1.1 Number and algebra - SL content

Normalize a number into scientific notation.

Move the decimal point until the coefficient is at least 1 and less than 10; the number of places becomes the integer exponent.

Worked example
0.00052 = 5.2 × 10⁻⁴ because the decimal moves four places right.

Worked example
Write 7.1 × 10⁵ as an ordinary number. 710000; a positive exponent moves the decimal right.

Common boundary
E notation on a calculator is a display format, not the required written form.

Use the common difference to model an arithmetic sequence.

An arithmetic sequence changes by the same additive amount each step: aₙ=a₁+(n−1)d, and a finite sum is n/2[2a₁+(n−1)d].

Worked example
For 4, 7, 10, the difference is 3, so a₆=4+5(3)=19.

Worked example
What stays constant? the difference, not the ratio.

Common boundary
Do not use a geometric formula when the change is additive.

Complete finite-sum and sigma example: for $4,7,10,\ldots$, $a_1=4$ and $d=3$, so $a_n=4+3(n-1)$. For six terms, $S_6=\frac{6}{2}[2(4)+5(3)]=69$. The same sum is $\sum_{k=1}^{6}(3k+1)=69$. Use $a_n$ for one term and $S_n$ or $\Sigma$ for a total.

Use the common ratio to model geometric growth.

A geometric sequence multiplies by the same ratio: aₙ=a₁rⁿ⁻¹; finite sums use repeated multiplication.

Worked example
For 5, 10, 20, r=2, so a₅=5·2⁴=80.

Worked example
Is 3, 6, 12 arithmetic or geometric? geometric, because each term is multiplied by 2.

Common boundary
Equal increases do not make a sequence geometric; check multiplication.

Complete finite-series example: for $3,6,12,24,48$, $a_1=3$, $r=2$ and $n=5$. $S_5=\frac{a_1(1-r^n)}{1-r}=\frac{3(1-2^5)}{1-2}=93$, equivalently $\sum_{k=1}^{5}3\cdot2^{k-1}=93$. This finite formula is valid for $r\ne1$; if $r=1$, $S_n=na_1$.

Model compound interest or depreciation with repeated multiplication.

A percentage change each period is geometric: A=P(1±r)ⁿ, with the sign chosen for growth or depreciation.

Worked example
$1000 at 5$1102.50.

Worked example
Why is 5% added twice not 10% of the original? the second 5% applies to the new balance.

Common boundary
Compound percentage change is not simple linear addition.

Compounding-frequency example: $1200$ dollars at a nominal annual rate of $6\%$ compounded monthly for two years gives A=1200(1+0.06/12)^{24}=\1352.59$to the nearest cent. For annual depreciation replace the growth factor by$1-r$. To compare purchasing power with annual inflation$i$, divide the nominal value after$n$years by$(1+i)^n$; nominal growth alone does not guarantee a gain in real value.

Use exponent laws and logarithms to solve exponential equations.

Exponent laws simplify powers; a logarithm reverses exponentiation, so aˣ=b is equivalent to x=logₐb for a>0,a≠1.

Worked example
2ˣ=16 gives x=4; 10ˣ=3 gives x=log(3).

Worked example
What operation undoes a power? logarithm, with a valid positive base and argument.

Common boundary
A logarithm is not an ordinary division or a power of ten only.

Core integer exponent laws, for non-zero $a$ where required, are $a^ma^n=a^{m+n}$, $a^m/a^n=a^{m-n}$, $(a^m)^n=a^{mn}$, $a^0=1$ and $a^{-n}=1/a^n$. For example, $5^3\cdot5^{-6}=5^{-3}=1/125$. Logarithms reverse exponentiation: $10^x=b\iff x=\log_{10}b$ and $e^x=b\iff x=\ln b$, with $b>0$.

Write a deductive proof as linked equalities.

A proof starts from a known expression and uses justified equalities until it reaches the target; each line preserves truth.

Worked example
To show (n+1)²−n²=2n+1, expand to n²+2n+1−n², then simplify.

Worked example
Which line needs justification? every transformation, such as expansion or cancellation, must preserve equality.

Common boundary
A few confirming examples do not prove a statement for all allowed values.

Apply rational exponent and logarithm laws consistently.

Fractional powers represent roots and powers, while log laws convert products, quotients and powers into sums, differences and coefficients.

Worked example
x^(3/2)=(√x)³ for x≥0; log(ab)=log a+log b for positive a,b.

Worked example
When can you split log(ab)? both arguments must be in the valid domain.

Common boundary
Do not apply log laws to sums or ignore domain restrictions.

Use all three log laws only for positive arguments: $\log_a(xy)=\log_ax+\log_ay$, $\log_a(x/y)=\log_ax-\log_ay$, and $\log_a(x^m)=m\log_ax$. Change base with $\log_ax=\ln x/\ln a$. Example: $2^{x-1}=10$ gives $(x-1)\ln2=\ln10$, so $x=1+\ln10/\ln2\approx4.322$. Never split $\log(x+y)$.

Test convergence before summing an infinite geometric series.

An infinite geometric series has a finite sum only when |r|<1; then S∞=a₁/(1−r).

Worked example
3+1.5+0.75+… converges because r=0.5, giving S∞=6.

Worked example
Does 4+8+16+… have a finite sum? no; |r|=2≥1.

Common boundary
A pattern continuing forever is not automatically summable.

Expand a power with binomial coefficients.

For a non-negative integer n, (a+b)ⁿ is the sum of terms whose coefficients are nCr and whose powers of a and b add to n.

Worked example
(x+2)²=x²+4x+4; coefficients 1,2,1 come from Pascal’s triangle.

Worked example
What should the powers add to? n in every term.

Common boundary
Do not omit the middle term or change the coefficient pattern.

Full binomial theorem: $(a+b)^n=\sum_{r=0}^{n}\binom{n}{r}a^{n-r}b^r$ for $n\in\mathbb{N}$, where $\binom{n}{r}=n!/[r!(n-r)!]$. Example: $(2x-1)^3=\binom30(2x)^3+\binom31(2x)^2(-1)+\binom32(2x)(-1)^2+\binom33(-1)^3=8x^3-12x^2+6x-1$. Each term's total power is $n$.