Numbers, Square Roots and Equations Study Guide

Updated on Oct 5, 2011

Introduction to Numbers, Square Roots and Equations

Arithmetic is one of the oldest branches, perhaps the very oldest branch, of human knowledge; and yet some of its most abstruse secrets lie close to its tritest truths.

—NORMAN LOCKYER, English scientist (1836–1920)

This lesson contains miscellaneous math items that don't fall into the other lessons. However, achieving a comfort level with some of these tidbits will certainly support your success in other areas, such as word problems.

This lesson covers a variety of math topics that often appear on standardized tests, as well as in life:

  • Positive and negative numbers
  • Sequence of mathematical operations
  • Working with length units
  • Squares and square roots
  • Solving algebraic equations

Positive and Negative Numbers

Positive and negative numbers, also called signed numbers, can be visualized as points along the number line shown below.

Numbers to the left of 0 are negative and those to the right are positive. Zero is neither negative nor positive. If a number is written without a sign, it is assumed to be positive. On the negative side of the number line, numbers with bigger values are actually smaller. For example,–5 is less than –2. You come into contact with negative numbers more often than you might think; for example, very cold temperatures are recorded as negative numbers.

As you move to the right along the number line, the numbers get larger. Mathematically, to indicate that one number, say 4, is greater than another number, say –2, the greater than sign ">" is used:

4 > –2

Conversely, to say that –2 is less than 4, we use the less than sign, "<":

–2 < 4

Arithmetic with Positive and Negative Numbers

The following table illustrates the rules for doing arithmetic with signed numbers. Notice that when a negative number follows an operation (as it does in the second example), it is enclosed in parentheses to avoid confusion.

Miscellaneous Math


Sometimes subtracting with negatives can be tricky. Remembering "keep-switch-switch" can be a helpful way to recall that you should keep the first sign the same, switch the minus to a plus, and switch the sign of the third term.

Examples: –5 – 4 would become –5 + –4, and 27 – (–9) would become 27 + 9


Miscellaneous Math


To help remember the sign rules of multiplication, think of the following: Let being on time/starting on time be metaphors for a positive number and being late/starting late be metaphors for a negative number.

Being on time to something that starts on time is a good thing. (+ × + = +)

Being late to something that starts on time is a bad thing. (– × + = –)

Being on time to something that starts late is a bad thing (because you'll have to wait around). (+ × – = –)

Being late to something that starts late is a good thing (because now you're on time!). (– × – = +)

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