Numbers and Operations Study Guide for the Math Basics

How to Prepare for the Numbers and Operations Questions on a Test

General Information

Almost every math test you take as an adult will assume you know certain facts about numbers, their properties, and the operations you perform with them. We’ve created this study guide as a one-stop resource for mathematical properties and definitions to help you succeed in whatever test you are preparing for.

Types of Numbers

As a species, human beings discovered numbers. But we didn’t automatically know about all of them. It might have been helpful for early traders to know how to count: “I’ll give you 1 chicken for 4 scoops of wheat.” But they wouldn’t know about more abstract numbers like \(\sqrt2\). Money didn’t always exist, so the idea of debt (negative numbers) also didn’t exist. Different types of numbers have gradually integrated into our way of thinking. Therefore, we have named these types of numbers and discovered properties about them.

Natural Numbers— Natural numbers or counting numbers are all of the numbers from 1 counting upwards indefinitely. The set looks like: {1, 2, 3, 4, 5, 6, …}. These are the first numbers you learn as a child.

Whole Numbers— Whole numbers are the complete set of natural numbers with the addition of zero: {0, 1, 2, 3, 4, 5, 6, …} . Believe it or not, zero wasn’t really used in mathematics until less than 2,000 years ago in India.

Integers— Integers are the set of all whole numbers and their opposites (negatives). {… -3, -2, -1, 0, 1, 2, 3, …}. Negatives came into mathematical existence around the same time as zero.

Rational Numbers— Any number that can be written as a fraction \(\frac{a}{b}\), where a and b are both integers and b is not zero, is a rational number. Examples include: \(\frac{1}{2}, \;- \frac{3}{5},\) and \(\frac{-16}{29}\).

It’s important to note that all integers are also rational numbers since they can be written as a fraction with \(1\) in the denominator. (Examples: \(2=\frac{2}{1}, \; -3 =\frac{-3}{1}, \;0=\frac{0}{1}\)) .

Also, if a number in decimal notation either terminates (ends) or repeats with a pattern indefinitely, then that number is rational. (Examples: \(0.3333… = \frac{1}{3}\) and \(1.247 = 1 \frac{247}{1000} = \frac{1247}{1000}\).)

Irrational Numbers— Irrational numbers are numbers that can’t be written as a fraction \(\frac{a}{b}\) where a and b are integers. Until the ancient Greeks, humans thought all numbers were rational. Common examples of irrational numbers are \(\pi,\;\sqrt{2},\) and \(e\). Any number in decimal notation that doesn’t end and doesn’t have a repeating pattern is irrational.

Real Numbers— All rational and irrational numbers make up the set of real numbers.

Imaginary Numbers— The number \(i\) is defined as \(\sqrt{-1}\). If you think really hard, you’ll see why \(i\) has been given the name of an imaginary number. What number times itself is \(-1\)? Mental conundrum aside, imaginary numbers are the product of any real number and \(i\). For example:

\(\sqrt{-4} = \sqrt{4 \cdot -1} = \sqrt{4} \cdot \sqrt{-1} = 2 \cdot i = 2i\) which is an imaginary number.

Complex Numbers— The set of real and imaginary numbers combined with addition make up the set of complex numbers. Examples:

{\(1+ 3i, -12 - i,\) and \(\frac{3}{7} + 12i\)}

Prime Numbers— Prime numbers are natural numbers that can be written as a product of only two whole numbers: \(1\) and itself. Examples include:

\[2, 3, 5, 7, 11, 13, …\]

Composite Numbers— Composite numbers are natural numbers that can be written as a product of two whole numbers more than one way. The number \(12\), for example, is composite because it can be written as \(1 \cdot 12, 2 \cdot 6,\) or \(3 \cdot 4\). Any number that isn’t prime is composite.

Even and Odd Numbers— Even numbers are integers that are divisible by two, whereas odd numbers aren’t. The numbers \(\{..., 2, 4, 6, 8, …\}\) are even and \(\{..., 1, 3, 5, 7, 9, …\}\) are odd.

Consecutive Numbers— If your teacher asks for a set of 4 consecutive integers, you could answer \(\{-5, \,-4,\, -3,\, -2\}, \,\{-2,\, -1,\, 0,\, 1\},\, \text{or} \,\{5,\, 6,\, 7,\, 8\}\), for example. Consecutive numbers are a set of integers separated by 1.

