Math Semester Review

1.1 Number Systems:

The Number system classifies numbers into sets and subsets based on their type and characteristics.

Categories of the Number System

Natural Numbers: The set of positive integers (it does NOT INCLUDE 0 ) **Ex: 1,7,9

Whole Numbers: The set of all positive counting numbers (includes 0 ) Ex: 0,1,2,3…

Integers: The set of whole numbers, positive or negatives (includes 0) Ex: 0, -1, 2, -3

Rational Numbers: Numbers that can be written as fractions, terminating decimals, and repeating decimals (they can be positive or negative)

The Real Number System Diagram:

The Real Number System Diagram is usually shown as a set of nested boxes or circles, where each larger set contains all the sets inside it. It visually shows how different types of numbers are related. If a number belongs to a smaller set, it automatically belongs to all the larger sets outside it. But if a number belongs to a larger set, it does not automatically belong to the smaller sets inside (Numbers can move outward, not inward).

Practice

Look at the list of numbers below:

–3, 0, 1, 4.75, 2/3, 12, –8.5

Sort each number into all the categories it belongs to:

Natural Numbers, Whole Numbers, Integers, Rational Numbers.

Write each number under every correct category.

• Answers

  Natural: 1,12

  Whole: 0, 1, 12

  Integer: 0, 1, 12, –3, –8

  Rational:–3, 0, 1, 12, 4.75, 2/3, –8.5

1.2 Adding and Subtracting Integers

Adding and subtracting integers is mostly about paying attention to the signs. When the signs are the same, you add the numbers and keep the sign. When the signs are different, you subtract the numbers and keep the sign of the bigger number. When subtracting, you can use keep-change-change: keep the first number, change the subtraction to addition, and change the sign of the second number. These rules make it easier to solve any integer problem.

• Key points

  • Pay attention to the signs of the numbers.

  • Same signs: add the numbers and keep the sign.

  • Different signs: subtract and keep the sign of the bigger number.

  • Subtracting rule: use keep-change-change (turn subtraction into addition and flip the second sign).

  • These rules help you solve integer problems quickly and correctly.

Adding and Subtracting Rules

1.3 Adding and Subtracting Rational Numbers

Adding (LCD):

1. Make sure the numbers are in fraction

2. Get the least common denominator (LCD)

3. Add the numerators (denominators stay the same).

4. Keep the sign rules for integers when adding positives and negatives.

5. Simplify the answer if needed.

• How to find LCD

  Steps to Find the LCD

  1. List the multiples of each denominator.

     • Example: denominators 4 and 6

       • Multiples of 4 → 4, 8, 12, 16, …

       • Multiples of 6 → 6, 12, 18, …

  2. Find the smallest common multiple.

     • For 4 and 6 → smallest number in both lists = 12

  3. Use the LCD to rewrite each fraction with the same denominator before adding or subtracting.

Subtracting (KCF):

1. Rewrite the subtraction as addition (use keep-change-flip).

2. Find common denominators if you are working with fractions.

3. Add the numerators (follow integer sign rules).

4. Keep the denominator the same.

5. Simplify the final answer.

1.4 Fractions and Decimals (Converting)

Fraction to Decimals:

A fraction is just “numerator ÷ denominator”. To get a decimal, divide the top number by the bottom number.

Steps to Solve

1. Take the numerator (top number).

2. Divide it by the denominator (bottom number) using long division or a calculator.

3. Write the result as a decimal.

Just remember: “Divide top by bottom” — that’s all you need!

• Types of Decimals

  A decimal that ends or terminates is know as a terminating decimal (8.25)

  A decimal that repeats is called a repeating decimal

Decimals to Fraction

1. Count the number of digits after the decimal.

2. Write the decimal as a fraction with that many zeros in the denominator:

   • 1 digit → 10

   • 2 digits → 100

   • 3 digits → 1000, etc.

