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.
Tip
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 |
Tip
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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Order of Operations - PEMDAS
When an expression contains more than one operation—like 2 + 3 × 4—you need to know the order in which to perform the operations. For example, if you add 2 to 3 before multiplying by 4, you'll get 20, which is wrong. The correct answer is 14: You must multiply 3 times 4 before adding 2.
Here is the order in which to perform calculations:
- Parentheses
Evaluate everything inside parentheses before doing anything else.
- Exponents
Next, evaluate all exponents.
- Multiplication and Division
Go from left to right, performing each multiplication and division as you come to it.
- Addition and Subtraction
Go from left to right, performing each addition and subtraction as you come to it.
Hook: The following sentence can help remind you of this order of operations: Please excuse my dear Aunt Sally.
Working with Length Units
The United States uses the English system to measure length; however, Canada and most other countries in the world use the metric system to measure length. Using the English system requires knowing many different equivalences, but you're probably used to dealing with these equivalences on a daily basis. Mathematically, however, it's simpler to work in metric units because their equivalences are all multiples of 10. The meter is the basic unit of length, with all other length units defined in terms of the meter.
Length Conversions
Math questions on tests, especially geometry word problems, may require conversions within a particular system. An easy way to convert from one unit of measurement to another is to multiply by an equivalence ratio. Such ratios don't change the value of the unit of measurement because each ratio is equivalent to 1.
Example: Convert 3 yards to feet.
Multiply 3 yards by the ratio 
. Notice that we chose rather than 
because the yards cancel during the multiplication:
Example: Convert 31 inches to feet and inches.
First, multiply 31 inches by the ratio 
Then, change the 
portion of 
ft. to inches:
Thus, 31 inches is equivalent to both 
ft. and 2 feet 7 inches.
Tip
Time Conversions: Word problems love giving information in mixed units of time. Since 5 minutes 42 seconds ≠ 5.42 minutes, it's always best to convert mixed information into the smallest unit. Change 5 minutes 42 seconds into seconds by multiplying the minutes by 60 and adding on the seconds: 5 × 60 + 42 = 342 seconds. You can change your answer back to minutes and seconds later by dividing it by 60. The whole number part of your answer will be the minutes, and the remainder will be the number of seconds. Example: 343 seconds: 343 ÷ 60 = 5, remainder 43, which is 5 minutes 43 seconds. (Use the same tips for working with minutes and hours.)
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Addition and Subtraction with Length Units
Finding the perimeter of a figure may require adding lengths of different units.
Example: Find the perimeter of the figure at right.
To add the lengths, add each column of length units separately:
Since 27 inches is more than 1 foot, the total of 16 ft. 27 in. must be simplified:
Finding the length of a line segment may require subtracting lengths of different units.
Example: Find the length of line segment 
in the following figure.
To subtract the lengths, subtract each column of length units separately, starting with the rightmost column. 
Warning: You can't subtract 8 inches from 3 inches because 8 is larger than 3!
As in regular subtraction, you have to borrow 1 from the column on the left. However, borrowing 1 ft. is the same as borrowing 12 inches; adding the borrowed 12 inches to the 3 inches gives 15 inches. Thus:
Thus, the length of 
is 5 feet 7 inches.
Squares and Square Roots
Squares and square roots are used in all levels of math. You'll use them quite frequently when solving problems that involve right triangles.
To find the square of a number, multiply that number by itself. For example, the square of 4 is 16, because 4 × 4 = 16. Mathematically, this is expressed as:
42 = 16
4 squared equals 16
To find the square root of a number, ask yourself, "What number times itself equals the given number?" For example, the square root of 16 is 4 because 4 × 4 = 16. Mathematically, this is expressed as:
√16 = 4
The square root of 16 is 4
Some square roots cannot be simplified. For example, there is no whole number that squares to 5, so just write the square root as √5.
Because certain squares and square roots tend to appear more often than others, the best course is to memorize the most common ones.
Arithmetic Rules for Square Roots
You can multiply and divide square roots, but you cannot add or subtract them:
√a + √b ≠ √a + b
√a – √b ≠ √a – b
√a × √b = √a × b
Solving Algebraic Equations
An equation is a mathematical sentence stating that two quantities are equal. For example:
2x = 10 y + 5 = 8
The idea is to find a replacement for the unknown that will make the sentence true. That's called solving the equation. Thus, in the first example, x = 5 because 2 × 5 = 10. In the second example, y = 3 because 3 + 5 = 8.
The general approach is to consider an equation like a balance scale, with both sides equally balanced. Essentially, whatever you do to one side, you must also do to the other side to maintain the balance. (You've already come across this concept in working with percentages.) Thus, if you were to add 2 to the left side, you'd also have to add 2 to the right side.
Example: Apply the previous concept to solve the following equation for the unknown n.
+ 1 = 3
The goal is to rearrange the equation so n is isolated on one side of the equation. Begin by looking at the actions performed on n in the equation:
- n was added to 2.
- The sum was divided by 4.
- That result was added to 1.
To isolate n, we'll have to undo these actions in reverse order:
| 3. |
Undo the addition of 1 by subtracting 1 from both sides of the equation: |
 |
| 2. |
Undo the division by 4 by multiplying both sides by 4: |
 |
| 1. |
Undo the addition of 2 by subtracting 2 from both sides: That gives us our answer: |
 |
Notice that each action was undone by the opposite action:
Check your work! After you solve an equation, check your work by plugging the answer back into the original equation to make sure it balances. Let's see what happens when we plug 6 in for n:
+ 1 = 3 ?
+ 1 = 3 ?
2 + 1 = 3 ?
3 = 3 
Tip
Do you know how tall you are? If you don't, ask a friend to measure you. Write down your height in inches using the English system. Then convert it to feet and inches (for example, 5' 6"). If you're feeling ambitious, measure yourself again using the metric system. Wouldn't you like to know how many centimeters tall you are?
Next, find out how much taller or shorter you are than a friend by subtracting your heights. How much shorter are you than the ceiling of the room you're in? (You can estimate the height of the ceiling, rounding to the nearest foot.)
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Find practice problems and solutions for these concepts at Numbers, Square Roots and Equations Practice Questions.
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