Mathematics Beginner

Every whole number is built from the digits 0–9. The position of each digit tells us its place value. In the number 3 457, the digit 3 is in the thousands place, 4 in the hundreds, 5 in the tens, and 7 in the ones.

• Ones (1s) — the rightmost digit.
• Tens (10s) — one step to the left.
• Hundreds (100s) — two steps to the left.
• Thousands (1 000s) — three steps to the left.

3 457  =  3 × 1 000  +  4 × 100  +  5 × 10  +  7 × 1
       =  3 000 + 400 + 50 + 7

We can compare numbers by looking at the highest place value first. 4 201 > 3 999 because 4 thousands > 3 thousands.

Addition (+) combines two amounts to find a total. Subtraction (−) finds the difference between two amounts. Both operations work column by column, starting from the ones place.

• When a column adds up to 10 or more, carry 1 to the next column (addition).
• When a column would give a negative digit, borrow 1 from the next column (subtraction).

  Addition:         Subtraction:
    3 4 7             5 0 0
  + 2 8 5           −  1 7 4
  -----             -----
    6 3 2             3 2 6

Check addition by subtracting one addend from the sum: 632 − 285 = 347 ✓. The commutative property means a + b = b + a, so order does not matter in addition.

Multiplication (×) is repeated addition. Division (÷) splits a number into equal groups. Knowing your times tables (1–10) makes both operations fast.

• 6 × 7 = 42  (six groups of seven)
• 42 ÷ 7 = 6  (42 shared into groups of 7)
• Commutative: 4 × 9 = 9 × 4 = 36.
• Division check: quotient × divisor + remainder = dividend.

Long multiplication:    47 × 6
  47 × 6:
    7 × 6 = 42 → write 2, carry 4
    4 × 6 = 24, + 4 = 28
  Answer: 282

Long division:   95 ÷ 4
  4 goes into 9 twice (8); remainder 1.
  Bring down 5 → 15; 4 goes into 15 three times (12); remainder 3.
  Answer: 23 remainder 3

A fraction shows a part of a whole. The bottom number (the denominator) tells how many equal parts the whole is cut into; the top number (the numerator) tells how many parts we have.

• ½ — one out of two equal parts.
• ¼ — one out of four equal parts.  ¾ — three out of four.
• Equivalent fractions: ½ = 2⁄4 = 4⁄8 — same size, different notation.
• To compare fractions with the same denominator, compare numerators: ¾ > ¼.

Compare ½ and ¾:
  Write with a common denominator of 4:
  ½ = 2⁄4
  2⁄4 < ¾   →   ½ < ¾

A proper fraction has a numerator smaller than the denominator (value < 1). An improper fraction like ⁵⁄₄ = 1¼ has numerator ≥ denominator.

Decimals extend place value to the right of a decimal point. The first place is tenths (÷10), the second is hundredths (÷100).

• 0.5 = ½ = 5 tenths.
• 0.25 = ¼ = 25 hundredths.
• Money uses two decimal places: $3.75 = 3 dollars and 75 cents.

Add money:   $2.49 + $1.85
    $2.49
  + $1.85
  ------
    $4.34

Line up decimal points before adding or subtracting decimals. Multiplying a decimal by 10 moves the decimal point one place to the right: 0.7 × 10 = 7.

Rounding makes numbers easier to work with. To round to the nearest ten, look at the ones digit:

• If the ones digit is 0–4, round down (keep the tens digit).
• If the ones digit is 5–9, round up (add 1 to the tens digit).

Round to the nearest 10:
  43  →  40   (ones digit 3 < 5, round down)
  78  →  80   (ones digit 8 ≥ 5, round up)
  85  →  90   (ones digit 5 ≥ 5, round up)

Estimation uses rounded numbers to get a quick approximate answer. 49 + 32 ≈ 50 + 30 = 80. Estimation is useful for checking whether an exact answer is reasonable.

A number is even if it can be divided by 2 with no remainder. A number is odd if dividing by 2 leaves a remainder of 1.

• Even numbers end in 0, 2, 4, 6, or 8.
• Odd numbers end in 1, 3, 5, 7, or 9.
• Even + Even = Even; Odd + Odd = Even; Even + Odd = Odd.

Skip-counting patterns:
  Count by 2s: 2, 4, 6, 8, 10, 12 …
  Count by 5s: 5, 10, 15, 20, 25 …
  Count by 10s: 10, 20, 30, 40, 50 …

Look at a sequence and find the rule. In 3, 6, 9, 12, … the rule is "add 3 each time." The next term is 15.

A 2-D shape is flat (two-dimensional). Common shapes:

• Triangle — 3 sides, 3 angles.
• Rectangle — 4 sides; opposite sides are equal; all angles are 90°.
• Square — a rectangle with all 4 sides equal.
• Circle — a perfectly round shape; no sides.

The perimeter is the total distance around a shape — add up all the side lengths.

Perimeter of a rectangle:
  length = 8 cm,  width = 5 cm
  P = 8 + 5 + 8 + 5 = 26 cm
  (or: P = 2 × (8 + 5) = 2 × 13 = 26 cm)

We use standard units to measure the world around us.

• Length: millimetres (mm), centimetres (cm), metres (m), kilometres (km).  1 m = 100 cm; 1 km = 1 000 m.
• Weight (mass): grams (g), kilograms (kg).  1 kg = 1 000 g.
• Time: seconds, minutes, hours, days, weeks, months, years.  1 minute = 60 seconds; 1 hour = 60 minutes; 1 day = 24 hours.

Conversion examples:
  2.5 m  = 2.5 × 100 cm = 250 cm
  3 kg   = 3 × 1 000 g  = 3 000 g
  2 hours = 2 × 60 min  = 120 minutes