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1.2: Real Numbers - Algebra Essentials


Learning Objectives

  • Classify a real number as a natural, whole, integer, rational, or irrational number.
  • Perform calculations using order of operations.
  • Use the following properties of real numbers: commutative, associative, distributive, inverse, and identity.
  • Evaluate algebraic expressions.
  • Simplify algebraic expressions.

It is often said that mathematics is the language of science. The earliest use of numbers occurred (100) centuries ago in the Middle East to count, or enumerate items. Farmers, cattlemen, and tradesmen used tokens, stones, or markers to signify a single quantity—a sheaf of grain, a head of livestock, or a fixed length of cloth, for example. Doing so made commerce possible, leading to improved communications and the spread of civilization.

Three to four thousand years ago, Egyptians introduced fractions. They first used them to show reciprocals. Later, they used them to represent the amount when a quantity was divided into equal parts.

But what if there were no cattle to trade or an entire crop of grain was lost in a flood? How could someone indicate the existence of nothing? From earliest times, people had thought of a “base state” while counting and used various symbols to represent this null condition. However, it was not until about the fifth century A.D. in India that zero was added to the number system and used as a numeral in calculations.

Clearly, there was also a need for numbers to represent loss or debt. In India, in the seventh century A.D., negative numbers were used as solutions to mathematical equations and commercial debts. The opposites of the counting numbers expanded the number system even further.

Because of the evolution of the number system, we can now perform complex calculations using these and other categories of real numbers. In this section, we will explore sets of numbers, calculations with different kinds of numbers, and the use of numbers in expressions.

Classifying a Real Number

The numbers we use for counting, or enumerating items, are the natural numbers: (1, 2, 3, 4, 5) and so on. We describe them in set notation as ({1,2,3,...}) where the ellipsis ((cdots)) indicates that the numbers continue to infinity. The natural numbers are, of course, also called the counting numbers. Any time we enumerate the members of a team, count the coins in a collection, or tally the trees in a grove, we are using the set of natural numbers. The set of whole numbers is the set of natural numbers plus zero: ({0,1,2,3,...}).

The set of integers adds the opposites of the natural numbers to the set of whole numbers: ({cdots,-3,-2,-1,0,1,2,3,cdots}).It is useful to note that the set of integers is made up of three distinct subsets: negative integers, zero, and positive integers. In this sense, the positive integers are just the natural numbers. Another way to think about it is that the natural numbers are a subset of the integers.

[egin{array}{ccc} [4pt] ext{negative integers}& ext{zero}& ext{positive integers}[4pt] [4pt] cdots ,-3,-2,-1&0&1,2,3,cdots [4pt] end{array}]

The set of rational numbers is written as ({mnparallel ext{m and n are integers and } n eq 0}).Notice from the definition that rational numbers are fractions (or quotients) containing integers in both the numerator and the denominator, and the denominator is never (0). We can also see that every natural number, whole number, and integer is a rational number with a denominator of (1).

Because they are fractions, any rational number can also be expressed in decimal form. Any rational number can be represented as either:

  1. a terminating decimal: (frac{15}{8} =1.875), or
  2. a repeating decimal: (frac{4}{11} =0.36363636cdots = 0.ar{36})

We use a line drawn over the repeating block of numbers instead of writing the group multiple times.

Example (PageIndex{1}): Writing Integers as Rational Numbers

Write each of the following as a rational number. Write a fraction with the integer in the numerator and (1) in the denominator.

  1. (7)
  2. (0)
  3. (-8)

Solution

  1. (0= frac{0}{1})
  2. (-8= frac{8}{1})

Exercise (PageIndex{1})

Write each of the following as a rational number.

  1. (11)
  2. (3)
  3. (-4)
Answer
  1. (frac{11}{1})
  2. (frac{3}{1})
  3. (-frac{4}{1})

Example (PageIndex{2}): Identifying Rational Numbers

Write each of the following rational numbers as either a terminating or repeating decimal.

  1. (-frac{5}{7})
  2. (frac{15}{5})
  3. (frac{13}{25})

Solution

  1. (frac{15}{5} = 3)(or (3.0)), a terminating decimal
  2. (frac{13}{25} =0.52), a terminating decimal

Exercise (PageIndex{2})

Write each of the following rational numbers as either a terminating or repeating decimal.

  1. (frac{68}{17})
  2. (frac{8}{13})
  3. (-frac{13}{25})
Answer
  1. (4) (or (4.0)), terminating
  2. (0.overline{615384}),repeating
  3. (-0.85), terminating

Irrational Numbers

At some point in the ancient past, someone discovered that not all numbers are rational numbers. A builder, for instance, may have found that the diagonal of a square with unit sides was not (2) or even (32),but was something else. Or a garment maker might have observed that the ratio of the circumference to the diameter of a roll of cloth was a little bit more than (3), but still not a rational number. Such numbers are said to be irrational because they cannot be written as fractions. These numbers make up the set of irrational numbers. Irrational numbers cannot be expressed as a fraction of two integers. It is impossible to describe this set of numbers by a single rule except to say that a number is irrational if it is not rational. So we write this as shown.

[{hmid ext {h is not a rational number}}]

Example (PageIndex{3}): Differentiating Rational and Irrational Numbers

Determine whether each of the following numbers is rational or irrational. If it is rational, determine whether it is a terminating or repeating decimal.

  1. (sqrt{25})
  2. (frac{33}{9})
  3. (sqrt{11})
  4. (frac{17}{34})
  5. (0.3033033303333…)

Solution

  1. (sqrt{25}): This can be simplified as (sqrt{25} = 5).Therefore,(sqrt{25})is rational.
  2. (frac{33}{9}): Because it is a fraction,(frac{33}{9})is a rational number. Next, simplify and divide. [frac{33}{9}=cancel{frac{33}{9}} onumber] So,(frac{33}{9}) is rational and a repeating decimal.
  3. (sqrt{11}): This cannot be simplified any further. Therefore,(sqrt{11})is an irrational number.
  4. (frac{17}{34}): Because it is a fraction,(frac{17}{34})is a rational number. Simplify and divide. [frac{17}{34} = 0.5 onumber] So,(frac{17}{34}) is rational and a terminating decimal.
  5. (0.3033033303333…) is not a terminating decimal. Also note that there is no repeating pattern because the group of (3s) increases each time. Therefore it is neither a terminating nor a repeating decimal and, hence, not a rational number. It is an irrational number.

Exercise (PageIndex{3})

Determine whether each of the following numbers is rational or irrational. If it is rational, determine whether it is a terminating or repeating decimal.

  1. (frac{7}{77})
  2. (sqrt{81})
  3. (4.27027002700027…)
  4. (frac{91}{13})
  5. (sqrt{39})
Answer
  1. rational and terminating;
  2. rational and repeating;
  3. irrational

Real Numbers

Given any number (n), we know that (n) is either rational or irrational. It cannot be both. The sets of rational and irrational numbers together make up the set of real numbers. As we saw with integers, the real numbers can be divided into three subsets: negative real numbers, zero, and positive real numbers. Each subset includes fractions, decimals, and irrational numbers according to their algebraic sign (+ or –). Zero is considered neither positive nor negative.

The real numbers can be visualized on a horizontal number line with an arbitrary point chosen as (0), with negative numbers to the left of (0) and positive numbers to the right of (0). A fixed unit distance is then used to mark off each integer (or other basic value) on either side of (0). Any real number corresponds to a unique position on the number line.The converse is also true: Each location on the number line corresponds to exactly one real number. This is known as a one-to-one correspondence. We refer to this as the real number line as shown in Figure((PageIndex{1}))

Example (PageIndex{4}): Classifying Real Numbers

Classify each number as either positive or negative and as either rational or irrational. Does the number lie to the left or the right of (0) on the number line?