3. Simplify the fraction by dividing numerator and denominator by their greatest common factor (GCF).

Examples:

• 0.5 → 5/10 → simplify → 1/2

• 0.75 → 75/100 → simplify → 3/4

• 0.2 → 2/10 → simplify → 1/5

Tip: Count the digits after the decimal → put over 10, 100, 1000… → simplify. That’s it!

1.6 Squares and Square Roots

• Perfect Squares

  Number (n)	1	2	3	4	5	6	7	8	9	10	11	12	13	14	15	16	17	18	19	20	21	22	23	24	25
  Perfect Square (n²)	1	4	9	16	25	36	49	64	81	100	121	144	169	196	225	256	289	324	361	400	441	484	529	576	625

A square is a number multiplied by itself. It is written as n^2, where n is the number. For example,5^2 = 5 × 5 = 25. Numbers like 1, 4, 9, 16, 25, and so on are called perfect squares because their square roots are whole numbers. The square root of a number is the value that, when multiplied by itself, gives the original number. It is written as $√$. For example, √25 = 5 because 5×5=25

Not all square roots are whole numbers. Numbers that are not perfect squares, like 2, 3, or 5, have irrational square roots that go on forever without repeating (for example,2≈1.414). A quick way to estimate these is to find the two perfect squares it is between. Memorizing the first several perfect squares, like 1² up to 10² (1, 4, 9, 16, 25, 36, 49, 64, 81, 100), helps solve problems quickly and makes it easier to work with squares and square roots in higher-level math.

How to Estimate Square Roots:

1. Find the nearest perfect squares above and below the number.

2. Take the square roots of those perfect squares.

3. Your answer is between those two numbers.

4. Optional: Refine by seeing which one it is closer to.

Tip: Memorize the first 10 perfect squares for quick reference (1, 4, 9, 16, 25, 36, 49, 64, 81, 100). This makes estimation very fast!

How to Estimate Square Roots on a Number Line:

• Draw a number line that includes the perfect squares around your number.

• Mark the perfect squares on the line.

• See where your number falls between the squares.

• Estimate the square root based on distance.

Tip: Using a number line helps you see how close the number is to the lower or upper square and gives a quick visual estimate.

1.8 Multiplying and Dividing Integers:

When multiplying or dividing integers, the most important thing is the sign of the answer. The rules are simple: if the signs of the two numbers are the same (both positive or both negative), the answer is positive. If the signs are different (one positive, one negative), the answer is negative. The actual multiplication or division is done using the absolute values of the numbers, then you apply the sign rule. These rules work for both whole numbers and rational numbers. Understanding this helps solve problems quickly without making sign mistakes. Same signs → Positive | Different signs → Negative.

Multiplying & Dividing Integers

1.8 Multiplying & Dividing Rational Numbers

Multiplying and Dividing rational numbers are pretty much the same just with a few different rules:

Steps for Multiplying Rational Numbers (Fractions or Decimals)

1. Multiply the numerators together.

2. Multiply the denominators together.

3. Apply the sign rule (same signs → positive, different signs → negative).

4. Simplify the fraction if possible.

Steps for Dividing Rational Numbers (Fractions)

1. Keep the first fraction.

2. Change the division to multiplication.

3. Flip the second fraction (take the reciprocal).

4. Multiply as usual.

5. Apply the sign rule.

6. Simplify.

1.9 Comparing and Ordering Real Numbers:

• Identify the numbers you want to compare or order.

• Convert all numbers to decimals if they are fractions or percentages.

• Converting

  Fraction → Decimal

  • Divide the numerator by the denominator.

  Decimal → Fraction

  • Count the digits after the decimal and write over 10, 100, 1000….

  • Simplify if needed.

  Decimal → Percent

  • Multiply the decimal by 100 and add the % sign.

  • Example: 0.75 × 100 = 75%

  Percent → Decimal

  • Divide the percent by 100 or move the decimal two places left.

  • Example: 60% → 60 ÷ 100 = 0.6

  Fraction → Percent

  • First convert fraction → decimal, then decimal → percent.

