Arithmetic Reasoning and Mathematics Knowledge Review: Number Sense

Classifications of Numbers
Numbers are the basic building blocks of mathematics. Specific features of numbers are identified by the following terms:
Integer – any positive or negative whole number, including zero. Integers do not include fractions 

, decimals (0.56), or mixed numbers

.

Prime number – any whole number greater than 1 that has only two factors, itself and 1; that is, a number that can be divided evenly only by 1 and itself.
Composite number – any whole number greater than 1 that has more than two different factors; in other words, any whole number that is not a prime number. For example: The composite number 8 has the factors of 1, 2, 4, and 8.
Even number – any integer that can be divided by 2 without leaving a remainder. For example: 2, 4, 6, 8, and so on.
Odd number – any integer that cannot be divided evenly by 2. For example: 3, 5, 7, 9, and so on.
Decimal number – any number that uses a decimal point to show the part of the number that is less than one. Example: 1.234.
Decimal point – a symbol used to separate the ones place from the tenths place in decimals or dollars from cents in currency.
Decimal place – the position of a number to the right of the decimal point. In the decimal 0.123, the 1 is in the first place to the right of the decimal point, indicating tenths; the 2 is in the second place, indicating hundredths; and the 3 is in the third place, indicating thousandths.

The decimal, or base 10, system is a number system that uses ten different digits (0, 1, 2, 3, 4, 5, 6, 7, 8, 9). An example of a number system that uses something other than ten digits is the binary, or base 2, number system, used by computers, which uses only the numbers 0 and 1. It is thought that the decimal system originated because people had only their 10 fingers for counting.

Rational numbers include all integers, decimals, and fractions. Any terminating or repeating decimal number is a rational number.
Irrational numbers cannot be written as fractions or decimals because the number of decimal places is infinite and there is no recurring pattern of digits within the number. For example, pi (π) begins with 3.141592 and continues without terminating or repeating, so pi is an irrational number.
Real numbers are the set of all rational and irrational numbers.

The Number Line
A number line is a graph to see the distance between numbers. Basically, this graph shows the relationship between numbers. So, a number line may have a point for zero and may show negative numbers on the left side of the line. Also, any positive numbers are placed on the right side of the line. For example, consider the points labeled on the following number line:


We can use the dashed lines on the number line to identify each point. Each dashed line between two whole numbers is

. The line halfway between two numbers is

.



Numbers in Word Form and Place Value
When writing numbers out in word form or translating word form to numbers, it is essential to understand how a place value system works. In the decimal or base-10 system, each digit of a number represents how many of the corresponding place value – a specific factor of 10 – are contained in the number being represented. To make reading numbers easier, every three digits to the left of the decimal place is preceded by a comma. The following table demonstrates some of the place values:

Power of 10: 103 - 102 - 101 - 100 - 10-1 - 10-2 - 10-3
Value: 1000 - 100 - 10 - 1 - 0.1 - 0.01 - 0.001
Place: thousands - hundreds - tens - ones - tenths - hundredths - thousandths

For example, consider the number 4,546.09, which can be separated into each place value like this:
4: thousands
5: hundreds
4: tens
6: ones
0: tenths
9: hundredths

This number in word form would be four thousand five hundred forty-six and nine hundredths.

Absolute Value
A precursor to working with negative numbers is understanding what absolute values are. A number's absolute value is simply the distance away from zero a number is on the number line. The absolute value of a number is always positive and is written

. For example, the absolute value of 3, written as

, is 3 because the distance between 0 and 3 on a number line is three units. Likewise, the absolute value of –3, written as

 is 3 because the distance between 0 and –3 on a number line is three units. So,

.



Practice
P1. Write the place value of each digit in 14,059.826
P2. Write out each of the following in words:
(a) 29
(b) 478
(c) 98,542
(d) 0.06
(e) 13.113
P3. Write each of the following in numbers:
(a) nine thousand four hundred thirty-five
(b) three hundred two thousand eight hundred seventy-six
(c) nine hundred one thousandths
(d) nineteen thousandths
(e) seven thousand one hundred forty-two and eighty-five hundredths

Practice Solutions
P1. The place value for each digit would be as follows:
Digit - Place Value
1 ten-thousands
4 thousands
0 hundreds
5 tens
9 ones
8 tenths
2 hundredths
6 thousandths

P2. Each written out in words would be:
(a) twenty-nine
(b) four hundred seventy-eight
(c) ninety-eight thousand five hundred forty-two
(d) six hundredths
(e) thirteen and one hundred thirteen thousandths
P3. Each in numeric form would be:
(a) 9,435
(b) 302, 876
(c) 0.901
(d) 0.019
(e) 7,142.85

Operations

An operation is simply a mathematical process that takes some value(s) as input(s) and produces an output. Elementary operations are often written in the following form: value operation value.
For instance, in the expression

 the values are 1 and 2 and the operation is addition. Performing the operation gives the output of 3. In this way we can say that

 and 3 are equal, or

.

Addition
Addition increases the value of one quantity by the value of another quantity (both called addends). For example,

.
The result is called the sum. With addition, the order does not matter,

.
When adding signed numbers, if the signs are the same simply add the absolute values of the addends and apply the original sign to the sum.
For example,

 and

.
When the original signs are different, take the absolute values of the addends and subtract the smaller value from the larger value, then apply the original sign of the larger value to the difference. For instance,

 and

.

Subtraction
Subtraction is the opposite operation to addition; it decreases the value of one quantity (the minuend) by the value of another quantity (the subtrahend). For example,

.
The result is called the difference. Note that with subtraction, the order does matter,

.
For subtracting signed numbers, change the sign of the subtrahend and then follow the same rules used for addition. For example,

.

