Numbers and Operations Study Guide for the Math Basics

Scientific Notation

Scientific notation is a way to represent really large or really small numbers. For instance, the diameter of the solar system is \(1.39 \times 10^6\;km\) \((1,390,000 \;km)\) and the diameter of an atom is around \(2 \times 10^{-14}\;m\) \((0.00000000000002 \;m)\).

The exponent of the \(10\) tells you how many times and which direction to move the decimal point to turn the number into standard form. In the first case (\(1.39 \times 10^6\)), the positive \(6\) means move the decimal to the right \(6\) places. Fill in any blank spaces with \(0\).

In the second case (\(2 \times 10^{-14}\)), the \(-14\) tells you to move the decimal to the left \(14\) places. Note the decimal is immediately after the \(2\), even though it’s not written.

For a number to be truly in scientific notation, \(a\times 10^m\), \(a\) must be greater than or equal to \(1\) and less than \(10\). To turn a number from standard form into scientific notation, move the decimal until \(a\) fits the restrictions.

Try it with the number \(2,341\):

You’ll have to move the decimal \(3\) places to make \(a \;= 2.341\), which is less than \(10\) and greater than or equal to \(1\).
Now decide if \(m\) is positive or negative. If the original number in standard form is more than \(1\), then \(m\) is positive, otherwise, \(m\) is negative. So, \(2,341 = 2.341 \times 10^3\).

Sets and Set Notation

In mathematics, a set is a collection of distinct objects. We write sets with curly braces { } in order to distinguish it from other common mathematicals objects like ordered pairs on a coordinate plane. To write a set in set notation just use the curly braces and list the elements. What is the set of integers between (but not including) -3 and 5? {-2, -1, 0, 1, 2, 3, 4}. Sometimes ellipses are necessary. What is the set of natural numbers greater than 7? {8, 9, 10, 11, …}

Number Sequences and Patterns

In essence, mathematics began when we started noticing patterns. If you are given the sequence 1, 2, 3, 4, … , you probably know that the next number in the pattern is 5. There are two main types of sequences: geometric and arithmetic.

Geometric Sequences

A geometric sequence is a series of numbers where each term is a multiple of the previous term and the common ratio r.

Here’s an example: \(2, 4, 8, 16, …\)
The common ratio is \(2\) because each term is equal to the previous term times two:

\(2 \cdot 2 = 4, \;4 \cdot 2 = 8, \;8 \cdot2 = 16\), and the next term is \(32\) because \(16 \cdot 2 = 32\).

In addition to just finding the next number in a sequence, you’ll often be asked to find a specific term. In this case, it’s helpful to know the formula for a geometric sequence:

\(a_n = a_1 \cdot r^{(n-1)}\), where \(a_n\) is the \(nth\) term, \(a_1\) is the first term, and \(r\) is the common ratio.

For example, find the 8th term of the sequence above. Substitute and see that \(n = 8,\; a_n = a_8,\; a_1 = 2, \;r = 2\).

The answer is:

\[a_8 = 2 \cdot 2^{(8-1)} = 2 \cdot 2^7 = 2^8 = 256\]

Arithmetic Sequences

An arithmetic sequence is a series of numbers where each term is the sum of the previous term and the common difference d. Here’s an example: 3, 6, 9, 12, … The common difference is 3 because each term is 3 more than the previous term: 3 + 3 = 6, 6 + 3 = 9, 9 + 3 = 12, and the next term is 12 + 3 = 15.

To find a specific term of an arithmetic sequence, use the formula \(a_n = a_1 + d(n - 1)\) where \(a_n\) is the nth term, \(a_1\) is the first term, and d is the common difference.

Now, find the 8th term of the arithmetic sequence above:

\[a_8 = 3 + 3(8 - 1) = 3 + 3 \cdot 7 = 3 + 21 = 24\]

Of course, you could just build the sequence up to the 8th term (3, 6, 9, 12, 15, 18, 21, 24), but sometimes the formula saves valuable time.

Ratio and Proportion

A ratio is a relationship between two quantities, often by division.
A proportion is a mathematical statement showing two equal ratios.

Equivalent Ratios

Two ratios are equivalent if they have the same simplest form.
The simplest form of a ratio is \(\frac{a}{b}\), where a and b have no common factors other than 1.

For example, \(\frac{6}{9}\) and \(\frac{12}{18}\) are equivalent because each simplifies to \(\frac{2}{3}\).

You can also check to see if two ratios are equivalent by cross multiplying the two.

Let’s see if \(\frac{3}{4} = \frac{9}{12}\)

Using cross multiplication, \(3 \cdot 12 = 9 \cdot 4\) is true (\(36=36\)), so the ratios are equivalent.

Often, it’s helpful to find an equivalent ratio in order to add or subtract fractions. To do this, multiply the top and the bottom numbers by the same value.

For example: \(\frac{2}{5} + \frac{3}{10}\)

In this case, \(10\) is the common denominator so I need to turn \(\frac{2}{5}\) into an equivalent ratio, with \(10\) as its denominator, by multiplying both the top and the bottom by \(2\).

\[\frac{2}{5} \cdot \frac{2}{2} = \frac{4}{10}\]

Now we can add:

\[\frac{4}{10} + \frac{3}{10} = \frac{7}{10}\]

Solving and Using Proportions

The tried and true method of solving a proportion is the cross multiplication method. In mathematics, this is specifically called the means-extremes property of proportions.

Solve this proportion:

\[\frac{6}{x} = \frac{9}{12}\]

Using cross multiplication, we get:

\[6 \cdot 12 = 9 \cdot x\] \[72 = 9x\]

Now divide both sides by 9 to isolate x and you get \(8 = x\).

Proportions can be used in most any word problem that gives you three pieces of information and asks for a fourth. Here’s one: A recipe for chocolate chips calls for 2 cups of flour to 3 cups of chocolate chips. How many cups of chocolate chips do you need if you want to make the same recipe using 5 cups of flour?

Three pieces of information: 2 cups flour, 3 cups chocolate chips, 5 cups flour.

Fourth piece to find: “x” cups chocolate chips.

Now, set up a proportion. There are many possibilities to set it up, just remember that items in the same row and same column should have something in common. Here are a couple ways we could set it up:

\[\begin{array}{|c|c|c|} \hline \text{} & \text{flour} & \text{choc. chips} \\ \hline \text{recipe} & \text{2} & \text{3} \\ \hline \text{to make} & \text{5} & \text{x} \\ \hline \end{array}\] \[\begin{array}{|c|c|c|} \hline \text{} & \text{recipe} & \text{to make} \\ \hline \text{choc. chips} & \text{3} & \text{x} \\ \hline \text{flour} & \text{2} & \text{5} \\ \hline \end{array}\]

So either proportion: \(\frac{2}{5} = \frac{3}{x}\; or\; \frac{3}{2} = \frac{x}{5}\) would work and yield the following when doing cross multiplication to solve:

\[2 \cdot x = 3\cdot 5\] \[2x = 15\]

Divide both sides by 2 and find that \(x = 7.5\) cups of chocolate chips.