  Percent → Fraction

  • Convert percent → decimal, then decimal → fraction.

  To convert:

  • Fraction → Decimal: divide numerator ÷ denominator

  • Decimal → Percent: ×100%

  • Percent → Decimal: ÷100

  • Decimal → Fraction: put over 10, 100, 1000… and simplify

  • Fraction → Percent: fraction → decimal → percent

  • Percent → Fraction: percent → decimal → fraction

  ---

• Visualize a number line: numbers to the right are bigger, numbers to the left are smaller.

• Compare two numbers at a time using >,<,=

• To order a list:

  • Least to greatest: start from the leftmost number.

  • Greatest to least: start from the rightmost number.

• Double-check by comparing decimals or fractions if needed and add the % sign.

• Example: 0.75 × 100 = 75%

1.91 Scientific Notation:

Scientific notation is a way to write very big or very small numbers in a shorter, easier form using powers of ten. The general form is:

                                         *a* × 10^*n*

• a is a number between 1 and 10 (can have decimals)

• n is an integer showing how many times to multiply or divide by 10

Reading Scientific Notation

• Look at a × 10^n and read it as: “a times ten to the power of n”

• Example: 3.5 × 10^4 → “three point five times ten to the fourth power”

Writing Numbers in Scientific Notation

1. Large Numbers (>1) (positives):

   • Move the decimal point after the first non-zero digit.

   • Count the places you moved → that is the positive exponent n.

   • Example: 45,000 → 4.5 × 10^4

   (move the decimal to the right depending on the number of times the exponent states)

2. Small Numbers (<1) (negatives):

   • Move the decimal after the first non-zero digit.

   • Count the places you moved → that is the negative exponent n.

   • Example: 0.0072 → 7.2 × 10^-3

   (move the decimal to the left depending on the number of times the exponent states

Tips:

• Positive exponent → number is greater than 1

• Negative exponent → number is less than 1

• Always keep one non-zero digit before the decimal

• Scientific notation makes very large or small numbers easier to read, compare, and calculate

2.1 Calculating Interest

Simple Interest

Key Concept:

Simple interest is the extra money earned or paid on an amount of money over time. It is called “simple” because the interest is only calculated on the original amount (principal), not on the interest that has already been added.

Simple Interest Formula:

I=prt

Where:

• I = Interest (the extra money earned or paid)

• P = Principal (the original amount of money)

• R = Rate (as a decimal, e.g., 5% = 0.05)

• T = Time (in years)

💡 Tip: Always convert the rate from a percentage to a decimal before using the formula.

Example 1: Finding Simple Interest

Problem: You deposit $500 in a bank account with a 4% annual interest rate for 3 years. How much interest will you earn?

Solution:

         I=prt

 I=500x0.04x3

           I=60

✅ You will earn $60 in interest.

Quick Tips:

• Always check if time is in years; if it’s in months, divide by 12.

• Convert percentages to decimals (divide by 100).

• Simple interest does not compound.

How to Find the Total Balance

1. Find how much interest you earned.

2. Add that interest to the money you started with.

In super simple words:

Total balance = starting money + interest

Compound Interest:

Compound Interest is earned both on the principle plus any previously earned intrest

The compound interest formula is:

A = p(2+r)^t

A is the total amount in the account

P is the starting amount (principal)

R is the interest rate as a decimal

T is time in years

To find the total amount, you plug the values into the formula and solve. The exponent makes the account grow each time interest is added.

Example:

You deposit 500 dollars at an annual rate of 4 percent, compounded once per year, for 3 years.

A = 500(1 + 0.04/1)^(1·3)

A = 500(1.04)^3

A ≈ 562.43

So after 3 years, your account balance is about 562 dollars and 43 cents.

Quick tips:

• Always change percent to a decimal before using the formula.

• If interest is compounded monthly, n = 12.

• If it is yearly, n = 1.

• Compound interest grows faster than simple interest because interest builds on top of interest.