Whole Numbers and Decimals

Whole numbers are the set of numbers: ${0, 1, 2, 3, 4, 5, …}$ .
Our number system is a decimal system (base $10$ ), using a place-value system.
Position of digits determines value (e.g., in $387$ , $3$ is hundreds, $8$ is tens, $7$ is ones).

Basic: addend + addend = sum.
Align numbers by place value; add digits from right to left.
Carry over to the next column when the sum of digits is $10$ or greater.
Properties of Addition:
- Associative Property: $(a + b) + c = a + (b + c)$
- Commutative Property: $a + b = b + a$
- Identity Property of Zero: $a + 0 = a$

Basic: minuend - subtrahend = difference.
Align numbers by place value; subtract digits from right to left.
Borrow from the next column to the left when a digit is smaller than the one being subtracted.
Checking: Subtraction is the inverse of addition ( $12 - 7 = 5$ translates to $5 + 7 = 12$ ).

Basic: factor * factor = product. Related to repeated addition.
Multiplying by a Single-Digit Number: Multiply each digit of the larger number by the single digit, carrying over as needed.
Multiplying by a Power of $10$ : Count the zeros in the power of $10$ and attach that many zeros to the right of the other whole number.
Multiplying by a Several-Digit Number: Multiply the first number by each digit of the second number (considering place value) to get partial products, then add the partial products.
Properties of Multiplication:
- Associative Property: $(a * b) * c = a * (b * c)$
- Commutative Property: $a * b = b * a$
- Identity Property of One: $a * 1 = a$
- Multiplication Property of Zero: $a * 0 = 0$
- Distributive Property: $a * (b + c) = (a * b) + (a * c)$

Basic: dividend / divisor = quotient.
Division by $1$ and $0$ :
- Any nonzero number divided by itself is $1$ .
- Any nonzero number divided by $1$ remains unchanged.
- Zero divided by any nonzero number is $0$ .
- Division by zero is undefined.

Exponents: A shorthand for repeated multiplication ( $base^{exponent}$ ).
- A number raised to the power of $0$ is $1$ (e.g., $10^0 = 1$ ), for any nonzero base.
Order of Operations (BEDMAS/PEMDAS):
1. Parentheses
2. Exponents
3. Multiplication and Division (from left to right)
4. Addition and Subtraction (from left to right)

Steps: Understand the problem, Solve and state the answer, Check the answer's reasonableness.

Place Values: Digits to the right of the decimal point represent tenths ( $1/10$ ), hundredths ( $1/100$ ), thousandths ( $1/1000$ ), etc.
Comparing Decimals: Compare digits from left to right. The number with the larger digit at the first differing place is greater.
Rounding Decimals:
1. Identify the rounding place value.
2. If the digit to its right is less than $5$ , drop it and all digits to the right.
3. If the digit to its right is $5$ or greater, increase the digit at the rounding place value by one and drop all digits to the right.

Align numbers vertically by their decimal points.
Add or subtract digits by place value from right to left, borrowing or carrying as needed.
Place the decimal point in the sum/difference directly in line with the others.

Multiply numbers as if they were whole numbers.
Count the total number of decimal places in both factors.
Place the decimal point in the product so it has the same total number of decimal places from the factors.
Multiplying by a Power of $10$ : Move the decimal point to the right by the number of zeros in the power of $10$ .

Dividing by a Whole Number: Place the decimal point in the quotient directly above the decimal point in the dividend, then divide normally.
Dividing by a Decimal:
1. Move the decimal point in the divisor to the right to make it a whole number.
2. Move the decimal point in the dividend the same number of places to the right.
3. Place the decimal point in the quotient directly above the new dividend's decimal point, then divide normally.

Follow the same order as for whole numbers: Parentheses, Exponents, Multiplication/Division (left to right), Addition/Subtraction (left to right).