  1. (-frac{10}{3})
  2. (-sqrt{5})
  3. (-6π)
  4. (0.615384615384…)

Solution

  1. (-frac{10}{3})is negative and rational. It lies to the left of (0) on the number line.
  2. (-sqrt{5})is positive and irrational. It lies to the right of (0).
  3. (-sqrt{289} = -sqrt{17^2} = -17) is negative and rational. It lies to the left of (0).
  4. (-6π) is negative and irrational. It lies to the left of (0).
  5. (0.615384615384…) is a repeating decimal so it is rational and positive. It lies to the right of (0).

Exercise (PageIndex{4})

Classify each number as either positive or negative and as either rational or irrational. Does the number lie to the left or the right of (0) on the number line?

  1. (sqrt{73})
  2. (-11.411411411…)
  3. (frac{47}{19})
  4. (-frac{sqrt{5}}{2})
  5. (6.210735)
Answer
  1. positive, irrational
  2. right negative, rational
  3. left positive, rational
  4. right negative, irrational
  5. left positive, rational; right

Sets of Numbers as Subsets

Beginning with the natural numbers, we have expanded each set to form a larger set, meaning that there is a subset relationship between the sets of numbers we have encountered so far. These relationships become more obvious when seen as a diagram, such as Figure((PageIndex{2})).

SETS OF NUMBERS

The set of natural numbers includes the numbers used for counting: ({1,2,3,...}).

The set of whole numbers is the set of natural numbers plus zero: ({0,1,2,3,...}).

The set of integers adds the negative natural numbers to the set of whole numbers: ({...,-3,-2,-1,0,1,2,3,...}).

The set of rational numbers includes fractions written as ({mnparallel ext{m and n are integers and }n eq 0}).

The set of irrational numbers is the set of numbers that are not rational, are nonrepeating, and are nonterminating: ({hparallel ext{h is not a rational number}}).

Example (PageIndex{5}): Differentiating the Sets of Numbers

Classify each number as being a natural number (N), whole number (W), integer (I), rational number (Q), and/or irrational number (Q′).

  1. (sqrt{36})
  2. (frac{8}{3})
  3. (sqrt{73})
  4. (-6)
  5. (3.2121121112…)

Solution

NWIQQ'
a. (sqrt{36} = 6)XXXX
b. (frac{8}{3} =2.overline{6})X
c. (sqrt{73})X
d. (-6)XX
e. (3.2121121112...)X

Exercise (PageIndex{5})

Classify each number as being a natural number (N), whole number (W), integer (I), rational number (Q), and/or irrational number (Q′).

  1. (-frac{35}{7})
  2. (0)
  3. (sqrt{169})
  4. (sqrt{24})
  5. (4.763763763...)
Answer
NWIQQ'
a. (-frac{35}{7})XX
b. (0)XXX
c. (sqrt{169})XXXX
d. (sqrt{24})X
e. (4.763763763...)X

Performing Calculations Using the Order of Operations

When we multiply a number by itself, we square it or raise it to a power of (2). For example, (4^2 =4 imes4=16). We can raise any number to any power. In general, the exponential notation an means that the number or variable (a) is used as a factor (n) times.

[a^n=acdot acdot acdots a qquad ext{ n factors} onumber ]

In this notation, (a^n) is read as the (n^{th}) power of (a), where (a) is called the base and (n) is called the exponent. A term in exponential notation may be part of a mathematical expression, which is a combination of numbers and operations. For example, (24+6 imes dfrac{2}{3} − 4^2) is a mathematical expression.

To evaluate a mathematical expression, we perform the various operations. However, we do not perform them in any random order. We use the order of operations. This is a sequence of rules for evaluating such expressions.

Recall that in mathematics we use parentheses ( ), brackets [ ], and braces { } to group numbers and expressions so that anything appearing within the symbols is treated as a unit. Additionally, fraction bars, radicals, and absolute value bars are treated as grouping symbols. When evaluating a mathematical expression, begin by simplifying expressions within grouping symbols.

The next step is to address any exponents or radicals. Afterward, perform multiplication and division from left to right and finally addition and subtraction from left to right.

Let’s take a look at the expression provided.

[24+6 imes dfrac{2}{3} − 4^2 onumber]

There are no grouping symbols, so we move on to exponents or radicals. The number (4) is raised to a power of (2), so simplify (4^2) as (16).

[24+6 imes dfrac{2}{3} − 4^2 onumber ]

[24+6 imes dfrac{2}{3} − 16 onumber]

Next, perform multiplication or division, left to right.

[24+6 imes dfrac{2}{3} − 16 onumber]

[24+4-16 onumber]

Lastly, perform addition or subtraction, left to right.

[24+4−16 onumber]

[28−16 onumber]

[12 onumber]

Therefore,

[24+6 imes dfrac{2}{3} − 4^2 =12 onumber]

For some complicated expressions, several passes through the order of operations will be needed. For instance, there may be a radical expression inside parentheses that must be simplified before the parentheses are evaluated. Following the order of operations ensures that anyone simplifying the same mathematical expression will get the same result.

ORDER OF OPERATIONS

Operations in mathematical expressions must be evaluated in a systematic order, which can be simplified using the acronym PEMDAS:

  • P(arentheses)
  • E(xponents)
  • M(ultiplication) and D(ivision)
  • A(ddition) and S(ubtraction)

HOW TO: Given a mathematical expression, simplify it using the order of operations.

  1. Simplify any expressions within grouping symbols.
  2. Simplify any expressions containing exponents or radicals.
  3. Perform any multiplication and division in order, from left to right.
  4. Perform any addition and subtraction in order, from left to right.

Example (PageIndex{6}): Using the Order of Operations

Use the order of operations to evaluate each of the following expressions.

  1. (dfrac{5^2-4}{7}- sqrt{11-2})
  2. (dfrac{14-3 imes2}{2 imes5-3^2})
  3. (7 imes(5 imes3)−2 imes[(6−3)−4^2]+1)

Solution

  1. [egin{align*} (3 imes2)^2-4 imes(6+2)&=(6)^2-4 imes(8) && qquad ext{Simplify parentheses} &=36-4 imes8 && qquad ext{Simplify exponent} &=36-32 && qquad ext{Simplify multiplication} &=4 && qquad ext{Simplify subtraction} end{align*}]
  2. [egin{align*} dfrac{5^2-4}{7}- sqrt{11-2}&= dfrac{5^2-4}{7}-sqrt{9} && qquad ext{Simplify grouping symbols (radical)} &=dfrac{5^2-4}{7}-3 && qquad ext{Simplify radical} &=dfrac{25-4}{7}-3 && qquad ext{Simplify exponent} &=dfrac{21}{7}-3 && qquad ext{Simplify subtraction in numerator} &=3-3 && qquad ext{Simplify division} &=0 && qquad ext{Simplify subtraction} end{align*}]

Note that in the first step, the radical is treated as a grouping symbol, like parentheses. Also, in the third step, the fraction bar is considered a grouping symbol so the numerator is considered to be grouped.

