Argand plane — common mistake

“Imagine drawing a wind vector on a weather map; the east-west component is the x-axis and the north-south component is the y-axis – just like real and imaginary parts.”

The correct plot is (a,b): a on the x-axis, b on the y-axis. Plotting the imaginary part on the x-axis rotates the point, altering the argument and possibly the quadrant, though the modulus √(a²+b²) remains unchanged. Trap: a wrong axis placement gives a wrong angle, not a wrong length.

Memory trick: i goes vertical

|z| = √(a²+b²)

“The modulus of a+bi is the diagonal of a right-triangle with sides a and b, just like the hypotenuse in Pythagoras.”

|z| = √(a²+b²) gives the distance from the origin to (a,b). It follows directly from the Pythagorean theorem. Remember that signs of a or b do not affect the modulus; only their squares matter.

|z| = √(a²+b²)

Memory trick: Pythagoras for complex

z·z̄ = |z|²

“To simplify (3+4i)/(2−i), you multiply top and bottom by 2+i; the denominator becomes (2)²+(1)² = |2−i|², a pure real number.”

For any z = a+bi, z·z̄ = (a+bi)(a−bi)=a²+b² = |z|², a real quantity. This identity is used to rationalise denominators and to find the modulus. Trap: forgetting the minus sign in the conjugate gives (a+bi)², which is not real.

z·z̄ = |z|²

Memory trick: Conjugate flips sign → a²+b²

real part vs imaginary part

“Like a GPS coordinate, a+bi’s ‘a’ tells how far east-west you are, while ‘b’ tells the north-south height on the Argand map.”

In a+bi, a is the real part plotted on the horizontal axis and b is the imaginary part plotted on the vertical axis. The real part is the x-coordinate, the imaginary part the y-coordinate. Beware: the imaginary part is b, not bi; i belongs to the axis, not the value.

Memory trick: Real = R → Right-left axis; Imag = I → Up-ward axis.

modulus vs argument

“A boat’s rope length from the dock is the modulus, and the angle it makes with the dock’s line is the argument of the complex number.”

Modulus |z| = √(a²+b²) is the distance from the origin. Argument θ = atan2(b,a), which automatically accounts for the correct quadrant; using tan⁻¹(b/a) alone can give the wrong principal value when a<0 or a=0. Trap: ignore quadrant → wrong θ.

Memory trick: θ = atan2(b,a) – check quadrant

Myth: √(−1)·√(−1) = 1

“When you flip a coin twice, you still get heads or tails—not a neutral side; likewise √(-1)·√(-1) stays –1, not 1.”

The rule √a·√b = √(ab) holds only for non-negative real a and b. For negative radicands we define √(-1)=i (principal root), so i·i = i² = -1. Using the product rule with negatives gives the wrong answer, a frequent exam trap.

Memory trick: √a·√b = √(ab) only if a,b ≥0

Arithmetic progression — common mistake

“If you start at 5 m and move 3 m each minute, after 4 minutes you are at 5+3×(4-1)=14 m, not 5+3×4.”

In an AP the nth term is a+(n-1)d because the first term is counted as step 1. Using n instead of n-1 overestimates the term. Ensure you count the number of differences, not the number of terms.

Memory trick: n-1 = steps after the first term.

Geometric progression — common mistake

“Doubling a bacteria culture each hour multiplies the count by 2 every step, never adds 2.”

In a GP each term is obtained by multiplying the previous term by the common ratio r, so Tₙ = a·r^(n-1). Adding r each step is wrong. The trap is to treat a linear progression as geometric.

Memory trick: GP = start × (ratio)^(steps).

Infinite geometric series

“When you keep adding half the previous amount of water to a tank, the total water approaches a/(1-r).”

For a GP a, ar, ar²… the sum S = a/(1−r) only if |r|<1, because then the terms shrink to zero and the series converges. If |r|≥1 the terms do not vanish, the series diverges and the formula fails.

Memory trick: Finite sum ⇔ |r|<1 (|r| less than one, series settles)

Infinite geometric series — common mistake

“An echo that loses 30 % of its loudness each bounce has r=0.7; if r≥1 the total sound energy would diverge.”

A common error is using the infinite-GP sum formula when |r|≥1. The series a/(1-r) is valid only for |r|<1; otherwise the terms do not diminish and the sum is infinite.

Memory trick: Never apply a/(1-r) when |r|≥1.

aₙ = a + (n−1)d

“Like climbing stairs where each step is exactly 3 cm higher, the nth step height follows aₙ = a + (n−1)d.”

The nth term of an arithmetic progression (AP) is aₙ = a + (n−1)d, where a is the first term and d the constant difference. It applies only when consecutive terms differ by the same amount. Using it on a geometric pattern gives a wrong answer.

aₙ = a + (n−1)d

Memory trick: AP = Add Progression → add d each step

aₙ = a·r^(n−1)

“A bank account that doubles every year follows aₙ = a·r^(n−1).”