Number Notation

It’s important to have a common language when speaking mathematics. Numbers, and the way you write them, are an important component of the language of math.

Arabic Numerals and the Decimal System

The numbers most people in the world use are based on a system created 1500 years ago in the middle east. These are the digits from 0 to 9. It makes sense that there are only 10 digits because we have 10 fingers (also called digits). To denote numbers bigger than 9 we use more digits. Each additional digit represents a power of 10: 1 (one), 10 (ten), 100 (one hundred), 1000 (one thousand), etc. We call the position of the digit the place value. Here’s a chart showing the place values of the number 13,249.

\[\begin{array}{|c|c|c|} \hline \text{1} & \text{3} & \text{2} & \text{4} & \text{9}\\ \hline \text{ten thousands} & \text{thousands} & \text{hundreds} & \text{tens} & \text{ones} \\ \hline \end{array}\]

The digits are grouped in 3’s separated by a comma (if you live in the United States or the UK). Outside those countries, many people just separate the groups using spaces.

If you want to represent a fractional part of a number in the decimal system, you need to put a decimal point (again, if in the US or UK) and then more digits. Each represents a power of 10 less than one: 0.1 (one tenth), 0.01 (one hundredth), 0.001 (one thousandth), etc. Here’s a chart showing the place values of the number 3,865.912.

\[\begin{array}{|c|c|c|} \hline \text{3} & \text{8} & \text{6} & \text{5} & \text{.9} & \text{1} & \text{2} \\ \hline \text{thousands} & \text{hundreds} & \text{tens} & \text{ones} & \text{tenths} & \text{hundredths} & \text{thousandths} \\ \hline \end{array}\]

Roman Numerals

Before the Arabic system, the Roman Empire (and what followed) used Roman Numerals. Today, you might see it when talking about which SuperBowl it is. Here’s a chart showing which Roman Numeral relates to which Arabic number.

\[\begin{array}{|c|c|} \hline \mathbf{Roman \;Numeral} & \mathbf{Arabic \;Numeric\; Value} \\ \hline \text{I} & \text{1} \\ \hline \text{V} & \text{5} \\ \hline \text{X} & \text{10} \\ \hline \text{L} & \text{50} \\ \hline \text{C} & \text{100} \\ \hline \text{D} & \text{500} \\ \hline \text{M} & \text{1,000} \\ \hline \end{array}\]

To write the number 123, you would write CXXIII. This corresponds to 1 group of 100, 2 groups of 10 and 3 groups of 1. Pretty intuitive. Here are two other examples:

357 = CCCLVII

2,632 = MMDCXXXII

The only catch is that you can’t have more than 3 of the same symbol in any group. To write the number 4, you wouldn’t write IIII, because at a glance, it might look like III. To fix this, think of 4 as “5 - 1” and write IV. This is different from VI (6) because the order is switched. The smaller value (I) is placed in front of the larger value (V) to denote subtraction. This is a list of examples that might help you see the patterns and process this system:

9 = IX
11 = XI
40 = XL
60 = LX
79 = LXXIX
81 = LXXXI
344 = CCCXLIV
346 = CCCXLVI
364 = CCCLXIV
366 = CCCLXVI

While there are further complicated processes (like why is 99 = XCIX and not IC?), those shouldn’t show up on any tests for which you might be preparing.

Signs and Symbols

To comprehend the language of math, of course you’ll need to know how numbers are represented, but you also need to be able to instantly attach meaning to some basic symbols. Here is a list of some of the most common ones:

\[\begin{array}{|c|l|} \hline \mathbf{Symbol} & \mathbf{Meaning} \\ \hline {+} & \text{plus (or positive)} \\ \hline {-} & \text{minus (or negative)} \\ \hline {\times\; or \;\cdot} & \text{times} \\ \hline {\div} & \text{divided by} \\ \hline {\pm} & \text{plus or minus (or positive-negative)} \\ \hline {=} & \text{equals} \\ \hline {\lt} & \text{is less than} \\ \hline {\gt} & \text{is greater than} \\ \hline {\le} & \text{is less than or equal to} \\ \hline {\ge} & \text{is greater than or equal to} \\ \hline {\neq} & \text{is not equal to} \\ \hline {\cong} & \text{is congruent to} \\ \hline {\sim} & \text{is similar to} \\ \hline {\pi} & \text{pi (3.14)} \\ \hline \end{array}\]