  1. [egin{align*} 6-mid 5-8mid +3 imes(4-1)&=6-|-3|+3 imes3 && qquad ext{Simplify inside grouping symbols} &=6-3+3 imes3 && qquad ext{Simplify absolute value} &=6-3+9 && qquad ext{Simplify multiplication} &=3+9 && qquad ext{Simplify subtraction} &=12 && qquad ext{Simplify addition} end{align*}]
  2. [egin{align*} dfrac{14-3 imes2}{2 imes5-3^2}&=dfrac{14-3 imes2}{2 imes5-9} && qquad ext{Simplify exponent} &=dfrac{14-6}{10-9} && qquad ext{Simplify products} &=dfrac{8}{1} && qquad ext{Simplify differences} &=8 && qquad ext{Simplify quotient} end{align*}]

In this example, the fraction bar separates the numerator and denominator, which we simplify separately until the last step.

  1. [egin{align*} 7 imes(5 imes3)-2 imes[(6-3)-4^2]+1&=7 imes(15)-2 imes[(3)-4^2]+1 && qquad ext{Simplify inside parentheses} &=7 imes(15)-2 imes(3-16)+1 && qquad ext{Simplify exponent} &=7 imes(15)-2 imes(-13)+1 && qquad ext{Subtract} &=105+26+1 && qquad ext{Multiply} &=132 && qquad ext{Add} end{align*}]

Exercise (PageIndex{6})

Use the order of operations to evaluate each of the following expressions.

  1. (sqrt{5^2-4^2}+7 imes(5-4)^2)
  2. (1+dfrac{7 imes5-8 imes4}{9-6})
  3. (|1.8-4.3|+0.4 imessqrt{15+10})
  4. (dfrac{1}{2} imes[5 imes3^2-7^2]+dfrac{1}{3} imes9^2)
  5. ([(3-8^2)-4]-(3-8))
Answer
  1. (10)
  2. (2)
  3. (4.5)
  4. (25)
  5. (26)

Using Properties of Real Numbers

For some activities we perform, the order of certain operations does not matter, but the order of other operations does. For example, it does not make a difference if we put on the right shoe before the left or vice-versa. However, it does matter whether we put on shoes or socks first. The same thing is true for operations in mathematics.

Commutative Properties

The commutative property of addition states that numbers may be added in any order without affecting the sum.

[a+b=b+a]

We can better see this relationship when using real numbers.

((−2)+7 = 5 ext{ and } 7+(−2)=5)

Similarly, the commutative property of multiplication states that numbers may be multiplied in any order without affecting the product.

[a imes b=b imes a]

Again, consider an example with real numbers.

((−11) imes(−4)=44) and ((−4) imes(−11)=44)

It is important to note that neither subtraction nor division is commutative. For example, (17−5) is not the same as (5−17). Similarly, (20÷5≠5÷20).

Associative Properties

The associative property of multiplication tells us that it does not matter how we group numbers when multiplying. We can move the grouping symbols to make the calculation easier, and the product remains the same.

[a(bc)=(ab)c]

Consider this example.

((3 imes4) imes5=60 ext{ and } 3 imes(4 imes5)=60)

The associative property of addition tells us that numbers may be grouped differently without affecting the sum.

[a+(b+c)=(a+b)+c]

This property can be especially helpful when dealing with negative integers. Consider this example.

([15+(−9)]+23=29 ext{ and } 15+[(−9)+23]=29)

Are subtraction and division associative? Review these examples.

[egin{align*} 8-(3-15)overset{?}{=}&(8-3)-15 8-(-12)overset{?}{=}&5-15 20 eq &20-10 64div (8div 4)overset{?}{=}&(64div 8)div 4 64div 2overset{?}{=}&8div 4 32 eq &2 end{align*}]

As we can see, neither subtraction nor division is associative.

Distributive Property

The distributive property states that the product of a factor times a sum is the sum of the factor times each term in the sum.

[a imes(b+c)=a imes b+a imes c]

This property combines both addition and multiplication (and is the only property to do so). Let us consider an example.

Note that (4) is outside the grouping symbols, so we distribute the (4) by multiplying it by (12), multiplying it by (–7), and adding the products.

Example (PageIndex{7})

To be more precise when describing this property, we say that multiplication distributes over addition. The reverse is not true, as we can see in this example.

[egin{align*} 6+(3 imes5)overset{?}{=}&(6+3) imes(6 imes5) 6+(15)overset{?}{=}&(9) imes(11) 21 eq &99 end{align*}]

Multiplication does not distribute over subtraction, and division distributes over neither addition nor subtraction.

A special case of the distributive property occurs when a sum of terms is subtracted.

[a−b=a+(−b)]

For example, consider the difference (12−(5+3)). We can rewrite the difference of the two terms (12) and ((5+3)) by turning the subtraction expression into addition of the opposite. So instead of subtracting ( (5+3)), we add the opposite.

(12+(−1) imes(5+3)])

Now, distribute (-1) and simplify the result.

[egin{align*} 12-(5+3)&=12+(-1) imes(5+3) &=12+[(-1) imes5+(-1) imes3] &=12+(-8) &=4 end{align*}]

This seems like a lot of trouble for a simple sum, but it illustrates a powerful result that will be useful once we introduce algebraic terms. To subtract a sum of terms, change the sign of each term and add the results. With this in mind, we can rewrite the last example.

[egin{align*} 12-(5+3)&=12+(-5-3) &=12-8 &=4 end{align*}]

Identity Properties

The identity property of addition states that there is a unique number, called the additive identity ((0)) that, when added to a number, results in the original number.

[a+0=a]

The identity property of multiplication states that there is a unique number, called the multiplicative identity ((1)) that, when multiplied by a number, results in the original number.

[a imes 1=a]

For example, we have ( (−6)+0=−6) and( 23 imes1=23). There are no exceptions for these properties; they work for every real number, including (0) and (1).

Inverse Properties

The inverse property of addition states that, for every real number a, there is a unique number, called the additive inverse (or opposite), denoted (−a), that, when added to the original number, results in the additive identity, (0).

[a+(−a)=0]

For example, if (a =−8), the additive inverse is (8), since ((−8)+8=0).

The inverse property of multiplication holds for all real numbers except (0) because the reciprocal of (0) is not defined. The property states that, for every real number (a), there is a unique number, called the multiplicative inverse (or reciprocal), denoted (1a), that, when multiplied by the original number, results in the multiplicative identity, (1).

[a imes dfrac{1}{a}=1]

For example, if (a =−dfrac{2}{3}), the reciprocal, denoted (dfrac{1}{a}), is (-dfrac{3}{2}) because

(a⋅dfrac{1}{a}=left(−dfrac{2}{3} ight) imesleft(−dfrac{3}{2} ight)=1)

PROPERTIES OF REAL NUMBERS

The following properties hold for real numbers (a), (b), and (c).

Table (PageIndex{1})
AdditionMultiplication
Commutative Property(a+b=b+a)(a imes b=b imes a)
Associative Property(a+(b+c)=(a+b)+c)(a(bc)=(ab)c)
Distributive Property

(a imes (b+c)=a imes b+a imes c)

Identity Property

There exists a unique real number called the additive identity, 0, such that, for any real number a

(a+0=a)

There exists a unique real number called the multiplicative identity, 1, such that, for any real number a

(a imes 1=a)

Inverse Property

Every real number a has an additive inverse, or opposite, denoted –a, such that

(a+(−a)=0)

Every nonzero real number a has a multiplicative inverse, or reciprocal, denoted 1a , such that

(a imes left(dfrac{1}{a} ight)=1)

Example (PageIndex{8}): Using Properties of Real Numbers

Use the properties of real numbers to rewrite and simplify each expression. State which properties apply.