For a geometric progression (GP) the nth term is aₙ = a·r^(n−1), with a the first term and r the common ratio. It works only when each term is obtained by multiplying the previous one by the same r. Mixing r with d is a common error.

aₙ = a·r^(n−1)

Memory trick: GP = Multiply Progression → multiply by r each step

S∞ = a/(1−r)

“A leaking bucket losing half its water each minute has total water = a/(1−½).”

An infinite geometric series sums to S∞ = a/(1−r) only if |r| < 1. If |r| ≥ 1 the terms do not shrink enough and the series diverges. Forgetting the |r| < 1 condition yields a false finite answer.

S∞ = a/(1−r)

Memory trick: Converge only when |r|<1 – ‘less than one’ keeps shrinking

arithmetic progression vs geometric progression

“Walking up a staircase adds a fixed height each step, while a car that doubles speed each hour multiplies – AP vs GP.”

In an AP each term increases by a fixed addition d; in a GP each term increases by a fixed multiplication r. The pattern of constant differences versus constant ratios distinguishes them. Using the wrong formula is a typical trap.

Memory trick: Add → AP, Multiply → GP

finite series vs infinite series

“Adding the prices of 5 books gives a total, but an endless stream of coins adds up only if each coin’s value keeps shrinking.”

A finite series has a definite number of terms, so its sum is always calculable. An infinite series has a sum only when it converges, e.g., a GP with |r| < 1; otherwise it diverges. Writing S = a/(1−r) without checking convergence is wrong.

Memory trick: Finite = countable, Infinite = need convergence check

Myth: Infinite GP always sums

“When a ball drops from height a and rebounds to r·a each bounce, the total distance is a(1+r)/(1−r), which only works because |r|<1.”

An infinite geometric series Σ a rⁿ converges to a/(1−r) only when the common ratio satisfies |r|<1; otherwise the terms never vanish and the series diverges. The exam trap is using a/(1−r) without first checking the |r|<1 condition.

Memory trick: Check |r|<1 first, then apply a/(1−r).

Permutation — common mistake

“Arranging 3 books on a shelf gives a different order each time you shuffle them.”

nPr = n!/(n−r)! counts ordered arrangements of r items from n. Use when the sequence matters, like seat assignments. The common trap is applying nCr, which ignores order and gives a smaller count.

Memory trick: P = order, C = no order.

Combination — common mistake

“Selecting 3 friends for a team, you don't care who stands left or right.”

nCr = n!/[r!(n−r)!] counts selections where order is irrelevant. It’s the right tool for committees or lottery tickets. Mistaking it for nPr overcounts by a factor of r!.

Memory trick: C = choose, ignore order.

Fundamental principle of counting

“Picking a shirt (4 colors) and shoes (3 styles) makes 4×3 =12 distinct outfits.”

If one task has m ways and another independent task has n ways, total ways = m×n (Fundamental Counting Principle). The exam trap is adding when both choices must happen together.

Memory trick: AND → ×, OR → +.

Fundamental principle of counting — common mistake

“A 2-letter then 2-digit password uses 26×26×10×10 possibilities, not a sum.”

When choices are independent and must all occur, multiply their counts; adding applies only to mutually exclusive options. Forgetting this inflates or deflates the answer dramatically.

Memory trick: AND → ×, OR → +.

nPr = n!/(n−r)!

“How many 4-digit PINs without repeating digits? Use nPr with n=10, r=4.”

nPr = n!/(n−r)! gives ordered arrangements without repetition, useful for codes, seating, or arranging objects in a line. Using nCr here ignores order and yields far fewer possibilities.

nPr = n!/(n−r)!

Memory trick: P = n!/(n−r)!.

nCr = n!/[r!(n−r)!]

“Choosing 3 toppings from 10 for a pizza, the order you place them doesn’t matter, so you use nCr.”

nCr = n!/[r!(n−r)!] gives the number of ways to select r objects from n without regard to order. It is obtained by dividing the total permutations nPr by r! because each selection can be arranged in r! ways. Use it only when order is irrelevant.

nCr = n!/[r!(n−r)!]

Memory trick: nCr = nPr ÷ r! → divide permutations by internal order

permutation vs combination

“A 4-digit password (1234) changes if you swap digits, but picking 4 players for a team does not.”

Permutation counts ordered arrangements; combination counts unordered selections. nPr = n!/(n−r)! and nCr = n!/[r!(n−r)!] = nPr / r!. Use permutation when sequence matters, combination when it doesn’t.

Memory trick: P = C × r! → order = combination × internal swaps

with repetition vs without repetition

“When setting a 4-digit PIN you can repeat digits, but assigning seats in a row to 4 distinct people you cannot repeat a person.”

With repetition each choice can be reused, giving n^r possibilities for r positions. Without repetition each item is used at most once, giving nPr = n!/(n−r)!. Pick the formula that matches whether reuse is allowed.

Memory trick: Rep → n^r; No rep → nPr