Numbers and Operations Study Guide for the Math Basics

Positive and Negative Numbers

We’ve already talked about some of the properties of real numbers, but here’s the section where we’ll go through specific rules when dealing with negative numbers (as promised).

Using a Number Line

Which number is greater: - 5 or - 4? This question trips up people all the time. I know it’s worse to owe someone 5 dollars rather than 4, so is -5 greater than - 4? This type of reasoning can get you into trouble. To help answer these types of questions, use a number line. Numbers to the right are always greater than numbers on the left.

Looking at the number line, it’s clear to see that - 4 is greater than - 5. So, \(-4 \gt -5\). Equivalently, \(-5 \lt -4\).

Using a number line also really helps with addition or subtraction. To add, move right. Try \(-3 + 5\) by starting at -3 and moving to the right 5.

You end at 2, so \(-3 + 5 = 2\).

Absolute Value

The absolute value of a number is its distance from 0. Distance is always positive, and so is absolute value. The absolute value of -6 is 6 and can be written: \(\left\vert -6 \right\vert = 6\).

Rules for Addition

There are simple rules when adding two numbers.

If they have the same sign, add and keep the sign.

Examples: \(3 + 5 = 8\) and \(-2 + (-7) = -9\)

If they have different signs, subtract the smaller absolute value from the larger and put the sign of the larger.

Examples:

\(-3 + 5 \rightarrow 5 - 3 = 2\) (positive because largest number, \(5\), was positive)

\(4 + (-11) \rightarrow 11 - 4 = 7 \rightarrow -7\) (negative because larger number, \(11\), was negative)

Rules for Subtraction

Any subtraction problem can be turned into an addition problem in the following manner: \(a - b = a + (-b)\). Note: the negative of a negative is positive, so if b was already negative, it would become positive. Once you’ve turned it into an addition problem, use the rules above.

Examples:

\[-4 - 3 = -4 + (-3)\]

This is now addition with the same sign, so just add 4 and 3 and keep the sign (negative). The answer is -7.

\[8 - (-4) = 8 + (-(-4))= 8 + 4 = 12\]

Rules for Multiplication and Division

The rules for multiplication and division are even easier. If the numbers have the same sign, the answer is positive. If they have different signs, the answer is negative. Simply put, determine the sign of the answer first, then do the problem as if both numbers are positive.

Examples:

\[-3 \cdot 4 = - (3 \cdot 4) = -12\] \[-15 \div (-5) = +(15 \div 5) = 3\]

Exponents and Roots

We know that multiplication is just repeated addition (multiple addition). Similarly, whole number exponents mean repeated multiplication.

Working with Exponents

If an exponent (or power) means repeated multiplication, then what does \(3^2\) mean?

\[3^2 = 3 \cdot 3 = 9\]

In words, \(3^2\) is “three squared,” \(7^3\) is “seven cubed,” and \(9^4\) is “nine to the fourth power.” Everything from the 4th power on is read this way.

One important thing to know is the difference between \(-3^2\) and \((-3)^2\). An exponent only applies to the number it directly follows, so it applies to just the 3 in the first case and to everything in the parentheses in the second case.

In the first case, \(-3^2 = -(3)^2 = -9\).

In the second case, \((-3)^2 = -3 \cdot -3 = 9\).

Working with Roots

A root is the opposite of an exponent. Some common roots you might see are:

\(\sqrt{x}\) (square root of x)

\(\sqrt[3]{x}\) (cube root of x)

\(\sqrt[4]{x}\) (fourth root of x)

When finding the square root of a number, \(\sqrt{25}\), for example, find the number that multiplies by itself (or “squares”) to get \(25\). In this case, the answer is \(5\), because \(5 \cdot 5\) or \(5^2 = 25\).

Similarly, the cube root of a number, such as \(\sqrt[3]{64}\), is the number that multiplies \(3\) times (or cubes) to get to \(64\). The answer is \(4\), because \(4^3 = 64\).

Not all numbers have whole number square roots. If a calculator isn’t allowed on a test, it will be important to know how to estimate them. To do this, it’ll be helpful to know common perfect squares: 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169 . . .

Now, estimate \(\sqrt{59}\) by looking at the perfect squares on either side: \(49\) and \(64\). So,

\[\sqrt{49} \lt \sqrt{59} \lt \sqrt{64}\]

or

\[7 \lt \sqrt{59} \lt 8\]

Therefore, \(\sqrt{59}\) is somewhere between 7 and 8.

