1. Types of numbers

• natural: 1 to infinity as whole numbers
• whole: 0 plus all natural numbers
• integer: positive and negative whole numbers
• rational: fraction of integers
• irrational number: all others that do not fit above patterns

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Perform calculations using order of operations.

Please excuse my dear Aunt Sally!

• Parenthesis
• Exponents
• Multiplication
• Division
• Addition
• Subtraction

Examples

\[3^2 - 8/2 + 4^2\] \[\sqrt{5^2 - 4^2} + 7(5-4)^2\]

Properties of real numbers:

• commutative: adding or multiplication in either order

\[a+b = b+a;\quad a\cdot b= b\cdot a\] - associative: regrouping of adding and multiplication is possible

\[a+(b+c) = (a+b)+c\ ; \quad a\cdot(b\cdot c) = (a\cdot b) \cdot c\]

• distributive: multiplying a sum is the same as the sum individual products

\[a(b+ c) = ab + ac\]

• inverse: a form of the number needed to create the identity

\[a - a = 0\ ;\quad a\frac{1}{a} = 1\]

• identity: a number that does not effect the result

\[ a + 0 = a\ ;\quad a\cdot 1 = a\]

Evaluate algebraic expressions.

exponential notation:

\[\frac{n^5}{n^3} = \left\{n^5\cdot n^{-3}\strut=n^2 \atop\strut \frac{n\cdot n\cdot n\cdot n\cdot n}{n\cdot n\cdot n }=n\cdot n\right\}= n^2\]

Simplify algebraic expressions.

Given a mathematical expression, simplify it using the order of operations.

1. Simplify any expressions within grouping symbols.
2. Simplify any expressions containing exponents or radicals.
3. Perform any multiplication and division in order, from left to right.
4. Perform any addition and subtraction in order, from left to right.

simplify

\[\eqalign{ 3x -2y + x -3y -7\\ 2r -5(3-r)+4\\ 12 / (36/9) + 6\\ 2 + 8 \times 7 / 4\\ -25/5^2 - 7\\ }\]

Example

• Fred earns $40 at the community garden. He spends $10 on a streaming subscription, puts half of what is left in a savings account, and gets another $5 for walking his neighbor’s dog.

  • Write the expression that represents the number of dollars Fred keeps (and does not put in his savings account). Remember the order of operations.

  • How much money does Fred keep?

3 Radicals and Rational Exponents

\[\eqalign{\sqrt{2}\cdot \sqrt{2} &=& 2^{\frac{1}{2}}\cdot 2^{\frac{1}{2}}=2^\frac{2}{2}=2^1= 2\\ \sqrt[3]{2}\cdot \sqrt[3]{2} \cdot \sqrt[3]{2}&=&2^{\frac{1}{3}}\cdot 2^{\frac{1}{3}}\cdot 2^{\frac{1}{3}} = 2^\frac{3}{3} = 2^1 = 2\\ \sqrt[4]{2}\cdot \sqrt[4]{2}\cdot \sqrt[4]{2} \cdot \sqrt[4]{2}&=&\left\{\eqalign{ 2^{\frac{1}{4}}\cdot 2^{\frac{1}{4}}\cdot 2^{\frac{1}{4}}\cdot 2^{\frac{1}{4}}\ =\ 2^\frac{4}{4}\\ 2^{\frac{2}{4}}\cdot 2^{\frac{2}{4}}= 2^{\frac{1}{2}}\cdot 2^{\frac{1}{2}}=2^\frac{2}{2}\\ }\right.\\&=& 2^1 = 2\\}\]

\[\eqalign{\frac{3}{1-\sqrt{2}}&=&\frac{3}{\left(1-\sqrt{2}\right)}\frac{\left(1+\sqrt{2}\right)}{\left(1+\sqrt{2}\right)}\\&=&\frac{3+3\sqrt{2}}{1-2}=\frac{3\left(1+\sqrt{2}\right)}{-1}\\&=&-3 + 3\sqrt{2}\\&=&3\sqrt{2} -3\\}\]