Basic Math & Pre-Algebra For Dummies Cheat Sheet

The English system of measurements is most commonly used in the United States. In contrast, the metric system is used throughout most of the rest of the world. Converting measurements between the English and metric systems is a common everyday reason to know math. This article gives you some precise metric-to-English conversions, as well as some easy-to-remember conversions that are good enough for most situations.

Here’s an easy temperature conversion to remember: 16°C = 61°F.

When arithmetic expressions get complex, use the order of operations (also called the order of precedence) to simplify them. Complex math problems require you to perform a combination of operations — addition, subtraction, multiplication, and division — to find the solution. The order of operations simply tells you what operations to do first, second, third, and so on.

Evaluate arithmetic expressions from left to right, according to the following order of precedence:

1. Parentheses

2. Exponents

3. Multiplication and division

4. Addition and subtraction

Following the order of operation is important; otherwise, you’ll end up with the wrong answer. Suppose you have the problem 9 + 5 × 7. If you follow the order of operations, you see that the answer is 44. If you ignore the order of operations and just work left to right, you get a completely different — and wrong — answer:

9 + 5 × 7 = 9 + 35 = 44    RIGHT

9 + 5 × 7 = 14 × 7 = 98     WRONG!

The Big Four math operations — addition, subtraction, multiplication, and division — let you combine numbers and perform calculations. Certain operations possess properties that enable you to manipulate the numbers in the problem, which comes in handy, especially when you get into higher math like algebra. The important properties you need to know are the commutative property, the associative property, and the distributive property. Understanding what an inverse operation is is also helpful.

Inverse operations

Inverse operations are pairs of operations that you can work “backward” to cancel each other out. Two pairs of the Big Four operations — addition, subtraction, multiplication, and division —are inverses of each other:

• Addition and subtraction are inverse operations of each other. When you start with any value, then add a number to it and subtract the same number from the result, the value you started with remains unchanged. For example:

  2 + 3 = 5    so    5 – 3 = 2

  7 – 1 = 6    so    6 + 1 = 7

• Multiplication and division are inverse operations of each other. When you start with any value, then multiply it by a number and divide the result by the same number (except zero), the value you started with remains unchanged. For example:

  3 × 4 = 12    so    12 ÷ 4 = 3

  10 ÷ 2 = 5    so    5 × 2 = 10

The commutative property

An operation is commutative when you apply it to a pair of numbers either forwards or backwards and expect the same result. The two Big Four that are commutative are addition and subtraction.

Addition is commutative because, for example, 3 + 5 is the same as 5 + 3. In other words

3 + 5 = 5 + 3

Multiplication is commutative because 2 × 7 is the same as 7 × 2. In other words

2 × 7 = 7 × 2

The associative property

An operation is associative when you can apply it, using parentheses, in different groupings of numbers and still expect the same result. The two Big Four operations that are associative are addition and multiplication.

Addition is associative because, for example, the problem (2 + 4) + 7 produces the same result as does the problem 2 + (4 + 7). In other words,

(2 + 4) + 7 = 2 + (4 + 7)

No matter which pair of numbers you add together first, the answer is the same: 13.

Multiplication is associative because, for example, the problem 3 × (4 × 5) produces the same result as the problem (3 × 4) × 5. In other words,

3 × (4 × 5) = (3 × 4) × 5

Again, no matter which pair of numbers you multiply first, both problems yield the same answer: 60.

The distributive property

The distributive property connects the operations of multiplication and addition. When multiplication is described as “distributive over addition,” you can split a multiplication problem into two smaller problems and then add the results.

For example, suppose you want to multiply 27 × 6. You know that 27 equals 20 + 7, so you can do this multiplication in two steps:

1. First multiply 20 × 6; then multiply 7 × 6.

   20 × 6 = 1207 × 6 = 42

2. Then add the results.

   120 + 42 = 162

Therefore, 27 × 6 = 162.

Exponents, radicals, and absolute value are mathematical operations that go beyond addition, subtraction, multiplication, and division. They are useful in more advanced math, such as algebra, but they also have real-world applications, especially in geometry and measurement.

Exponents (powers) are repeated multiplication: When you raise a number to the power of an exponent, you multiply that number by itself the number of times indicated by the exponent. For example:

7² = 7 × 7 = 49

2⁵ = 2 × 2 × 2 × 2 × 2 = 32

Square roots (radicals) are the inverse of exponent 2 — that is, the number that, when multiplied by itself, gives you the indicated value.

Absolute value is the positive value of a number — that is, the value of a negative number when you drop the minus sign. For example:

Absolute value is used to describe numbers that are always positive, such as the distance between two points or the area inside a polygon.

Fractions, decimals, and percents are the three most common ways to give a mathematical description of parts of a whole object. Fractions are common in baking and carpentry when you’re using English measurement units (such as cups, gallons, feet, and inches). Decimals are used with dollars and cents, the metric system, and in scientific notation. Percents are used in business when figuring profit and interest rates, as well as in statistics.

Use the following table as a handy guide when you need to make basic conversions among the three.