Mathematics employs many symbols. The first symbols discussed here are the ten numerals or digits we use to express numbers. Using only ten symbols, every whole number can be written. These symbols are 1, 2, 3, 4, 5, 6, 7, 8, 9, and 0. Because we use ten digits, we have what is called a base-ten system. We might consider ourselves lucky in this regard! The Babylonians used a base 60 system which employed the use of the numerals pictured below.

The Mayan numeral system was a base-twenty system, meaning that they had symbols for 20 numbers, and wrote numbers larger than twenty by using a “twenty’s-place” and then a “400’s place” much like we use a “10’s place” and a “100’s place” The first twenty numerals are pictured below.

Mayan numerals were, however, made more complicated by the fact that the second place value was often a base 18, while the rest were 20. This was presumably done to make the first three digits amount to 360 (1820=360) – A close approximation to the number of days in a year. However, evidence against this reasoning arises when it was found that the Mayans had a very accurate approximation of 365.2422 days in a year.

Many different numeral systems have been used through the ages. Some of these are more easily used than others. The Cyrillic numeral system, which is used in Russia and various surrounding areas until as late as the 18th century, was devised by assigning letters from the Cyrillic alphabet with numerical values. If reading a manuscript that used both words and letters and the author did not designate the numbers in special notation, the reader would only know he had come upon a number from context, or because the letters did not spell a word. Occasionally, the letters would spell a small word. In this case, it was important that the context of the statements indicated the number. The Cyrillic numerals are shown below.

Another system of numbers still widely used today (although primarily decoratively) are the Roman Numerals. You might see these numbers at the start of movies to indicate the year or to designate which Super Bowl you are watching. (Super Bowl 50 abandoned this numeration because while Super Bowl 49, stylized as Super Bowl XLIX looked neat, the NFL disliked the way Super Bowl 50 would by stylized: Super Bowl L). A sample of Roman Numerals is pictured below.

 

Place Values

Since we use a base-ten system, place values are used to write numbers larger than 9. For example, to write the next number after 9, we write 10. We place a 1 in the tens-place and our ones-place starts back at 0. We continue this again and again until we reach the number 99. 100, the next number after 99, places a one in our hundreds-place, and both the tens and ones place values go back to 0. There is no limit to the amount of times another place value can be added to a number to get a larger number. There is a periodic naming system for these numbers. The first few periods are shown here:

The periods have names such as Billions, Millions, Thousands and Ones. Each of these has a hundreds, tens and ones place. When writing numbers in this way, the periods are separated by a comma. This is known as standard form. For example, the number 48,359,117 is written in standard form. When a number written in this way, each digit can be identified. For example, the 2 in the number 123,456,789 is in the ten-millions place.

These periods continue forever. While many aren’t typically used in everyday speech, the names of the periods after Billions are Trillions, Quadrillions, Quintillions, Sextillions, Septillion, Octillion, Nonillion, Decillion and so on. The number one novemdecillion is a 1 followed by 60 zeros!

 

 

 

Expanded form

To write a number in expanded form, write each digit followed by its place-value name. Include a plus sign between each number. For example, the number in expanded form is:   4 ten thousands + 5 one thousands + 6 hundreds + 9 tens + 8 ones

 

 

Numbers in expanded form can also be converted into standard form:

 

Word Names for Numbers

The word name for a number is the way we say numbers. To write the word name for a number, we will separate each period. Start at the left-most period, and write the word name for the number in that period followed by the period name and follow this with a comma. Continue in this way throughout each period.

 

Note that we don’t include the “ones” period at the end of the word name. Also, note that the word and is not used in the word name. The word “and” can often be confused with “add”. For example, 3 and 4 make 7. To see why it is important to eliminate the word “and”, try to say the number 285,000. If you said “Two hundred and eighty-five thousand”, this might be taken to mean “two hundred plus eighty-five thousand” which of course, when added together is 85,200. This is quite different from the number we meant to say.

To write a number given in expanded form into standard form, write the digits, separating each period:

 

 

Graphing a whole number on a number line

Pictured below is the number line with whole numbers between 0 and 10. The arrow on the right of the line indicates that the numbers increase indefinitely.

To graph a number, place a dot on that number on the number line:

 

Order and Equality

Numbers can be compared with each other. Numbers that are equivalent are compared using the equal sign (). For example, 55. Numbers that are not the same can be compared by using the greater than sign(), or the less than sign (). A number is greater than a second number if it is further to the right on a number line than the second. An equation is a mathematical relationship that uses an equal sign. An inequality is a mathematical relation that uses an inequality, that is or . Statements using these symbols have a truth value associated with them.

 

 

Rounding Numbers

Often, numbers can become tedious to say and/or work with. For example, the distance from the earth to the moon is 238,855 miles. For most practical purposes, we won’t be interested in this level of a precision. We might state the number of miles to the nearest thousands of miles. In this case, the distance is nearest 239,000 miles.

To round a number to a designated place, look at the number following that place. If the following digit is 0-4, then you will round down. If the following digit is 5-9, then you will round up.

According to the US National Debt Clock (www.usdebtclock.org) the U.S. national debt as of the moment I write this sentence (in 2015) is $18,604,770,740,256. This number is quite a mouthful, and as it is constantly changing, not very accurate for long. In fact, as of THIS sentence’s writing (10 seconds later), the debt increased by more than one million dollars. Thus, we might want to round this number to the nearest billion. We would say that the national debt is nearest $18,605,000,000,000. Because the number after the billions digit is a seven, we increase the billions digit by 1.

If we wanted to round the national debt number given above ($18,604,770,740,256) to the
nearest trillions, we again examine the digit following the trillions. The hundred billions digit is a 6, so we increase the trillions digit. We then say this number is nearest $19,000,000,000,000.

 

Note that in the last example, rounding to the nearest million increases the millions from 569 million to 570. You will want to exercise care when the digit you are rounding is a nine.