Number (AQA)

Types of Numbers

• Integers: Whole numbers, including positive numbers, negative numbers, and zero.

  • Examples: -3, -2, -1, 0, 1, 2, 3

• Natural Numbers: Positive whole numbers (counting numbers).

  • Examples: 1, 2, 3, 4... (Note: 0 is not included)

• Rational Numbers: Numbers that can be expressed as a fraction fracpq where p and q are integers, and q is not zero.

  • Includes terminating decimals (e.g., 0.5 = frac12) and recurring decimals (e.g., 0.333... = frac13).

  • Example: frac34, -2, 0.75, 0.1666...

• Irrational Numbers: Numbers that cannot be expressed as a fraction. They are non-terminating and non-recurring decimals.

  • Examples: pi (pi), sqrt2 (square root of 2)

Number Operations

• Mathematical Operations: Addition, subtraction, multiplication, and division.

• Order of Operations (BIDMAS/BODMAS): A set of rules that dictate the order in which operations should be performed in a mathematical expression.

  • Brackets

  • Indices (Powers, Square Roots, etc.) or Orders

  • Division

  • Multiplication

  • Addition

  • Subtraction

  • Example: Calculate 2+3times(6−4)2

    • Brackets: 6−4=2

    • Indices: 22=4

    • Multiplication: 3times4=12

    • Addition: 2+12=14

    • Therefore, 2+3times(6−4)2=14

• Negative Numbers: Numbers less than zero, indicated with a minus sign (-).

  • Rules for addition and subtraction:

    • Same signs: Replace with a positive sign.

    • Different signs: Replace with a negative sign.

    • Same signs (answer): Positive result.

    • Different signs (answer): Negative result.

  • Example: −5+(−3)=−8 (Same signs, add the numbers and keep the negative sign)

  • Example: 7+(−2)=5 (Different signs, subtract the smaller number from the larger number and use the sign of the larger number)

Factors, Multiples, and Primes

• Multiples: The numbers you get when you multiply a number by an integer.

  • Example: Multiples of 3 are 3, 6, 9, 12, 15...

• Factors: Numbers that divide exactly into another number without leaving a remainder.

  • Example: Factors of 12 are 1, 2, 3, 4, 6, 12.

• Prime Numbers: Numbers that have only two factors: 1 and themselves.

  • Examples: 2, 3, 5, 7, 11, 13, 17, 19, 23...

• Prime Factor Decomposition: Expressing a number as a product of its prime factors.

  • Example: Prime factor decomposition of 12: 12=2times2times3=22times3

• Highest Common Factor (HCF): The largest factor that is common to two or more numbers.

• Lowest Common Multiple (LCM): The smallest multiple that is common to two or more numbers.

Powers and Roots

• Powers (Indices): A way of writing repeated multiplication of a number by itself.

  • Example: 23=2times2times2=8

• Roots: The inverse operation of raising to a power.

  • Example: sqrt9=3 (square root of 9 is 3 because 32=9)

• Laws of Indices: Rules for simplifying expressions with powers.

  • Multiplication: amtimesan=am+n

  • Division: amdivan=am−n

  • Power of a power: (am)n=amtimesn

  • Fractional indices: afracmn=(sqrt[n]a)m

  • Negative indices: a−n=frac1an

Standard Form

• A way of writing very large or very small numbers in the form Atimes10n, where 1 ≤ A < 10 and n is an integer.

  • Example: 25,000 in standard form is 2.5times104

  • Example: 0.0003 in standard form is 3times10−4

• Operations with Standard Form: Rules for adding, subtracting, multiplying, and dividing numbers in standard form.

Fractions

• Basic Fractions: Represent parts of a whole.

  • Numerator (top number): Indicates how many parts you have.

  • Denominator (bottom number): Indicates how many parts the whole is divided into.

• Equivalent Fractions: Fractions that represent the same value, even though they have different numerators and denominators.

  • Example: frac12=frac24=frac36

• Simplified Fractions: Fractions reduced to their lowest terms by dividing the numerator and denominator by their highest common factor.

• Mixed Numbers: A whole number and a fraction combined.

  • Example: 2frac12

• Improper Fractions: Fractions where the numerator is greater than or equal to the denominator.

  • Example: frac52

• Adding and Subtracting Fractions: Requires a common denominator.

• Multiplying Fractions: Multiply numerators together and denominators together.

• Dividing Fractions: Multiply the first fraction by the reciprocal of the second fraction.

Percentages

• "Percent" means "out of one hundred."

• Percentages of Amounts: Calculating a percentage of a given quantity.

  • Example: 20% of 50 = frac20100times50=10

• Percentage Increase and Decrease: Calculating the percentage change when a quantity increases or decreases.

• Percentage Change: fractextChangetextOriginalAmounttimes100

• Reverse Percentages: Finding the original amount after a percentage increase or decrease.

Simple and Compound Interest

• Simple Interest: Interest calculated only on the principal amount.

• Compound Interest: Interest calculated on the principal amount and also on the accumulated interest of previous periods.

• Depreciation: A decrease in the value of an asset over time.

• Exponential Growth and Decay: Growth or decay at a constant percentage rate over time.

Fractions, Decimals, and Percentages

• Converting between Fractions, Decimals, and Percentages: Being able to convert fluently between these three forms.

  • Fraction to Decimal: Divide the numerator by the denominator.

  • Decimal to Percentage: Multiply by 100.

  • Percentage to Decimal: Divide by 100.

  • Decimal to Fraction and Percentage to Fraction also require understanding of place value and simplification.

• Ordering Fractions, Decimals, and Percentages: Being able to arrange a set of fractions, decimals, and percentages in ascending or descending order (often easiest to convert them all to decimals or percentages first).

Rounding, Estimation, and Error Intervals

• Rounding: Approximating a number to a given place value or number of significant figures.

  • Rounding to decimal places, rounding to significant figures, rounding to the nearest whole number.

• Estimation: Making a reasonable guess or approximation of a value.

• Error Intervals (Bounds): The range of possible values within which the true value lies after rounding.

  • Upper bound and lower bound.

Surds

• An expression that includes a root (usually a square root) that cannot be simplified to a whole number.

  • Examples: sqrt2, sqrt3, sqrt5

• Simplifying Surds: Factoring out perfect square factors from under the root.

  • Example: sqrt12=sqrt4times3=sqrt4timessqrt3=2sqrt3

• Rationalising Denominators: Removing surds from the denominator of a fraction.