Number operations and integers Revision Notes

Addition: Combining Quantities

Addition is the process of combining two or more numbers to find their total sum. When adding numbers, especially larger ones, it's often helpful to align them vertically by place value (ones, tens, hundreds, etc.) and add columns from right to left, 'carrying over' to the next column when a sum exceeds 9. This systematic approach ensures accuracy. For example, to add 245 + 178: 245 + 178 ----- 423 (5+8=13, write 3 carry 1; 4+7+1=12, write 2 carry 1; 2+1+1=4). Basic addition facts are the foundation.

Subtraction: Finding the Difference

Subtraction is used to find the difference between two numbers or to take one quantity away from another. Similar to addition, aligning numbers by place value is crucial for larger numbers. When a digit in the top number is smaller than the digit below it, you 'borrow' from the digit in the next higher place value. For example, to subtract 321 - 187: 3 ¹11 ¹1

• 1 8 7

------------ 1 3 4 (1-7 not possible, borrow 1 from 2 to make 11; 11-7=4. 1-8 not possible, borrow 1 from 3 to make 11; 11-8=3. 2-1=1). Understanding borrowing is key.

Multiplication: Repeated Addition

Multiplication is a faster way to add the same number multiple times. It involves finding the product of two or more numbers. For single-digit multiplication, recall your times tables. For multi-digit numbers, you multiply each digit of one number by each digit of the other, typically using a grid method or long multiplication. For example, 12 × 3: 12 is 10 + 2 3 × 10 = 30 3 × 2 = 6 30 + 6 = 36. For 2-digit by 2-digit, like 23 × 14, multiply 23 by 4, then 23 by 10, and add the results. Place value alignment is still important during the addition step.

Division: Sharing and Grouping

Division is the process of splitting a number into equal parts or seeing how many times one number fits into another. It's the inverse operation of multiplication. Short division (or bus stop method) is commonly used for dividing by a single digit. Long division can be used for dividing by larger numbers. The result of division is called the quotient, and any leftover amount is the remainder. For example, 15 ÷ 3 = 5, as 3 goes into 15 exactly 5 times. For 37 ÷ 5, 5 goes into 37 seven times with a remainder of 2, so 7 R 2. Practice with mental division facts builds confidence for more complex problems.

Integers, Fractions, and Decimals

An 'integer' is any whole number (positive, negative, or zero), like -5, 0, or 10. This distinguishes them from 'fractions' (like 1/2 or 3/4) and 'decimals' (like 0.5 or 0.75) which represent parts of whole numbers. Recognising these categories is essential for number classification and understanding different number systems. For example, when a question asks for an 'integer solution', decimal or fractional answers are not valid.

Prime and Composite Numbers, Factors, Multiples

A 'prime number' is a whole number greater than 1 with only two factors: 1 and itself. Examples include 2, 3, 5, 7, 11. The number 1 is not prime. A 'composite number' is a whole number greater than 1 that is not prime (e.g., 4, 6, 8, 9, 10). 'Factors' are numbers that divide evenly into another number (e.g., factors of 10 are 1, 2, 5, 10). 'Multiples' are the results of multiplying a number by an integer (e.g., multiples of 4 are 4, 8, 12, 16...). These concepts are fundamental for topics like prime factorisation, LCM, and HCF.

Square, Cube, Even, and Odd Numbers

A 'square number' (or perfect square) is the product of an integer multiplied by itself (e.g., 1, 4, 9, 16...). A 'cube number' (or perfect cube) is the product of an integer multiplied by itself three times (e.g., 1, 8, 27, 64...). 'Even numbers' are integers divisible by 2 (e.g., -4, 0, 2, 6). 'Odd numbers' are integers not divisible by 2 (e.g., -3, 1, 5, 9). Knowing these terms helps in number properties and pattern recognition.

Expressions, Equations, and Formulae

An 'expression' is a combination of numbers, variables, and operations (e.g., 3x + 7, 5y^2). It does not contain an equals sign. An 'equation' is a statement that two expressions are equal (e.g., 3x + 7 = 10, y - 4 = 1). The goal with an equation is often to find the value(s) of the variable(s) that make the statement true. A 'formula' is a type of equation that expresses a rule or relationship between specific quantities, often using letters to represent them (e.g., Area (A) = Length (L) x Width (W)).

Definition of Prime Numbers

A prime number is a natural number (a positive whole number) greater than 1 that has exactly two positive divisors: 1 and the number itself. This means no other positive integer can divide it without leaving a remainder. For instance, 5 is a prime number because its only divisors are 1 and 5. Numbers such as 6 are not prime because they have divisors 1, 2, 3, and 6.

Key Characteristics and Exceptions

The number 1 is NOT a prime number, as it only has one divisor (itself), not two distinct divisors. The number 2 is the smallest and only even prime number. All other even numbers (4, 6, 8, etc.) are divisible by 2, and therefore have at least three divisors (1, 2, and the number itself), meaning they cannot be prime. To test if a number is prime, you can try dividing it by prime numbers less than or equal to its square root. If none of these divide it evenly, the number is prime.

Identifying Prime Numbers

To identify if a small number is prime, list its divisors. If there are exactly two (1 and itself), it's prime. For example, to check 13: Divisors are 1 and 13. So, 13 is prime. To check 15: Divisors are 1, 3, 5, 15. Since it has more than two divisors, 15 is not prime. Start with small prime numbers: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, and so on.

Definition of a Prime Number

A prime number is a natural number (a positive whole number) greater than 1 that has no positive divisors other than 1 and itself. This means it cannot be formed by multiplying two smaller positive integers. For example, 5 is prime because its only positive divisors are 1 and 5. Numbers such as 4 are not prime because they can be divided by 1, 2, and 4.

Key Facts about Prime Numbers

The number 1 is not a prime number because it only has one positive divisor (itself), not two distinct divisors. The number 2 is the smallest and only even prime number. All other even numbers are divisible by 2, meaning they have more than two divisors (1, 2, and themselves). The list of the first few prime numbers is 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31...

Identifying Prime Numbers

To identify if a number is prime, you can systematically test for divisibility by all integers greater than 1 and less than or equal to the square root of that number. If the number is not divisible by any of these, then it is prime. For instance, to check if 37 is prime, calculate the square root of 37, which is approximately 6.08. You only need to test for divisibility by prime numbers up to 5 (2, 3, 5). Since 37 is not divisible by 2, 3, or 5, it is a prime number.