  1. (3 imes 6+3 imes 4)
  2. ((5+8)+(−8))
  3. (6−(15+9))
  4. (dfrac{4}{7} imesleft(dfrac{2}{3} imes dfrac{7}{4} ight))
  5. (100 imes[0.75+(−2.38)])

Solution

  1. [egin{align*} 3 imes6+3 imes4&=3 imes(6+4)qquad ext{Distributive property} &=3 imes10qquad ext{Simplify} &=30qquad ext{Simplify} end{align*}]
  2. [egin{align*} (5+8)+(-8)&=5+[8+(-8)]qquad ext{Associative property of addition} &=5+0qquad ext{Inverse property of addition} &=5qquad ext{Identity property of addition} end{align*}]
  3. [egin{align*} 6-(15+9)&=6+[(-15)+(-9)]qquad ext{Distributive property} &=6+(-24)qquad ext{Simplify} &=-18qquad ext{Simplify} end{align*}]
  4. [egin{align*} dfrac{4}{7} imesleft(dfrac{2}{3} imesdfrac{7}{4} ight)&=dfrac{4}{7} imesleft(dfrac{7}{4} imesdfrac{2}{3} ight)qquad ext{Commutative property of multiplication} &=left(dfrac{4}{7} imesdfrac{7}{4} ight) imesdfrac{2}{3}qquad ext{Associative property of multiplication} &=1 imesdfrac{2}{3}qquad ext{Inverse property of multiplication} &=dfrac{2}{3}qquad ext{Identity property of multiplication} end{align*}]
  5. [egin{align*} 100 imes[0.75+(-2.38)]&=100 imes0.75+100 imes(-2.38)qquad ext{Distributive property} &=75+(-238)qquad ext{Simplify} &=-163qquad ext{Simplify} end{align*}]

Exercise (PageIndex{7})

Use the properties of real numbers to rewrite and simplify each expression. State which properties apply.

  1. (left(-dfrac{23}{5} ight) imesleft[11 imesleft(-dfrac{5}{23} ight) ight])
  2. (5 imes(6.2+0.4))
  3. (18-(7-15))
  4. (dfrac{17}{18}+left[dfrac{4}{9}+left(-dfrac{17}{18} ight) ight])
  5. (6 imes(-3)+6 imes3)
Answer
  1. (33), distributive property
  2. (dfrac{4}{9}), commutative property of addition, associative property of addition, inverse property of addition, identity property of addition
  3. (0), distributive property, inverse property of addition, identity property of addition

Evaluating Algebraic Expressions

So far, the mathematical expressions we have seen have involved real numbers only. In mathematics, we may see expressions such as (x +5), (dfrac{4}{3}pi r^3), or (sqrt{2m^3 n^2}). In the expression (x +5), (5) is called a constant because it does not vary and (x) is called a variable because it does. (In naming the variable, ignore any exponents or radicals containing the variable.) An algebraic expression is a collection of constants and variables joined together by the algebraic operations of addition, subtraction, multiplication, and division.

We have already seen some real number examples of exponential notation, a shorthand method of writing products of the same factor. When variables are used, the constants and variables are treated the same way.

[egin{align*} (-3)^5 &=(-3) imes(-3) imes(-3) imes(-3) imes(-3)Rightarrow x^5=x imes x imes x imes x imes x (2 imes7)^3&=(2 imes7) imes(2 imes7) imes(2 imes7)qquad ; ; Rightarrow (yz)^3=(yz) imes(yz) imes(yz) end{align*}]

In each case, the exponent tells us how many factors of the base to use, whether the base consists of constants or variables.

Any variable in an algebraic expression may take on or be assigned different values. When that happens, the value of the algebraic expression changes. To evaluate an algebraic expression means to determine the value of the expression for a given value of each variable in the expression. Replace each variable in the expression with the given value, then simplify the resulting expression using the order of operations. If the algebraic expression contains more than one variable, replace each variable with its assigned value and simplify the expression as before.

Example (PageIndex{9}): Describing Algebraic Expressions

List the constants and variables for each algebraic expression.

  1. (x + 5)
  2. (dfrac{4}{3}pi r^3)
  3. (sqrt{2m^3 n^2})

Solution

ConstantsVariables
a. (x + 5)(5)(x)
b. (dfrac{4}{3}pi r^3)(dfrac{4}{3}), (pi)(r)
c. (sqrt{2m^3 n^2})(2)(m),(n)

Exercise (PageIndex{8})

List the constants and variables for each algebraic expression.

  1. (2(L + W))
  2. (4y^3+y)
Answer
ConstantsVariables
a. (2pi r(r+h))(2),(pi)(r),(h)
b. (2(L + W))(2)(L), (W)
c. (4y^3+y)(4)(y)

Example (PageIndex{10}): Evaluating an Algebraic Expression at Different Values

Evaluate the expression (2x−7) for each value for (x).
  1. (x=0)
  2. (x=1)
  3. (x=12)
  4. (x=−4)

Solution

  1. Substitute (0) for (x). [egin{align*} 2x-7 &= 2(0)-7 &= 0-7 &= -7 end{align*}]
  2. Substitute (1) for (x). [egin{align*} 2x-7 &= 2(1)-7 &= 2-7 &= -5 end{align*}]
  3. Substitute (dfrac{1}{2}) for (x). [egin{align*} 2x-7 &= 2left (dfrac{1}{2} ight )-7 &= 1-7 &= -6 end{align*}]
  4. Substitute (-4) for (x). [egin{align*} 2x-7 &= 2(-4)-7 &= -8-7 &= -15 end{align*}]

Exercise (PageIndex{9})

Evaluate the expression (11−3y) for each value for (y).

  1. (y=2)
  2. (y=0)
  3. (y=dfrac{2}{3})
  4. (y=−5)
Answer
  1. (11)
  2. (26)

Example (PageIndex{11}): Evaluating Algebraic Expressions

Evaluate each expression for the given values.

  1. ​(x+5) for (x=-5)
  2. (dfrac{t}{2t-1}) for (t=10)
  3. (dfrac{4}{3}pi r^3) for (r=5)
  4. (a+ab+b) for (a=11), (b=-8)
  5. (sqrt{2m^3 n^2}) for (m=2), (n=3)

Solution

  1. Substitute (-5) for (x). [egin{align*} x+5 &= (-5)+5 &= 0 end{align*}]
  2. Substitute (10) for (t). [egin{align*} dfrac{t}{2t-1} &= dfrac{(10)}{2(10)-1} &= dfrac{10}{20-1} &= dfrac{10}{19} end{align*}]
  3. Substitute (5) for (r). [egin{align*} dfrac{4}{3} pi r^3 &= dfrac{4}{3}pi (5)^3 &= dfrac{4}{3}pi (125) &= dfrac{500}{3}pi end{align*}]
  4. Substitute (11) for (a) and (-8) for (b). [egin{align*} a+ab+b &= (11)+(11)(-8)+(-8) &= 11-88-8 &= -85 end{align*}]
  5. Substitute (2) for (m) and (3) for (n). [egin{align*} sqrt{2m^3 n^2} &= sqrt{2(2)^3 (3)^2} &= sqrt{2(8)(9)} &= sqrt{144} &= 12 end{align*}]

Exercise (PageIndex{10})

Evaluate each expression for the given values.

