Fi & Lu Area Algebra Trigonometry Functions Complex Numbers Calculus
After seeing Mom use algebra to talk about triangles, Lu wanted to know more. Would Fi be able to understand as well? Mom began to explain why algebra is so useful.
“Algebra is the written language of mathematics. It allows us to express ideas about numbers and other abstract concepts in a very precise manner. We wouldn’t have modern technology without algebra.
“When we learn arithmetic, we learn to write numbers with digits, and we learn a few symbols to express operations on those numbers. We use the plus sign (\(+\)) for addition, the minus sign (\(-\)) for subtraction, the times sign (\(\times\)) for multiplication, and the obelus (\(\div\)) for division. There’s also comparison signs: \(<\) (less than), \(\leq\) (less than or equal to), \(=\) (equals), \(\geq\) (greater than or equal to), and \(>\) (greater than). Algebra adds a few more symbols, but, more importantly, uses letters to represent generic or unknown numbers. That allows us to express ideas about numbers and their relationships.
“And, just as natural language has rules about how to interpret a sequence of words, algebra has rules about how to interpret a string of symbols. Some of the first rules tell us in which order to do operations. These rules are themselves applied in a particular order determined by their precedence, with the highest precedence rules applied first. In order of precedence, from high to low, these rules involve parentheses, exponentiation, unary negation, multiplication and division, addition and subtraction:
The parenthesis rule says that the sequence of symbols delimited by a pair of matching parentheses is interpreted first. If there are nested pairs, the innermost pair should be handled first. \(3\times(5+7) = 3\times 12 = 36\) vs \(3\times 5+7 = 15+7=22\).
The exponent rule says that the sequence of symbols that is the exponent should be interpreted before doing the exponentiation. If there are multiple levels of exponents, the topmost one should be interpreted first. \(2^{1+2} = 2^3 = 8\). \(2^{3^2} = 2^9 = 512\).
The unary negation rule tells us to treat a unary minus as if it were \((-1)\times\). \(-2^4=-1\times 2^4 = -1\times 16 = -16\).
The multiplication/division rule tells us to apply \(\times\) and \(\div\) left to right. \(2\times 3\times 4\times 5\div 6\times 7 = 6\times 4\times 5\div 6\times 7 = 24\times 5\div 6\times 7 = 120\div 6\times 7 = 20\times 7 = 140\).
The addition/subtraction rule tells us to apply \(+\) and \(-\) left to right. \(1+2+3-4+5 = 3+3-4+5 = 6-4+5 = 2+5 = 7\).
“Here’s another example, shown one rule application at a time and results of the previous rule application: \begin{align*} & 6+5\times 4-3+2^{1+2}\times\textcolor{red}{(3+4)}-5 \\ &= 6+5\times 4-3+2^\textcolor{red}{1+2}\times \textcolor{green}{7}-5 \\ &= 6+5\times 4-3+\textcolor{red}{2}^\textcolor{green}{3}\times 7-5 \\ &= 6+\textcolor{red}{5\times 4}-3+\textcolor{green}{8}\textcolor{red}{\times 7}-5 \\ &= \textcolor{red}{6+}\textcolor{green}{20}-3+\textcolor{green}{56}-5 \\ &= \textcolor{green}{26}\textcolor{red}{-3}+56-5 \\ &= \textcolor{green}{23}\textcolor{red}{+56}-5 \\&= \textcolor{green}{79}\textcolor{red}{-5} \\ &= \textcolor{green}{74} \end{align*}
“When we use (italic) letters to represent numbers, we usually elide the times sign (\(\times\)). When a number is followed by a letter, or two letters are juxtaposed, like \(69x\) or \(ab\), we treat the pair as if they had a times sign between them—they are to be multiplied together. But sometimes we use a center dot (\(\cdot\)) to indicate multiplication, like \(a\cdot b\) or \(5\cdot 7\). And we hardly ever see a division sign (\(\div\)); instead we just see a slash, or we indicate division with a fraction bar, like \(a/b\) or \(\frac{a}{b}\). Particularly in the latter case, the numerator and denominator are separately interpreted before the division is applied. One other way to indicate division is with a negative exponent: \(a/b = ab^{-1}\).
“Even with numbers, letters used for numbers, symbols for operations, parentheses, superscripts used for exponentiation, and subscripts as we saw for sequence elements, mathematicians found that they had to overload the meanings of certain glyphs. As we’ll see when we talk more about functions, the string \(f(x)\) might interpreted as the function \(f\) applied to the argument \(x\), where \(f\) could be any single letter representing a function and where \(x\) might instead be an expression that is evaluated in the usual way. This leads to an ambiguity when interpreting something like \(a(b+c)\)—it might mean \(a\times(b+c)\) or it might mean apply the function \(a\) to the result of adding \(b\) and \(c\). Experience and context should tell you which interpretation to use.
