IB Math AI SL – Topic 1: Number & Algebra

What does PEMDAS stand for, and what is it used for?

Parentheses, Exponents, Multiplication/Division, Addition/Subtraction — the order to solve a math expression in

In PEMDAS, do multiplication/division or addition/subtraction happen in a fixed order relative to each other?

No, whichever comes first left to right within each pair

What is a system of equations?

A set of two or more equations that share variables and are solved together

What does the solution to a system of two linear equations represent graphically?

The point where the two lines intersect

What are the two main algebraic methods for solving a system of equations without a calculator?

Substitution and elimination

Briefly, how does the substitution method work?

Solve one equation for one variable, then plug that expression into the other equation

Briefly, how does the elimination method work?

Add or subtract the two equations so one variable cancels out, leaving one variable to solve for

Before adding or subtracting equations in elimination, what must be true about one variable's coefficients?

Same number, opposite sign, so they cancel to zero when combined

If no variable already has matching opposite coefficients, what do you do before eliminating?

Multiply one or both equations by a number so one variable's coefficients become equal and opposite

Worked example: solve x + y = 10 and x - y = 2 using elimination. What's the first step?

Add the two equations directly since y and -y already cancel: (x+y)+(x-y)=10+2, giving 2x=12

After solving for one variable in a system, how do you find the other variable?

Substitute that value back into either original equation and solve

What does LCM stand for and mean?

Least Common Multiple, the smallest number that two numbers both divide into evenly

Why is the LCM useful when adding or subtracting fractions?

It gives a common denominator so the fractions can be combined

What are the steps to multiply two fractions?

Multiply the numerators together, then multiply the denominators together

How do you multiply a fraction by a whole number, like 1/3 times 6?

Put the whole number over 1, so 6/1, then multiply straight across: (1x6)/(3x1) = 6/3 = 2

How do you divide by a fraction?

Multiply by its reciprocal instead

What is the reciprocal of a fraction?

The fraction flipped upside down, numerator and denominator swapped

Do you need to memorize the laws of exponents for AI SL, or are they given?

You must memorize them — they are not on the formula sheet

What is the product rule for exponents?

a^m times a^n = a^(m+n): add the exponents when multiplying same-base powers

What is the quotient rule for exponents?

a^m divided by a^n = a^(m-n): subtract the exponents when dividing same-base powers

What is the power rule for exponents?

(a^m)^n = a^(m times n): multiply the exponents when raising a power to a power

What does any nonzero number raised to the power of 0 equal?

1

What does a negative exponent mean, like a^-n?

It means 1 over a^n, i.e. flip it into the denominator

Your formula sheet defines a logarithm as log_a(b) = x if and only if a^x = b. Do you need to memorize this relationship?

No, it's given on the formula sheet, but you need to know how to use it

The compound interest formula is given on your formula sheet as FV = PV(1 + r/100k)^(kn). What does each variable stand for?

FV is future value, PV is present value, n is number of years, k is compounding periods per year, r% is the annual interest rate

The percentage error formula is given as |(approximate - exact) / exact| times 100%. What does it measure?

How far an estimated value is from the true value, as a percentage

What is scientific notation (also called standard form)?

A way to write numbers as a x 10^k, where 1 is less than or equal to a, which is less than 10, and k is an integer

When multiplying two numbers in scientific notation, what do you do?

Multiply the two decimal parts together, and add the exponents

When dividing two numbers in scientific notation, what do you do?

Divide the two decimal parts, and subtract the exponents

When adding or subtracting numbers in scientific notation, what must you do first?

Rewrite both numbers so they have the same power of 10, then add or subtract the decimal parts

If multiplying or dividing in scientific notation gives you a decimal part outside the 1-10 range, like 0.9 x 10^17, what should you do?

Adjust it back into proper form by shifting the decimal and changing the exponent, e.g. 0.9 x 10^17 becomes 9 x 10^16

What are significant figures?

The digits in a number that carry meaning toward its precision, starting from the first nonzero digit

When rounding a number like 8350 to 1 significant figure, which digit do you look at to decide rounding direction?

The digit right after the one you're keeping (so for 8350 to 1 sig fig, you look at the 3, which rounds down, giving 8000)

What does it mean for a measurement to be "accurate to 3 significant figures"?

The true value was rounded to the nearest unit at that level of precision, so the real value could be slightly higher or lower

What is the general idea behind upper and lower bounds for a rounded number?

The true value lies within half of the rounding precision above or below the rounded value

If 1170 is correct to 3 significant figures, what does that tell you about its rounding precision?

It was rounded to the nearest 10, since the first 3 significant digits are 1, 1, and 7

How do you find the lower and upper bounds of a rounded value?

Subtract half the rounding unit for the lower bound, and add half the rounding unit for the upper bound

Why would a word problem ask for a "lower bound" of a calculated value like volume?

Because using the smallest possible input value gives the smallest possible result, showing the full range of what the true answer could be

When you have an expression like 2^(x+4) - 2^(x+2) + 2^(x-1), what's the first move to simplify it?

Factor out the smallest power of 2 raised to x (2^x), rewriting each term as 2^x times a plain number

Why does factoring out 2^x work when the terms have different exponents like x+4, x+2, and x-1?

Because each exponent is x plus or minus a number, so 2^(x+n) always equals 2^x times 2^n by the product rule

When rounding a decimal like 3.6495 to 2 decimal places, what's the rule?