  1. (dfrac{y+3}{y-3}) for (y=5)
  2. (7-2t) for (t=-2)
  3. (dfrac{1}{3}pi r^2) for (r=11)
  4. ((p^2 q)^3) for (p=-2), (q=3)
  5. (4(m-n)-5(n-m)) for (m=dfrac{2}{3}) (n=dfrac{1}{3})
Answer
  1. (4)
  2. (11)
  3. (dfrac{121}{3}pi)
  4. (1728)
  5. (3)

Formulas

An equation is a mathematical statement indicating that two expressions are equal. The expressions can be numerical or algebraic. The equation is not inherently true or false, but only a proposition. The values that make the equation true, the solutions, are found using the properties of real numbers and other results. For example, the equation (2x +1= 7) has the unique solution of (3) because when we substitute (3) for (x) in the equation, we obtain the true statement (2(3)+1=7).

A formula is an equation expressing a relationship between constant and variable quantities. Very often, the equation is a means of finding the value of one quantity (often a single variable) in terms of another or other quantities. One of the most common examples is the formula for finding the area (A) of a circle in terms of the radius (r) of the circle: ( A= pi r^2). For any value of (r), the area (A) can be found by evaluating the expression (pi r^2).

Example (PageIndex{12}): Using a Formula

A right circular cylinder with radius (r) and height (h) has the surface area (S) (in square units) given by the formula (S=2pi r(r+h)). See Figure (PageIndex{3}). Find the surface area of a cylinder with radius (6) in. and height (9) in. Leave the answer in terms of (pi).

Evaluate the expression (2pi r(r+h)) for (r=6) and (h=9).

Solution

[egin{align*} S &= 2pi r(r+h) &= 2pi (6)[(6)+(9)] &= 2pi(6)(15) &= 180pi end{align*}]

The surface area is (180pi) square inches.

Exercise (PageIndex{11})

A photograph with length (L) and width (W) is placed in a matte of width (8) centimeters (cm). The area of the matte (in square centimeters, or (cm^2) is found to be (A=(L+16)(W+16) - L)⋅W.See Figure (PageIndex{4}). Find the area of a matte for a photograph with length (32)cm and width (24)cm.

Answer

(1152cm^2)

Simplifying Algebraic Expressions

Sometimes we can simplify an algebraic expression to make it easier to evaluate or to use in some other way. To do so, we use the properties of real numbers. We can use the same properties in formulas because they contain algebraic expressions.

Example (PageIndex{13}): Simplifying Algebraic Expressions

Simplify each algebraic expression.

  1. (3x-2y+x-3y-7)
  2. (2r-5(3-r)+4)
  3. (left(4t-dfrac{5}{4}s ight)-left(dfrac{2}{3}t+2s ight))
  4. (2mn-5m+3mn+n)

Solution

  1. [egin{align*} 3x-2y+x-3y-7&=3x+x-2y-3y-7 && qquad ext{Commutative property of addition} &=4x-5y-7 && qquad ext{Simplify} end{align*}]
  2. [egin{align*} 2r-5(3-r)+4&=2r-15+5r+4 && qquad qquad qquad ext {Distributive property} &=2r+5y-15+4 && qquad qquad qquad ext{Commutative property of addition} &=7r-11 && qquad qquad qquad ext{Simplify} end{align*}]
  3. [egin{align*} left(4t-dfrac{5}{4}s ight)-left(dfrac{2}{3}t+2s ight)&=4t-dfrac{5}{4}s-dfrac{2}{3}t-2s && qquad ext{Distributive property} &=4t-dfrac{2}{3}t-dfrac{5}{4}s-2s && qquad ext{Commutative property of addition} &=dfrac{10}{3}t-dfrac{13}{4}s && qquad ext{Simplify} end{align*}]
  4. [egin{align*} 2mn-5m+3mn+n&=2mn+3mn-5m+n && qquad ext{Commutative property of addition} &=5mn-5m+n && qquad ext{Simplify} end{align*}]

Exercise (PageIndex{12})

Simplify each algebraic expression.

  1. (dfrac{2}{3}y−2left(dfrac{4}{3}y+z ight))
  2. (dfrac{5}{t}−2−dfrac{3}{t}+1)
  3. (4p(q−1)+q(1−p))
  4. (9r−(s+2r)+(6−s))
Answer
  1. (−2y−2z) or (−2(y+z))
  2. (dfrac{2}{t}−1)
  3. (3pq−4p+q)
  4. (7r−2s+6)

Example (PageIndex{14}): Simplifying a Formula

A rectangle with length (L) and width (W) has a perimeter (P) given by (P =L+W+L+W). Simplify this expression.

Solution

[egin{align*} P &=L+W+L+W P &=L+L+W+W && qquad ext{Commutative property of addition} P &=2L+2W && qquad ext{Simplify} P &=2(L+W) && qquad ext{Distributive property} end{align*}]

Exercise (PageIndex{13})

If the amount (P) is deposited into an account paying simple interest (r) for time (t), the total value of the deposit (A) is given by (A =P+Prt). Simplify the expression. (This formula will be explored in more detail later in the course.)

Answer

(A=P(1+rt))

Access these online resources for additional instruction and practice with real numbers.

  • Simplify an Expression
  • Evaluate an Expression1
  • Evaluate an Expression2

Key Concepts

  • Rational numbers may be written as fractions or terminating or repeating decimals. See Example and Example.
  • Determine whether a number is rational or irrational by writing it as a decimal. See Example.
  • The rational numbers and irrational numbers make up the set of real numbers. See Example. A number can be classified as natural, whole, integer, rational, or irrational. See Example.
  • The order of operations is used to evaluate expressions. See Example.
  • The real numbers under the operations of addition and multiplication obey basic rules, known as the properties of real numbers. These are the commutative properties, the associative properties, the distributive property, the identity properties, and the inverse properties. See Example.
  • Algebraic expressions are composed of constants and variables that are combined using addition, subtraction, multiplication, and division. See Example. They take on a numerical value when evaluated by replacing variables with constants. See Example,Example, and Example
  • Formulas are equations in which one quantity is represented in terms of other quantities. They may be simplified or evaluated as any mathematical expression. See Example and Example.

Verbal

Is (sqrt{2}) an example of a rational terminating, rational repeating, or irrational number? Tell why it fits that category.

irrational number. The square root of two does not terminate, and it does not repeat a pattern. It cannot be written as a quotient of two integers, so it is irrational.

What is the order of operations? What acronym is used to describe the order of operations, and what does it stand for?

What do the Associative Properties allow us to do when following the order of operations? Explain your answer.

The Associative Properties state that the sum or product of multiple numbers can be grouped differently without affecting the result. This is because the same operation is performed (either addition or subtraction), so the terms can be re-ordered.

Numeric

For the following exercises, simplify the given expression.

Algebraic

For the following exercises, solve for the variable.

For the following exercises, simplify the expression.

Real-World Applications

For the following exercises, consider this scenario: Fred earns $40 mowing lawns. He spends $10 on mp3s, puts half of what is left in a savings account, and gets another $5 for washing his neighbor’s car.

Write the expression that represents the number of dollars Fred keeps (and does not put in his savings account). Remember the order of operations.

How much money does Fred keep?

For the following exercises, solve the given problem.

According to the U.S. Mint, the diameter of a quarter is 0.955 inches. The circumference of the quarter would be the diameter multiplied byπ.Is the circumference of a quarter a whole number, a rational number, or an irrational number?

Jessica and her roommate, Adriana, have decided to share a change jar for joint expenses. Jessica put her loose change in the jar first, and then Adriana put her change in the jar. We know that it does not matter in which order the change was added to the jar. What property of addition describes this fact?

For the following exercises, consider this scenario: There is a mound ofgpounds of gravel in a quarry. Throughout the day, 400 pounds of gravel is added to the mound. Two orders of 600 pounds are sold and the gravel is removed from the mound. At the end of the day, the mound has 1,200 pounds of gravel.

