a chart showing the prime numbers that are less than 100:
The factors of a number are the values that the number can be evenly divided by. For example, The factors of 10 are 1, 2, 5, and 10. The factors of a number include 1 and itself, and may include other numbers as well if it is a composite number. The greatest common factor (GCF) of a set of numbers is the largest factor that each of the numbers share. We can find this by listing the factors of each number in a set.
Finding the Greatest Common Factor
We can find the greatest common factor by listing out the factors of each number.
For example, find the greatest common factor of 15 and 35:
What if the numbers are much larger, and contain many factors? It would be easy to forget a factor, and writing out each factor would be time consuming…
Prime Factor Tree
We can create a factor tree for each number, and multiply each prime factor they have in common to find the GCF.
For example, find the greatest common factor of 144 and 160:
We can then circle the prime factors the two numbers have in common.
144 and 160 share four 2s as prime factors. To find the GCF, we multiply the common prime factors together: 2 × 2 × 2 × 2 =1 6.
The greatest common factor of 144 and 160 is 16.
A multiple is a number that can be divided by another without a remainder. When we think of multiples, we think of our multiplication tables. Multiples of 5 include 5, 10, 15, 20, etc. Multiples are like the opposite of factors - multiples of a number include itself and numbers greater, while factors include itself and numbers that are less than itself.
The least common multiple (LCM) of a set of numbers, is the smallest multiple that a set of numbers have in common. It can be found in the way we found the GCF, but instead of listing out the factors, we can list out the multiples.
Finding the Least Common Multiple (LCM)
We can find the least common multiple by listing out the multiples of each number.
For example, find the least common multiple of 4 and 5:
In this case, the product of the two numbers, 20, is the first multiple they both have in common. Therefore, the least common multiple of 4 and 5 is 20.
What if we are dealing with large numbers that are more difficult to list the multiples of? We can make a prime factor tree, just like with the GCF!
Prime Factor Tree
We can create a factor tree for each number, and multiply each prime factor together the largest number of times it occurs between the two numbers we are comparing to find the LCM.
For example, find the least common multiple of 35 and 25.
5 is a prime factor of both 35 and 25, but since it occurs twice as a prime factor of 25, we will list 5 twice when writing out our product:
𝐿𝐶𝑀 = 5 × 5 × 7 = 175
The parity of a number is whether it is even or odd.
The parity of a number, n, is determined by dividing n by 2 (n ÷ 2) and finding its remainder:
Rules of Parity
We can prove these rules by doing the operation on the parities.
For example, even + even = even. Even numbers have a parity of 0, so 0 + 0 = 0. Two even numbers added together equal an even number.
Answers to Practice Problems