“We can use algebraic notation to express properties of addition. For example, when we add two numbers, it doesn’t matter what order we add them in: \(a+b=b+a\). This is called commutativity. When we add three numbers, it doesn’t matter which addition we do first: \((a+b)+c=a+(b+c)\). This is called associativity.
“Multiplication (at least of numbers) is also commutative and associative: \(ab = ba\) and \((ab)c=a(bc)\). Multiplication is also distributive over addition: \(a(b+c) = ab+ac\) and \((a+b)c = ac+bc\).
“Exponentiation interacts with addition and multiplication: \(a^{b+c} = a^ba^c\) and \(a^{bc} = (a^b)^c\).
“Zero is the additive identity: \(a+0 = 0+a = a\). Also \(a\times 0 = 0\times a = 0\).
“One is the multiplicative identity: \(a\times 1 = 1\times a = a\). Also \(a^0 = 1\).
“Up to now, we’ve used comparison signs like = to show relationships that are always true. But there is another use. We can use comparison signs to indicate relationships that we want to be true, and for which we want to know what values to assign to the letters (called variables) so that those relationships are true.
“For example, suppose we want \(x+5=7\). One of the first techniques we use to solve such equations is to perform the same operation to both sides of the equality. In this case, we can subtract 5 from both sides to get \(x+5-5=7-5\) or \(x=2\). We can verify that we’ve successfully solved the equation by substituting the value we’ve found for \(x\): \(2+5 = 7\).
“A slightly harder example is \(3x+2=5x+8\). In this case we can subtract \(3x\) from both sides to get \(3x+2-3x=5x+8-3x\) or \(2=2x+8\), subtract 8 from both sides to get \(2-8 = 2x+8-8\) or \(-6 = 2x\), and then divide both sides by 2 to get \(-6/2 = 2x/2\) or \(-3 = x\). You may have noticed that we used some properties of addition and multiplication when we did this, in particular we used the commutativity of addition to change \(5x+8-3x\) to \(5x-3x+8\) and then the distributivity of multiplication over addition to change \(5x-3x\) to \((5-3)x\) to \(2x\). A similar transformation occurs on the left side, where we also use the properties of 0 to change \((3-3)x+2\) to \(0x+2\) to \(0+2\) and finally to \(2\). If you wondered about subtraction and division, you should realize that subtraction is just addition of the negative, and division is just multiplication of the inverse, so we just use the properties of addition and multiplication.
“Sometimes we have more than one variable and equations relating them. To solve such a system of equations, one useful technique is to use the equations to express one of the variables in terms of the others. Then eliminate that variable by substituting it with the equivalent expression and continue eliminating variables until only one is left. Once we solve for that variable, we can work backwards to find the values of the other variables. For example, suppose we are given \begin{align*} 0 &= x+3y+2z-3 \\ 0 &= -x-2y+z-9 \\ 0 &= 2x+y+3z-16 \end{align*} “We can solve for \(x\) in the first equation to get \(x = 3-3y-2z\). Substituting this for \(x\) in the other two equations, we get \begin{align*} 0 &= -\textcolor{red}{(3-3y-2z)} -2y + z - 9 = \textcolor{blue}{y + 3z -12} \\ 0 &= 2\textcolor{red}{(3-3y-2z)}+y+3z-16 = \textcolor{blue}{-5y -z -10} \end{align*} “Now we can solve for \(y\) in the first equation above to get \(y=12-3z\). Substituing this for \(y\) in the remaining equation, we get \begin{align*} 0 &= -5\textcolor{red}{(12-3z)} -z -10 = \textcolor{blue}{14z -70} \end{align*} so \(z = 70\div 14 = 5\), which then gives \(y=12-3z=12-15=-3\) and then \(x=3-3y-2z=3+9-10=2\).
Exponentiation is the process of raising a base to a power called the exponent. If the exponent is a nonnegative integer, it’s the number of times 1 is multiplied by the base. So, for example, \(2^3 = 1\times 2\times 2\times 2 = 8\), while \(2^0 = 1\). If the exponent is a negative integer, its magnitude is the number of times 1 is divided by the base. So \(2^{-3} = 1\div 2\div 2\div 2 = \frac{1}{8}\).
It’s fairly clear that for integer exponents \(b^c\times b^d = b^{c+d}\) and not hard to see that \((b^c)^d = b^{c\times d}\). If we want to maintain those equalities for rational exponents, we have to have \(b^{1/c} = \sqrt[c]{b}\), and in particular \(b^{1/2} = \sqrt{b}\). This can be problematic if the base isn’t positive, but for positive bases, it’s not hard to extend the definition to arbitrary real exponents.