Write the equation that describes the situation.

For the following exercise, solve the given problem.

Ramon runs the marketing department at his company. His department gets a budget every year, and every year, he must spend the entire budget without going over. If he spends less than the budget, then his department gets a smaller budget the following year. At the beginning of this year, Ramon got $2.5 million for the annual marketing budget. He must spend the budget such that2,500,000−x=0.What property of addition tells us what the value of x must be?

inverse property of addition

Technology

For the following exercises, use a graphing calculator to solve for x. Round the answers to the nearest hundredth.

Extensions

If a whole number is not a natural number, what must the number be?

Determine whether the statement is true or false: The multiplicative inverse of a rational number is also rational.

Determine whether the statement is true or false: The product of a rational and irrational number is always irrational.

Determine whether the simplified expression is rational or irrational:−18−4(5)(−1)‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾√.

Determine whether the simplified expression is rational or irrational:−16+4(5)+5‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾√.

The division of two whole numbers will always result in what type of number?

What property of real numbers would simplify the following expression:4+7(x−1)?

Glossary

algebraic expression
constants and variables combined using addition, subtraction, multiplication, and division
associative property of addition
the sum of three numbers may be grouped differently without affecting the result; in symbols,a+(b+c)=(a+b)+c
associative property of multiplication
the product of three numbers may be grouped differently without affecting the result; in symbols,a⋅(b⋅c)=(a⋅b)⋅c
base
in exponential notation, the expression that is being multiplied
commutative property of addition
two numbers may be added in either order without affecting the result; in symbols,a+b=b+a
commutative property of multiplication
two numbers may be multiplied in any order without affecting the result; in symbols,a⋅b=b⋅a
constant
a quantity that does not change value
distributive property
the product of a factor times a sum is the sum of the factor times each term in the sum; in symbols,a⋅(b+c)=a⋅b+a⋅c
equation
a mathematical statement indicating that two expressions are equal
exponent
in exponential notation, the raised number or variable that indicates how many times the base is being multiplied
exponential notation
a shorthand method of writing products of the same factor
formula
an equation expressing a relationship between constant and variable quantities
identity property of addition
there is a unique number, called the additive identity, 0, which, when added to a number, results in the original number; in symbols,a+0=a
identity property of multiplication
there is a unique number, called the multiplicative identity, 1, which, when multiplied by a number, results in the original number; in symbols,a⋅1=a
integers
the set consisting of the natural numbers, their opposites, and 0:{…,−3,−2,−1,0,1,2,3,…}
inverse property of addition
for every real numbera,there is a unique number, called the additive inverse (or opposite), denoted−a,which, when added to the original number, results in the additive identity, 0; in symbols,a+(−a)=0
inverse property of multiplication
for every non-zero real numbera,there is a unique number, called the multiplicative inverse (or reciprocal), denoted1a,which, when multiplied by the original number, results in the multiplicative identity, 1; in symbols,a⋅1a=1
irrational numbers
the set of all numbers that are not rational; they cannot be written as either a terminating or repeating decimal; they cannot be expressed as a fraction of two integers
natural numbers
the set of counting numbers:{1,2,3,…}
order of operations
a set of rules governing how mathematical expressions are to be evaluated, assigning priorities to operations
rational numbers
the set of all numbers of the formmn,wheremandnare integers andn≠0.Any rational number may be written as a fraction or a terminating or repeating decimal.
real number line
a horizontal line used to represent the real numbers. An arbitrary fixed point is chosen to represent 0; positive numbers lie to the right of 0 and negative numbers to the left.
real numbers
the sets of rational numbers and irrational numbers taken together
variable
a quantity that may change value
whole numbers
the set consisting of 0 plus the natural numbers:{0,1,2,3,…}

Mathematical Reasoning

You don’t have to have a “math mind” to pass the GED ® Math test -- you just need the right preparation.

Here's what you need to know:

  • You should be familiar with math concepts, measurements, equations, and applying math concepts to solve real-life problems.
  • You don’t have to memorize formulas and will be given a formula sheet in the test center as well as on the screen in the test.
  • Use the free Math Study Guide to start studying. It will help you understand the skills being tested. Log in to start using the study guide.
  • The GED Ready ® practice test for Math can help you determine if you are ready to take the actual test. Log in to take the practice test.

Try a Sample Question

Question Overview

This question requires you to order a set of fractions and decimals from smallest to largest. First, the numbers must all be converted to the same format--either all fractions or all decimals--then the resulting numbers are placed in order. (NOTE: On the GED ® Mathematical Reasoning test, a calculator would not be available to you on this question.)

A list of numbers is shown

Which list shows the numbers arranged from smallest to largest?

Option A is correct. This response is the result of either converting the decimals to fractions, or the fractions to decimals, then comparing the results, and placing the original numbers in the correct order.

Option B is incorrect. If you selected this response, you incorrectly compared the non-zero numbers in the decimals with the denominators in the fractions, before attempting to place them in numerical order.

Option C is incorrect. If you selected this response, you made a mistake with the 0.07, incorrectly reading it as 0.7, and placing it between 0.6 and 4⁄5.

Option D is incorrect. If you selected this response, you made a calculation error in converting, obtaining an answer of 0.4 and placing it between 1⁄8 and 1⁄2.


ALGEBRA

The four operations and their signs.
The function of parentheses.
Terms versus factors .
Powers and exponents.
The order of operations.
Values and evaluations.
Evaluating algebraic expressions.

The integers.
The algebraic sign and the absolute value.
Subtracting a larger number from a smaller.
The number line.
The negative of any number.

"Adding" a negative number.
Naming terms. The rule for adding terms.
Subtracting a negative number.

The rule of symmetry. Commutative rules. Inverses.
Two rules for equations.

The definition of reciprocals. The definition of division. Rules for 0.

Parentheses. Brackets. Braces.
The relationship of b &minus a to a &minus b .

The law of inverses.
Transposing.
A logical sequence of statements.
Simple fractional equations.

Absolute value equations.
Absolute value inequalities.

Powers of a number.
Rules of exponents: When to add, when to multiply.

The definition of a polynomial in x .
Factoring polynomials.
Factoring by grouping.
Equations in which the unknown is a common factor.

Quadratics in different arguments.

Perfect square trinomials.
The square of a trinomial.
Completing the square.
Geometrical algebra.

Summary of Multiplying/Factoring.
Factoring by grouping.
The sum and difference of any two powers: a n ± b n .

Negative exponents. Exponent 0. Scientific notation.

Rational expressions. The principle of equivalent fractions. Reducing to lowest terms.

The Lowest Common Multiple (LCM) of a series of terms.

The whole is equal to the sum of the parts.
Same time problem: Upstream-downstream.
Total time problem. Job problem.

Square roots.
Equations x ² = a , and the principal square root.
Rationalizing a denominator.
Real numbers.

Roots of numbers. The index of a radical.
Fractional exponents.
Negative exponents.

The square root of a negative number.
The real and imaginary components.
Conjugate pairs.

The distance of a point from the origin.
The distance between any two points.
A proof of the Pythagorean theorem.

The equation of the first degree and its graph.
Vertical and horizontal lines.

The slope intercept form of the equation of a straight line. The general form.
Parallel and perpendicular lines.
The point-slope formula. The two-point formula.

The method of addition. The method of substitution. Cramer's Rule: The method of determinants.
Three equations in three unknowns.

Investment problems. Mixture problems.
Upstream-downstream problems.

The roots of a quadratic.
Solution by factoring.
Completing the square.
The quadratic formula.
The discriminant.
The graph of a quadratic: A parabola.