Look at the digit right after the place you're rounding to; 5 or higher rounds up, so 3.6495 rounds up to 3.65

What is the reciprocal of a fraction?

Flip it: swap the numerator and denominator

What is the reciprocal of a whole number, like 5?

1 over that number (so 5 becomes 1/5), since a whole number is secretly that number over 1

How do you divide by a fraction?

Multiply by its reciprocal instead

What is 6 divided by 1/2?

12 — multiply 6 by the reciprocal of 1/2, which is 2

When you rewrite a term like 32x^12 so it shares a denominator with -4/x^6, what are you actually doing?

Multiplying it by a form of 1 (like x^6/x^6) so it becomes an equivalent fraction with that denominator

What happens when you multiply two negative numbers?

The result is positive

What happens when you multiply a positive number and a negative number?

The result is negative

What happens when you divide two negative numbers?

The result is positive

What happens when you subtract a negative number, like 5 - (-3)?

It becomes addition: 5 + 3 = 8

What does the distributive property say about a(b + c)?

It equals ab + ac — multiply a by each term inside the parentheses

What are "like terms" in algebra?

Terms with the exact same variable(s) raised to the exact same power(s), e.g. 3x and 7x

Can you combine 3x and 5x^2 into one term?

No — they're not like terms, since the exponents on x are different

How do you simplify an expression like 3x + 5 + 2x - 1?

Combine the like terms separately: (3x+2x) gives 5x, and (5-1) gives 4, so the result is 5x + 4

What is a factor of a number?

A whole number that divides into it evenly with no remainder

What is the greatest common factor (GCF) of two numbers?

The largest number that divides evenly into both of them

How do you simplify a fraction like 8/12 to lowest terms?

Divide both the numerator and denominator by their greatest common factor (4 here), giving 2/3

How do you add two fractions with different denominators, like 1/3 + 1/4?

Rewrite both over a common denominator (12 here: 4/12 + 3/12), then add the numerators to get 7/12

What is cross-multiplication used for?

Solving an equation where two fractions are set equal to each other, by multiplying diagonally across the equals sign

What is an arithmetic sequence?

A sequence where each term increases or decreases by the same constant amount, called the common difference

What is the common difference (d) in an arithmetic sequence?

The fixed amount added (or subtracted) to get from one term to the next

Your formula sheet gives the nth term of an arithmetic sequence as un = u1 + (n-1)d. Do you need to memorize this?

No, it's given on the formula sheet

Your formula sheet gives two forms for the sum of an arithmetic series. When do you use Sn = n/2(u1+un) versus Sn = n/2(2u1+(n-1)d)?

Use the first when you know the first and last term; use the second when you only know the first term and common difference

If you're given two non-consecutive terms of an arithmetic sequence, how do you find the common difference?

Subtract the two terms and divide by the number of steps (term-number gap) between them

What does it mean for real-world data to be "approximately arithmetic"?

The differences between consecutive data points aren't perfectly equal, but close enough to model as arithmetic

How do you choose a common difference to model data that is only approximately arithmetic?

Average the individual differences between consecutive data points

How do you find the number of terms in an arithmetic sequence given the first term, last term, and common difference?

Rearrange un = u1 + (n-1)d to solve for n

If you know two terms of an arithmetic sequence (like u3 and u10), how do you find u1 and d?

Write each term as un = u1 + (n-1)d, giving two equations with two unknowns, then solve as a system of equations

What is a geometric sequence?

A sequence where each term is multiplied by the same constant ratio to get the next term

What is the common ratio (r) in a geometric sequence?

The fixed number each term is multiplied by to get the next term

Your formula sheet gives the nth term of a geometric sequence as un = u1 r^(n-1). Do you need to memorize this?

No, it's given

Your formula sheet gives the sum of n terms of a geometric series as Sn = u1(r^n - 1)/(r-1). Do you need to memorize this?

No, it's given

If you're given two non-consecutive terms of a geometric sequence, how do you find the common ratio?

Divide the later term by the earlier term, then take the appropriate root based on the gap between them

How do you model something growing by a percentage each year, like 8% growth, as a geometric sequence?

The common ratio is 1 plus the percentage as a decimal, so 8% growth gives r = 1.08

How do you model something losing value by a percentage each year, like 8% depreciation?

The common ratio is 1 minus the percentage as a decimal, so 8% depreciation gives r = 0.92

What does sigma (Σ) notation represent?

A compact way to write the sum of a sequence of terms following a pattern, from a starting index to an ending index

In sigma notation for a sum, what do the numbers above and below the Σ mean?

The bottom number is the starting term number, the top number is the ending term number

When writing a geometric series in sigma notation, why does the exponent usually appear as (n-1) instead of n?

So the first term (n=1) uses the ratio raised to the power 0, which equals 1, matching the original first term unchanged

What is simple interest?

Interest calculated only on the original amount invested, so the same fixed amount is added every period

How do you calculate total value after simple interest over several years?

Original amount plus (original amount times rate times number of years)

What's the key difference between simple interest and compound interest as sequences?

Simple interest adds a fixed amount each period (arithmetic); compound interest multiplies by a fixed ratio each period (geometric), so it grows faster over time

When simplifying an expression like (3y^4)^3 + 2y^7 - 4y^12, what's the first move?

Apply the power rule to expand (3y^4)^3 into a single term, then check which terms share the same exponent so they can combine

Why can't 2y^7 and -4y^12 be combined into a single term?

They aren't like terms — y^7 and y^12 are different powers, even though they share the same base