OSAT (182)

The base-ten numeration system is the foundation of all arithmetic. In this system, each digit's value depends on its position within the number. The rightmost position represents ones, the next position to the left represents tens, then hundreds, thousands, and so forth. Each position is exactly ten times the value of the position immediately to its right.

The digit 0 serves as a placeholder, indicating that a particular place value has no units. For example, in 305, the zero indicates no tens, distinguishing this number from 35 or 350.

Place value understanding extends in both directions. Moving left, values increase by powers of ten. Moving right of the ones place (into decimals), values decrease by powers of ten: tenths, hundredths, thousandths.

Standard form presents numbers using digits (4,582). Word form expresses numbers in words (four thousand five hundred eighty-two). Expanded form shows the value of each digit (4,000 + 500 + 80 + 2). Expanded notation using powers of ten represents this as (4 × 10³) + (5 × 10²) + (8 × 10¹) + (2 × 10⁰).

When reading large numbers, group digits in sets of three from right to left, separated by commas. Each group of three has a period name: ones, thousands, millions, billions, trillions. Within each period, read the three-digit number followed by the period name.

To compare whole numbers, first examine the number of digits. The number with more digits is greater. When digit counts are equal, compare digits from left to right. The first position where digits differ determines which number is greater.

Symbols for comparison: greater than (>), less than (<), equal to (=), not equal to (≠), greater than or equal to (≥), less than or equal to (≤).

Ordering numbers means arranging them from least to greatest (ascending) or greatest to least (descending). Number lines provide visual representations for comparing and ordering, with numbers increasing from left to right.

Rounding replaces a number with a nearby number that has fewer significant digits, making calculations easier to perform or estimate. To round to a given place: identify the rounding digit, examine the digit immediately to its right, if this digit is 5 or greater round up (increase the rounding digit by 1), if less than 5 round down (keep the rounding digit unchanged), replace all digits to the right with zeros.

Rounding 3,847 to the nearest hundred: the hundreds digit is 8, the digit to its right is 4, since 4 < 5, round down, result is 3,800.

Factors of a number divide it evenly with no remainder. The number 12 has factors 1, 2, 3, 4, 6, and 12. Multiples of a number are products of that number with whole numbers. Multiples of 4 include 4, 8, 12, 16, 20, and so on.

Prime numbers have exactly two factors: 1 and the number itself. The first several primes are 2, 3, 5, 7, 11, 13, 17, 19, 23, 29. Note that 2 is the only even prime. The number 1 is neither prime nor composite.

Composite numbers have more than two factors. Every composite number can be expressed as a unique product of prime factors (Fundamental Theorem of Arithmetic). Finding the prime factorization involves repeatedly dividing by prime numbers until only 1 remains.

The Greatest Common Factor (GCF) of two or more numbers is the largest number that divides all of them evenly. Find GCF by listing factors of each number and identifying the largest common one, or by using prime factorization and multiplying common prime factors with lowest exponents.

The Least Common Multiple (LCM) of two or more numbers is the smallest positive number that is a multiple of all of them. Find LCM by listing multiples until a common one appears, or by using prime factorization and multiplying all prime factors with highest exponents.

Quick tests determine whether a number is divisible by small integers without performing division. Divisible by 2: last digit is even (0, 2, 4, 6, 8). Divisible by 3: sum of digits is divisible by 3. Divisible by 4: last two digits form a number divisible by 4. Divisible by 5: last digit is 0 or 5. Divisible by 6: divisible by both 2 and 3. Divisible by 8: last three digits form a number divisible by 8. Divisible by 9: sum of digits is divisible by 9. Divisible by 10: last digit is 0.