Rules for Operations with Exponents and Roots

There are several operations that will be important to know when working with exponents or simplifying expressions that deal with exponents.

\[\begin{array}{|c|c|} \hline {} & {} \\ \text{Property} & \text{Example} \\ \hline {} & {} \\ {x^n \cdot \,x^m = x^{(n + m)}} & {b^5 \cdot b^3 = b^8} \\ \hline {} & {} \\ {(x^n)^m = x^{(n \cdot m)}} & {(7^3)^4 = 7^{12}} \\ \hline {} & {} \\ {\frac{x^m}{x^n} = x^{(m - n)}} & {\frac{r^4}{r^1} = r^{(4 - 1)} = r^3} \\ \hline {} & {} \\ {\quad (x \cdot y)^m = x^m \cdot y^m\quad} & {\quad(3m)^2 = 3^2 \cdot m^2 = 9m^2\quad} \\ \hline {} & {} \\ {x^{-m} = \frac{1}{x^m} } & {3^{-2} = \frac{1}{3^2} = \frac{1}{9}} \\ {} & {} \\ {\text{or} \;\frac{1}{x^{-m}} = x^m} & {\text{or}\; \frac{1}{4^{-3}} = 4^3 = 64} \\ \hline {} & {} \\ {x^0 = 1} & {(5,430)^0 = 1} \\ \hline \end{array}\]

When it comes to roots, there is one main property:

\[\sqrt{\mathstrut{x \cdot y}} = \sqrt{\mathstrut{x}} \cdot \sqrt{\mathstrut{y}}\]

It’s helpful when simplifying a square root:

\[\sqrt{\mathstrut{12}} = \sqrt{\mathstrut{4 \cdot 3}} = \sqrt{\mathstrut{4}} \cdot \sqrt{\mathstrut{3}} = 2 \cdot \sqrt{\mathstrut{3}} = 2\sqrt{\mathstrut{3}}\]

It sometimes also helps when two or more roots (or radicals) are multiplied:

\[\sqrt{\mathstrut{30}} \cdot \sqrt{\mathstrut{10}} \cdot \sqrt{\mathstrut{3}} = \sqrt{\mathstrut{30 \cdot 10 \cdot 3}} = \sqrt{\mathstrut{900}} = 30\]

When You Don’t Need an Exact Answer

When you get the answer to a math problem, it’s nice to have a bit of “number sense” to see if your answer is even in the ballpark of the correct answer. Estimation and rounding will help, because sometimes you just don’t need the exact answer.

Estimation

Sometimes a question will ask you specifically to “estimate”, but estimation also comes in handy on multiple choice questions. For example:

What is \(3.2 \cdot 8.89\)?

Answer choices:

\(24.38\)
\(28.448\)
\(31.7\)
\(29.318\)

Now you could just do this problem by hand, but that’s time-consuming. Estimate that \(3.2\) is close to \(3\) and \(8.89\) is close to \(9\). In your head, you know \(3 \cdot 9 = 27\), so the answer should be close to \(27\). The best choice is the second one, \(28.448\), which turns out to be the correct answer.

Estimation also helps you determine if you made an error. Perhaps you messed up the decimal point when doing \(42.3 - 12.58\) and you got \(2.972\) as your answer. A quick estimate shows \(42 - 13 = 29\), pretty far off from what you got. Hopefully, this will help you catch your mistake and turn \(2.972\) into the correct answer: \(29.72\).

Rounding

The rules for rounding numbers are pretty universal. Look at the number to the right of the digit you are rounding to. If the number is below 5, keep the digit to its left unchanged. If it’s 5 or above, bump the digit up one.

Example: Round 3.214 to the nearest hundredth.
The 1 is in the hundredths place, so look at the next number: 4.
This is less than 5, so keep 1 as it is.
The answer is 3.21.

Example: Round 2.987 to the nearest hundredth.
8 is in the hundredths place, so look at the next number: 7.
This is greater than or equal to 5, so bump 8 up to 9.
The answer is 2.99.

When dealing with word problems, you might automatically need to round even when not asked to so the answer makes sense. For instance, If you make lemonade and sell it for $2.25 per glass, how many glasses do you have to sell to afford a $19 video game? Mathematically,

\[19 \div 2.25 = 8.444\]

However, you can’t sell a partial glass of lemonade, so you need to decide on either 8 glasses or 9 glasses.
8 glasses will leave you a little short (\(8 \cdot 2.25 = 18\)), so you’ll have to sell 9 glasses of lemonade.