Definition. The three laws of logarithms.
Common logarithms.

Direct variation. The constant of proportionality.
Varies as the square. Varies inversely. Varies as the inverse square.


Example 2: Solving a Real World Problem Given Two Points

The mathematics department sponsors Math Family Fun Night every year. In the first year, there were 35 participants. In the third year there were 57 participants.

  • Write an equation that can be used to predict the amount of participants, y, for any given year, x.
  • Based on your equation, how many participants are predicted for the fifth year?

Step 1: Identify your two points.

Let y = number of participants

We know that the first year, there were 35 participants. This can be written as (1,35)

In the third year, there were 57 participants. This can be written as (3,57).

Therefore, our two points are (1,35) and (3,57)

Let's enter this information into our chart.

Now that we have an equation, we can use this equation to determine how many participants are predicted for the 5th year.

All we need to do is substitute!

We will substitute 5 for x (x is the year) and solve for y.

There are 79 participants predicted for the 5th year.

And that's it! Not too bad, is it? I hope that you are learning how to recognize points, slope and y-intercepts when reading real world problems.

You may also want to visit our lesson on writing equations using point-slope form.

Comments

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3rd Grade Math Games

The 3rd grade math games on this webpage focus on several important topics such as place value, addition and subtraction of whole numbers and decimals, multiplication and division of whole numbers, concepts of length, perimeter, area, and time, characteristics of geometric figures, as well as collecting, organizing, displaying, and interpreting data.

Learning math has never been so much fun!

Are you a math magician? Make 20 bunnies disappear by solving addition, subtraction, and multiplication problems very quickly.

Be a part of the excitement of playing car racing with this great Math Racing Game Addition with Regrouping. Here you will add numbers correctly to continue in the race to the finish line. Work quickly so you can cross the finish line first.

This is a fun and interactive Tic Tac Toe game about classifying whole numbers as even or odd.

Round numbers correctly in this fun online math pirates game to search for the treasure chest.

Play this spooky Halloween math game and practice your math measurement skills to destroy a lot of monsters. For each correct answer, you will enter a bonus round where you can earn points by smashing monsters. The math problems are about measuring time, volume, and mass.

Enjoy car racing fun with this great Math Racing Game Number Facts. You must solve problems of the four operations to continue in the race to the finish line. Work quickly so you can cross the finish line first.

The fun, the excitement, the roar of the engines are all here in this Math Racing Game Multiply within 100. Multiply numbers quickly and accurately in order to get to the checkered flag.

Hear the roar of the engines, see the hairpin turns as they quickly approach in this Math Racing Game Divide within 100. You must divide numbers quickly to continue in the race to the finish line or your "racecar" could spin-out and your race is done.

Multiplication Facts up to 12 Baseball Game
Have fun practicing your multiplication skills by playing this exciting Baseball Math Multiplication Facts up to 12 Game.

3rd grade students will have fun identifying important math terms when playing this interactive vocabulary game. For each definition, the students will have only 60 seconds to identify the correct word.

Soar into great math skills by playing this 3rd Grade Rounding Halloween Math Game and get loads of practice rounding numbers to the nearest ten and hundred.

Play this fun and interactive game and make 20 bunnies disappear by quickly matching different division problems with the correct answer.

Division up to 100 Baseball Game
Make learning and improving your multiplication skills by playing this exciting Baseball Math Division up to 100 Game.

In this fast-paced car racing game, students will practice multiplication facts to 10 times 10.

In this fun basketball game, young students will have fun multiplying one-digit numbers.

Match the rounding problems with the correct solutions on these little bunnies in this fun Math Magician Rounding Game.

In this game students will multiply one-, two-, and three-digit numbers by 5, 6, 7, 8, and 9. Kids can play this game alone or in teams.

In this fast-paced racing game students will use repeated addition to model multiplication problems.

In this fast-paced math game, students will identify and use the commutative, associative, and identity property of multiplication.

In this game students will multiply 2-digit numbers by 1-digit numbers. They can play it alone or in pairs.

In this game students will count various US coins and match the pictures of the coins with the correct amounts.

In this fun soccer game, students will add two-digit numbers to get a chance to kick the ball and score points.

In this interactive soccer game, 2nd grade students will practice adding 2-digit numbers.

Third grade students will have fun dividing small numbers when playing this math basketball game.

In this fun place value game, students must pass the ball to the receiver in order to be given the chance to answer a problem and earn points.

This game is suitable for 3rd grade students and English language learners of all ages. The object of each problem is to match the analog clocks with the correct phrase.

In this interactive basketball game, 3rd grade students will practice telling time from analog clocks to the nearest minutes.

Who has? I have!

This is a printable time game that can be used as a classroom activity with elementary students.

Match the multiplication problems with the correct solutions on these little bunnies in this fun Math Magician Multiplication Game.

Return from the 3rd Grade Math Games page to the Elementary Math Games page or to the Math Play homepage.


Classical mechanics as generative modeling

Let’s remember physics classes. The very first topics you were taught were force, motion, time, velocity, acceleration in the simple 1D cases. Even before learning differentiation or integration in calculus classes, you could easily calculate the velocity of the object based on time and related positions. The same strategy could be applied to more complicated mechanics as spring force, pendulum, multi-dimensional mechanics, etc. You only substitute formulas for the appropriate ones and re-do the calculus routines. The simplified process of such modeling would be the following:

  • Identify whatobject is moving, what are the positions, the origin of the coordinate system, initial conditions
  • Find the appropriatemodel that describes the behavior of the given object under the given conditions
  • Solve theequation of the model with the given conditions and find velocity, acceleration, or another variable
  • Analyze the solution and its validity

For example, in the case of a pendulum system (on illustrations above), you can define the object dynamics model as an equilibrium between its kinetic and potential energy in a Lagrangian, and if you solve it for a single degree of freedom (the angle of the oscillation), you get the equation of the motion trajectory, which you can solve for different conditions.

The same as we sample faces, cats, and songs with GANs, we could sample complex motions of physical objects movements just via solving equations. For centuries. It literally got us to the moon. Without gigabytes of data and GPUs for deep neural nets.

I bet that they didn’t teach you this angle in the physics class. If you have the exact formula for this pendulum, you have your “pendulum-GAN”: you just need to sample length, gravity, amplitude, etc and insert them into the formula: this way you can generate as many pendulums as you want. The only difference that GAN has some quasi-random vector as the input and the formula is a black-box neural net trained with data. Below are a couple of illustrations of sampled trajectories and the code you can find here.


Math Insight

where the variable var successively takes on each value in sequence . For each such value, the code represented by code is run with var having that value from the sequence.

Here, we show some simple examples of using a for-loop in R.

Printing a list of numbers

Let's say we wanted to print a list of numbers from 0 to 3, inclusive. In R, the command 0:3 will create a vector with the numbers from 0 to 3, as you can see by entering that command at the R > command prompt:

(At the beginning of the output, R prints a [1] to let you know that lines starts with the first entry of the vector.)

We could create a simple for-loop that iterates through the four numbers of 0:3 and prints each number.

R outputs four lines, one for each number. (When typing the for-loop at the R > command prompt, R adds a + at the beginning of the line to indicate the command is continuing. We omit those + signs for clarity.)

If you don't want R to print the [1] at the beginning of the line, you could use the cat (concatenate) command instead, but you need to explicitly add a newline character to print each number on its own line.

We could assign the vector of numbers to a variable and then reference the variable in the for-loop. It would work exactly the same way.

Using for-loops with vectors

For-loops are especially convenient when working with vectors. Often we want to iterate over each element in a vector and do some computation with each element of the vector. We can also use for-loops to create or extend vectors, as R will automatically make a vector larger to accommodate values we assign to it.

First, lets create a vector using the c (combine) command is illustrated in the page on vector creation.

For any integer $i$ between 1 and 4, x[i] denotes the $i$th element of the vector.

We can use a for-loop to add one to the first element of x , add two to the second element of x , etc. We let use the variable n to store the number of elements in x (i.e., 4). In the loop, we will use the variable i to loop through the numbers 1, 2, 3, 4.

The for-loop is equivalent to running the four commands:

This for-loop creates a vector with five components where each component is double the previous.


1.2: Real Numbers - Algebra Essentials

Question: what is the sum of the first 100 whole numbers?? how am i supposed to work this out efficiently? thanks

The question you asked relates back to a famous mathematician, Gauss. In elementary school in the late 1700&rsquos, Gauss was asked to find the sum of the numbers from 1 to 100. The question was assigned as &ldquobusy work&rdquo by the teacher, but Gauss found the answer rather quickly by discovering a pattern. His observation was as follows:

Gauss noticed that if he was to split the numbers into two groups (1 to 50 and 51 to 100), he could add them together vertically to get a sum of 101.

1 + 2 + 3 + 4 + 5 + &hellip + 48 + 49 + 50

100 + 99 + 98 + 97 + 96 + &hellip + 53 + 52 + 51

1 + 100 = 101
2 + 99 = 101
3 + 98 = 101
.
.
.
48 + 53 = 101
49 + 52 = 101
50 + 51 = 101

Gauss realized then that his final total would be 50(101) = 5050.

The sequence of numbers (1, 2, 3, &hellip , 100) is arithmetic and when we are looking for the sum of a sequence, we call it a series. Thanks to Gauss, there is a special formula we can use to find the sum of a series:

S is the sum of the series and n is the number of terms in the series, in this case, 100.

There are other ways to solve this problem. You can, for example, memorize the formula

This is an arithmetic series, for which the formula is:
S = n[2a+(n-1)d]/2
where a is the first term, d is the difference between terms, and n is the number of terms.
For the sum of the first 100 whole numbers:
a = 1, d = 1, and n = 100
Therefore, sub into the formula:
S = 100[2(1)+(100-1)(1)]/2 = 100[101]/2 = 5050

You can also use special properties of the particular sequence you have.

An advantage of using Gauss' technique is that you don't have to memorize a formula, but what do you do if there are an odd number of terms to add so you can't split them into two groups, for example "what is the sum of the first 21 whole numbers?" Again we write the sequence "forwards and backwards" but using the entire sequence.

1 + 2 + 3 + . + 19 + 20 + 21
21 + 20 + 19 + . + 3 + 2 + 1


Math for the do-it-yourselfer

Math may not have been your favorite subject in school, but like it or not it's a part of just about every home improvement project.

You might be trying to figure out how many square feet are in a room so you can buy the right amount of paint or flooring, or maybe you're doing something a bit more complicated, like calculating a room's volume for sizing a bath fan, or figuring out the number of board feet in a piece of lumber.

But don't let your eyes glaze over just yet, because all you need to handle all this and more is a basic calculator and a few simple formulas.

Area measurements

Area calculations take two of the three dimensions into account. It might be width and length, as when measuring a floor, or width and height, as when measuring a wall. For the following examples, we'll use a rectangular room that is 10 feet wide, 12 feet long, and 8 feet high.

Area of the floor or ceiling: Multiply the length by the width (10 feet x 12 feet = 120 square feet of area).

Area of a wall: Multiply the width of the wall by its height. So one of the walls is 80 square feet (10 feet wide x 8 feet high) and the other is 96 square feet (12 feet x 8 feet). If you need the total square footage of the walls - for figuring paint or wallpaper for example - you can simplify the calculation by first adding all the wall lengths together, then multiplying by the height (10 + 12 + 10 + 12 = 44 x 8 = 352 square feet of total wall area).

Area in square yards: There are 3 feet in a yard, so there are a total of 9 square feet in a square yard (3 x 3). To calculate the number of square yards in our example room, which you might want to do when ordering carpet, divide the total square footage of the floor by 9 (120 square feet / 9 = 13.33 square yards).

Area in square inches: There are 12 inches in a foot, so there are 144 square inches in a square foot (12 x 12). To convert an area from square feet into square inches, simply multiply by 144. (Our room has 17,280 square inches).

Area of a triangle: If you want to figure out the area of a triangular space, such as a gable end, you need a simple formula: 1/2 x base x height. This means you multiply .5 x the base of the triangle x the height of the triangle. So, if your gable end is 18 feet wide at the base and 6 feet high from the base to the peak, it contains 54 square feet (.5 x 18 feet x 6 feet = 54 square feet).

Area of a circle: The formula for this is: pi x radius2 (pi = 3.1416). Let's say you have a circular space that's 22 feet across. That distance across is the diameter, and half of that, or 11 feet, is the radius. So the calculation would be : (3.1416 x 11 x 11 = 380.13 square feet).

Circumference of a circle: The circumference of a circle is the total distance around it, which is often a handy thing to be able to calculate. To do that, you need this formula: pi x diameter. For our 22-foot diameter circle, the circumference would be 69.12 feet (3.1416 x 22).

Cubic measurements

Where area measurements were two-dimensional, cubic measurements take all three dimensions into consideration. This will tell you the volume of a given area, for anything from sizing a fan to ordering concrete for a foundation.

Volume of a room: For the volume in cubic feet of our example room from above, simply multiply the width by the length by the height: (10 feet x 12 feet x 8 feet = 960 cubic feet).

Volume in cubic yards: There are 27 cubic feet in a cubic yard (3 x 3 x 3). So if you would like to convert cubic feet into cubic yards, which is necessary when ordering dirt, gravel, and concrete, simply divide the number of cubic feet by 27. For example, if you have a foundation form that is 2 feet wide, 20 feet long, and 1-1/2 feet high, first figure the cubic feet, then convert to cubic yards: (2 feet x 20 feet x 1.5 feet = 60 cubic feet / 27 = 2.22 cubic yards).

Volume in cubic inches: There are 1,728 cubic inches in a cubic foot (12 x 12 x 12). To convert the cubic feet in the above example into cubic inches, you would multiply by 1,728 (60 cubic feet x 1,728 = 103,680 cubic inches).

Many types of lumber are sold by the board foot. This unique unit of measurement refers to a board that's 1 foot long, 1 foot wide, and 1 inch thick. Any time you would like to know how many board feet are in a given piece of lumber, use the following formula: T x W x L / 12, where T = the thickness of the board in inches, W = the width of the board in inches, and L = the length of the board in feet.

For example, suppose you had a piece of 2 x 8 lumber that was 16 feet long. Using the formula, you can determine that the board contains 21.33 board feet (2 inches x 8 inches x 16 feet / 12 = 21.33).


Multiplication Coloring

We hope you like these multiplication worksheets. If you enjoy them, check out Coloring Squared: Multiplication and Division. It collects our basic and advanced multiplication and division pages into an awesome coloring book.

Coloring Squared: Multiplication and Division $9.95

There are a few different difficulty levels for each function. Hover over an image to see what the PDF looks like. Then you can click on any one of the images to pull up the PDF. You can then print the PDF.

Jr. Multiplication



Multiplication Worksheets- Basic



Multiplication Worksheets- Advanced



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Watch the video: The Real Number System (October 2021).