Multiplication
Multiplication can be thought of as repeated addition. One number (the multiplier) indicates how many times to add the other number (the multiplicand) to itself. For example,

. With multiplication, the order does not matter:

 or

, either way the result (the product) is the same.
If the signs are the same the product is positive when multiplying signed numbers. For example,

 and

.
If the signs are opposite, the product is negative. For example,

 and

. When more than two factors are multiplied together, the sign of the product is determined by how many negative factors are present. If there are an odd number of negative factors then the product is negative, whereas an even number of negative factors indicates a positive product. For instance,

 and

.

Division
Division is the opposite operation to multiplication; one number (the divisor) tells us how many parts to divide the other number (the dividend) into. The result of division is called the quotient.
For example,

; if 20 is split into
4 equal parts, each part is 5. With division, the order of the numbers does matter,

.
The rules for dividing signed numbers are similar to multiplying signed numbers. If the dividend and divisor have the same sign, the quotient is positive. If the dividend and divisor have opposite signs, the quotient is negative. For example,

.

Parentheses
Parentheses are used to designate which operations should be done first when there are multiple operations. Example:

; the parentheses tell us that we must add 2 and 1, and then subtract the sum from 4, rather than subtracting 2 from 4 and then adding 1 (this would give us an answer of 3).

Exponents
An exponent is a superscript number placed next to another number at the top right. It indicates how many times the base number is to be multiplied by itself. Exponents provide a shorthand way to write what would be a longer mathematical expression, for example:

. A number with an exponent of 2 is said to be 'squared,' while a number with an exponent of 3 is said to be 'cubed.' The value of a number raised to an exponent is called its power. So,

 is read as '8 to the 4th power,' or '8 raised to the power of 4.'
The properties of exponents are as follows:



Any number to the power of 1 is equal to itself



The number 1 raised to any power is equal to 1



Any number raised to the power of 0 is equal to 1



Add exponents to multiply powers of the same base number



Subtract exponents to divide powers of the same base



When a power is raised to a power, the exponents are multiplied






Multiplication and division operations inside parentheses can be raised to a power. This is the same as each term being raised to that power.



A negative exponent is the same as the reciprocal of a positive exponent

Note that exponents do not have to be integers. Fractional or decimal exponents follow all the rules above as well. Example:

.


Roots
A root, such as a square root, is another way of writing a fractional exponent. Instead of using a superscript, roots use the radical symbol (

. The two special cases of

 and

 are called square roots and cube roots. If there is no number to the upper left, it is understood to be a square root (

). Nearly all of the roots you encounter will be square roots. A square root is the same as a number raised to the one-half power. When we say that a is the square root of b
(

).
A perfect square is a number that has an integer for its square root. There are 10 perfect squares from 1 to 100: 1, 4, 9, 16, 25, 36, 49, 64, 81, 100 (the squares of integers 1 through 10).


Order of Operations
Order of operations is a set of rules that dictates the order in which we must perform each operation in an expression so that we will evaluate it accurately. If we have an expression that includes multiple different operations, order of operations tells us which operations to do first. The most common mnemonic for order of operations is PEMDAS, or "Please Excuse My Dear Aunt Sally." PEMDAS stands for parentheses, exponents, multiplication, division, addition, and subtraction. It is important to understand that multiplication and division have equal precedence, as do addition and subtraction, so those pairs of operations are simply worked from left to right in order.
For example, evaluating the expression

 using the correct order of operations would be done like this:
P: Perform the operations inside the parentheses:


·        
E: Simplify the exponents.
oThe equation now looks like this:


· MD:
Perform multiplication and division from left to right:

; then


o    
The equation now looks like this:


·     
AS: Perform addition and subtraction from left to right:

; then



Subtraction with Regrouping
A great way to make use of some of the features built into the decimal system would be regrouping when attempting longform subtraction operations. When subtracting within a place value, sometimes the minuend is smaller than the subtrahend, regrouping enables you to ‘borrow' a unit from a place value to the left in order to get a positive difference. For example, consider subtracting 189 from 525 with regrouping.

First, set up the subtraction problem in vertical form:
525
– 189
________________
Notice that the numbers in the ones and tens columns of 525 are smaller than the numbers in the ones and tens columns of 189. This means you will need to use regrouping to perform subtraction:
5 2 5
1 8 9
________________
To subtract 9 from 5 in the ones column you will need to borrow from the 2 in the tens columns:
5 1 15
– 1 8 9
________________
6
Next, to subtract 8 from 1 in the tens column you will need to borrow from the 5 in the hundreds column:
4 11 15
- 1 8 9
________________
3 6
Last, subtract the 1 from the 4 in the hundreds column:
4 11 15
- 1 8 9
________________
3 3 6

P1. Demonstrate how to subtract 477 from 620 using
P2. Simplify the following expressions with exponents:
(a)



(b)



(c)



(d)



(e)


P1. First, set up the subtraction problem in vertical form:
6 2 0
– 4 7 7
______________

o subtract 7 from 0 in the ones column you will need to borrow from the 2 in the tens column:
6 1 10
– 4 7 7
______________
3

Next, to subtract 7 from the 1 that’s still in the tens column you will need to borrow from the 6 in the hundreds column:
5 11 10

– 4 7 7
______________
4 3

Lastly, subtract 4 from the 5 remaining in the hundreds column:
5 11 10

– 4 7 7
______________
1 4 3

P2. Using the properties of exponents and the proper order of operations:
(a) Any number raised to the power of 0 is equal to 1:



(b) The number 1 raised to any power is equal to 1:



(c) Add exponents to multiply powers of the same base:



(d) When a power is raised to a power, the exponents are multiplied:



(e) Perform the operation inside the parentheses first: