Welcome to the number sense page at Math-Drills.com where we've got your number! This page includes Number Worksheets such as counting charts, representing, comparing and ordering numbers worksheets, and worksheets on expanded form, written numbers, scientific numbers, Roman numerals, factors, exponents, and binary numbers. There are literally hundreds of number worksheets meant to help students develop their understanding of numeration and number sense.

In the first few sections, there are some general use printables that can be used in a variety of situations. Hundred charts, for example, can be used for counting, but they can just as easily be used for learning decimal hundredths. Rounding worksheets help students learn this important skill that is especially useful in estimation.

Comparing and ordering numbers worksheets help students further understand place value and the ordinality of numbers. Continuing down the page are a number of worksheets on number forms: written, expanded, standard, scientific, and Roman numerals. Near the end of the page are a few worksheets for older students on factors, factoring, exponents and roots and binary numbers.

Most Popular Number Sense Worksheets this Week

**540 views this week**) Converting Between Standard, Expanded and Written Forms (5-Digit) U.S./U.K. Version (

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## Learning Numbers

### Number Posters.

There are a few different number posters in this section. The first two, with bird and butterfly themes include various ways of representing numbers from 0 to 9. Two versions of the numerals are used to demonstrate different printing styles, a Braille version and an American Sign Language version are also included to make students aware of different ways of representing each number. A linear representation and a ten-frame representation follow which is then followed by a pictorial representation using the theme. The poster sized numbers are just that ... made for printing and putting up in your classroom or home school.

**Writing Numerals and Numbers**

In the writing numerals to 20 worksheets, you will find that the A version includes all numbers, B to E versions have about half the numbers included, F to I versions have about a third of the numbers included and the J version includes no numbers... just the lines to write them on. All versions include dashes under the numbers, so students have a reference for where to place the numbers. You can access the other versions (B to J) once you select the A version you want below.

## Counting Worksheets

**Ten Frames**

Ten frames help students visualize numbers in relation to 10. They are used for many purposes, but the worksheets below are introductory and familiarize students with ten frames and give them practice using them.

**Completed**Ten Frames with the Numbers in

**Order**

**Completed**Ten Frames with the Numbers in

**Reverse Order**

**Completed**Ten Frames with the Numbers in

**Random Order**(10 versions)

**Blank**Ten Frames with the Numbers in

**Order**

**Blank**Ten Frames with the Numbers in

**Reverse Order**

**Blank**Ten Frames with the Numbers in

**Random Order**(10 versions)

**Skip counting** with a car theme

These skip counting worksheets include pictorial representations of the items the student is counting. For example, in the counting by 3's worksheet, students will see groups of three cars. This allows students to develop a mental image of skip counting. With larger numbers, including groups of items become impractical, so numbers are instead printed on the cars.

**by 1's**with Cars Skip Counting

**by 2's**with Cars Skip Counting

**by 3's**with Cars Skip Counting

**by 4's**with Cars Skip Counting

**by 5's**with Cars Skip Counting

**by 6's**with Cars Skip Counting

**by 7's**with Cars Skip Counting

**by 8's**with Cars Skip Counting

**by 9's**with Cars Skip Counting

**by 10's**with Cars Skip Counting

**by 11's**with Cars Skip Counting

**by 12's**with Cars Skip Counting

**by 25's**with Cars Skip Counting

**by 50's**with Cars Skip Counting

**by 100's**with Cars

**Hundred Charts**

Hundred charts are useful not only for learning counting but for many other purposes in math. For example, a hundred chart can be used to model fractions and to convert fractions into decimals. Modeling 1/4 on a hundred chart would require coloring every fourth square. After coloring every fourth square, there would be 25 squares colored in which is 25/100 or 0.25. Not magic, just math. Hundred charts can also be used as graph paper for graphing, learning long multiplication and division or any other purpose. A common use for hundred charts in older grades is to use it to find prime and composite numbers using the sieve of Eratosthenes.

**Completed**Hundred Chart

**Completed**Hundred Charts

**(4 Charts)**

**Left-Handed Completed**Hundred Chart

**Left-Handed Completed**Hundred Charts

**(4 Charts)**

**Blank**Hundred Chart

**Blank**Hundred Charts

**(4 Charts)**Hundred Chart with

**Even Numbers**Hundred Chart with

**Odd Numbers**Hundred Chart with

**Multiples of 3**Hundred Chart with

**Multiples of 4**Hundred Chart with

**Multiples of 5**Hundred Chart with

**Multiples of 6**Hundred Chart with

**Multiples of 7**Hundred Chart with

**Multiples of 8**Hundred Chart with

**Multiples of 9**Hundred Chart with

**Multiples of 10**

**Partially Completed**Hundred Chart (About 25%)

**Partially Completed**Hundred Charts (About 25%)

**(4 Charts)**

**120 Charts**

120 charts are very similar to hundred charts except they include the numbers from 101 to 120. 120 is a nice number for many reasons. One reason is that it has a lot of divisors—16 in fact. This makes the number 120 useful for many different grouping activities. Another reason is the Common Core Curriculum in the United States requires first graders to count to 120. A third reason is that 120 includes some three-digit numbers which could be a good introduction for some students into the hundreds place.

**Completed**120 Chart

**Completed**120 Charts

**(4 Charts)**

**Left-Handed Completed**120 Chart

**Left-Handed Completed**120 Charts

**(4 Charts)**

**Blank**120 Chart

**Blank**120 Charts

**(4 Charts)**120 Chart with

**Even Numbers**120 Chart with

**Odd Numbers**120 Chart with

**Multiples of 3**120 Chart with

**Multiples of 4**120 Chart with

**Multiples of 5**120 Chart with

**Multiples of 6**120 Chart with

**Multiples of 7**120 Chart with

**Multiples of 8**120 Chart with

**Multiples of 9**120 Chart with

**Multiples of 10**

**Partially Completed**120 Chart (About 25%)

**Partially Completed**120 Charts (About 25%)

**(4 Charts)**

### Backwards or **Counting Down 120 Charts** (**1** to a page)

**99 Charts**

Ninety-nine charts include zero and have no three-digit numbers. Each row starts with a multiple of ten rather than ending with a multiple of ten.

**Completed**99 Chart

**Completed**99 Charts

**(4 Charts)**

**Left-Handed Completed**99 Chart

**Left-Handed Completed**99 Charts

**(4 Charts)**

**Blank**99 Chart

**Blank**99 Charts

**(4 Charts)**99 Chart with

**Even Numbers**99 Chart with

**Odd Numbers**99 Chart with

**Multiples of 3**99 Chart with

**Multiples of 4**99 Chart with

**Multiples of 5**99 Chart with

**Multiples of 6**99 Chart with

**Multiples of 7**99 Chart with

**Multiples of 8**99 Chart with

**Multiples of 9**99 Chart with

**Multiples of 10**

**Partially Completed**99 Chart (About 25%)

**Partially Completed**99 Charts (About 25%)

**(4 Charts)**

**Counting** patterns

Counting collections of things in various patterns helps students develop shortcuts and strategies for counting. For example, when students count collections of items in rectangular patterns, they may use skip counting or multiplying to speed up their counting.

**Counting using number lines**

There are much better number line worksheets on the Number Line Worksheets page.

### Continue Counting Worsheets

**1**From Various Starting Numbers Continue Counting by

**2**From Various Starting Numbers Continue Counting by

**3**From Various Starting Numbers Continue Counting by

**4**From Various Starting Numbers Continue Counting by

**5**From Various Starting Numbers Continue Counting by

**6**From Various Starting Numbers Continue Counting by

**7**From Various Starting Numbers Continue Counting by

**8**From Various Starting Numbers Continue Counting by

**9**From Various Starting Numbers Continue Counting by

**10**From Various Starting Numbers Counting Backwards with Numbers to 120 Starting at Random Numbers

## Rounding Numbers Worksheets

**Rounding numbers** worksheets

Not only does rounding further an understanding of numbers, it can also be quite useful in estimating and measuring. There are many every day situations where a precise number isn't needed. For example if you needed to paint your basement floor, you don't really need to find out the area to exact square inch since you don't buy paint that way. You get a good idea of the floor space (e.g. it is roughly 20 feet by 15 feet) then read the label on the can to see how many square feet the can of paint covers (which, by the way is also a rounded number and variable depending on the roller used, the porosity of the floor, etc.) and buy enough cans to cover your floor.

**Tens**(U.S./U.K. Version) Rounding to

**Hundreds**(U.S./U.K. Version) Rounding to

**Thousands**(U.S./U.K. Version) Rounding to

**Ten Thousands**(U.S./U.K. Version) Rounding to

**Hundred Thousands**(U.S./U.K. Version) Rounding to

**Millions**(U.S./U.K. Version) Rounding to

**Tens**(SI Version) Rounding to

**Hundreds**(SI Version) Rounding to

**Thousands**(SI Version) Rounding to

**Ten Thousands**(SI Version) Rounding to

**Hundred Thousands**(SI Version) Rounding to

**Millions**(SI Version) Rounding to

**Tens**(Euro Format Version) Rounding to

**Hundreds**(Euro Format Version) Rounding to

**Thousands**(Euro Format Version) Rounding to

**Ten Thousands**(Euro Format Version) Rounding to

**Hundred Thousands**(Euro Format Version) Rounding to

**Millions**(Euro Format Version)

## Comparing & Ordering Numbers Worksheets

**Even and Odd** numbers

Distinguishing between even and odd numbers is an important skill for young students to learn. The vocabulary of even and odd is used throughout their math education, so it is necessary to learn it as soon as possible. Connecting cubes can help a great deal in visually demonstrating odd and even numbers. Create the numbers from 1 to 10 (or more) using connecting cubes in pairs and students will quickly see that the odd numbers have an unpaired cube that can be thought of as the "odd cube out." Once they have seen this pattern, they may be able to extend the pattern without making cube models. Ask them about 11 and 12 and 35 and so on.

**Comparing numbers** worksheets

There are many situations where it is important to know the relative size of one number to another. Several words are used to describe the relative sizes of one number to another, but it is probably best to use lesser than, greater than and equal to, although other words are more appropriate in certain situations. For example, if you were comparing two groups of candies, you would probably say, "there are **fewer** candies in that pile than in that one." The use of **"Tight"** in the worksheet titles means the numbers to be compared are close to one another.

**9**Comparing Numbers to

**25**Comparing Numbers to

**50**Comparing Numbers to

**50 (tight)**Comparing Numbers to

**100**Comparing Numbers to

**100 (tight)**Comparing Numbers to

**1000**Comparing Numbers to

**1000 (tight)**Comparing Numbers to

**10,000**(U.S./U.K. Version) Comparing Numbers to

**10,000 (tight)**(U.S./U.K. Version) Comparing Numbers to

**100,000**(U.S./U.K. Version) Comparing Numbers to

**100,000 (tight)**(U.S./U.K. Version) Comparing Numbers to

**1,000,000**(U.S./U.K. Version) Comparing Numbers to

**1,000,000 (tight)**(U.S./U.K. Version) Comparing Numbers to

**10,000,000**(U.S./U.K. Version) Comparing Numbers to

**10,000,000 (tight)**(U.S./U.K. Version) Comparing Numbers to

**10 000**(SI Version) Comparing Numbers to

**10 000 (tight)**(SI Version) Comparing Numbers to

**100 000**(SI Version) Comparing Numbers to

**100 000 (tight)**(SI Version) Comparing Numbers to

**1 000 000**(SI Version) Comparing Numbers to

**1 000 000 (tight)**(SI Version) Comparing Numbers to

**10 000 000**(SI Version) Comparing Numbers to

**10 000 000 (tight)**(SI Version) Comparing Numbers to

**10.000**(Euro Format Version) Comparing Numbers to

**10.000 (tight)**(Euro Format Version) Comparing Numbers to

**100.000**(Euro Format Version) Comparing Numbers to

**100.000 (tight)**(Euro Format Version) Comparing Numbers to

**1.000.000**(Euro Format Version) Comparing Numbers to

**1.000.000 (tight)**(Euro Format Version) Comparing Numbers to

**10.000.000**(Euro Format Version) Comparing Numbers to

**10.000.000 (tight)**(Euro Format Version)

**Ordering numbers** worksheets

## Expanded Form Worksheets

**Writing expanded form** worksheets

When writing numbers in expanded form, students might use three different formats which will be demonstrated using the number 9753. The first format is quite simple and combines both the place and the place value. For example, 9 is in the 1000's place which means the value of that 9 is 9000. The 7 is in the hundreds place which makes it 700. The 5 is in the tens place which makes it 50 and the 3 is in the ones place which makes it 3. To write in the simple expanded form, simply separate these four values with plus signs as in: 9000 + 700 + 50 + 3. In a more complex version of expanded form, the place and the place value are separated with multiplication signs as in (9 × 1000) + (7 × 100) + (5 × 10) + (3 × 1). Parentheses are included for clarity. In the third format, the place values are expressed as powers of ten: (9 × 10^{3}) + (7 × 10^{2}) + (5 × 10^{1}) + (3 × 10^{0}). All three formats are included in the answer keys for the writing expanded form worksheets. If you are only interested in one format, instruct your students accordingly.

**3-Digit**Numbers in Expanded Form Writing

**4-Digit**Numbers in Expanded Form Writing

**5-Digit**Numbers in Expanded Form (U.S./U.K. Version) Writing

**6-Digit**Numbers in Expanded Form (U.S./U.K. Version) Writing

**7-Digit**Numbers in Expanded Form (U.S./U.K. Version) Writing

**8-Digit**Numbers in Expanded Form (U.S./U.K. Version) Writing

**9-Digit**Numbers in Expanded Form (U.S./U.K. Version) Writing

**5-Digit**Numbers in Expanded Form (SI Version) Writing

**6-Digit**Numbers in Expanded Form (SI Version) Writing

**7-Digit**Numbers in Expanded Form (SI Version) Writing

**8-Digit**Numbers in Expanded Form (SI Version) Writing

**9-Digit**Numbers in Expanded Form (SI Version) Writing

**5-Digit**Numbers in Expanded Form (Euro Version) Writing

**6-Digit**Numbers in Expanded Form (Euro Version) Writing

**7-Digit**Numbers in Expanded Form (Euro Version) Writing

**8-Digit**Numbers in Expanded Form (Euro Version) Writing

**9-Digit**Numbers in Expanded Form (Euro Version) Write Expanded Form (range 1 000 to 9 999) (SI Version) (OLD) Write Expanded Form (range 1.000 to 9.999) (Euro Version) (OLD)

## Written Numbers Worksheets

Writing and reading numbers worksheets for students to learn how to write numbers in words and vice-versa.

The main idea of learning to write numbers in words is to be able to say numbers correctly. In the past it might also have been useful for writing checks/cheques, but there isn't a lot of that going on any more.

**Writing numbers in words** worksheets

**Reading written numbers** worksheets

Now, let's see if students can write the numbers that are written! The reading numbers written as words worksheets do not have format options as the student question sheets are all written in words. The answer keys are formatted with a comma thousands separator when necessary.

**Converting** between **standard, expanded and written forms**

The standard, expanded and written forms conversion worksheets sum up the previous sections by including all three number forms on the same page.

**3-Digit**) Converting Between Standard, Expanded and Written Forms (

**4-Digit**) Converting Between Standard, Expanded and Written Forms (

**5-Digit; U.S./U.K. Version**) Converting Between Standard, Expanded and Written Forms (

**3-Digit to 5-Digit; U.S./U.K. Version**) Converting Between Standard, Expanded and Written Forms (

**6-Digit; U.S./U.K. Version**) Converting Between Standard, Expanded and Written Forms (

**7-Digit; U.S./U.K. Version**) Converting Between Standard, Expanded and Written Forms (

**8-Digit; U.S./U.K. Version**) Converting Between Standard, Expanded and Written Forms (

**9-Digit; U.S./U.K. Version**) Converting Between Standard, Expanded and Written Forms (

**6-Digit to 9-Digit; U.S./U.K. Version**) Converting Between Standard, Expanded and Written Forms (

**5-Digit; SI Version**) Converting Between Standard, Expanded and Written Forms (

**3-Digit to 5-Digit; SI Version**) Converting Between Standard, Expanded and Written Forms (

**6-Digit; SI Version**) Converting Between Standard, Expanded and Written Forms (

**7-Digit; SI Version**) Converting Between Standard, Expanded and Written Forms (

**8-Digit; SI Version**) Converting Between Standard, Expanded and Written Forms (

**9-Digit; SI Version**) Converting Between Standard, Expanded and Written Forms (

**6-Digit to 9-Digit; SI Version**)

## Scientific Notation Worksheets

Scientific notation worksheets for learning how to write and interpret numbers in this format.

**Converting** Standard Numbers **to Scientific Numbers**

**Converting** Scientific Numbers **to Standard Numbers**

**Converting Between** Standard Numbers and Scientific Numbers

## Roman Numerals Worksheets

**Converting between** Roman numerals and standard numbers

This is about as "old school" as you can get. Put on your tunica and pick up your scutum to tackle the worksheets on Roman Numerals. Below, you will see options for standard and compact forms. The standard form Roman Numeral math worksheets include numerals in the commonly-taught version where 999 is CMXCIX (i.e. write the numeral one place value at a time). The compact versions are for those who want more of a challenge where the Roman numerals are written in as concise a version as possible. In the compact version, 999 is written as IM (i.e. one less than 1000).

**Compact**Roman Numerals up to C

**Compact**Roman Numerals up to M

**Compact**Roman Numerals up to MMMIM

**Operations** with Roman numerals

**Adding**Roman Numerals up to XXV

**Adding**Roman Numerals up to C

**Adding**Roman Numerals up to M

**Adding**Roman Numerals up to MMMCMXCIX

**Subtracting**Roman Numerals up to XXV

**Subtracting**Roman Numerals up to C

**Subtracting**Roman Numerals up to M

**Subtracting**Roman Numerals up to MMMCMXCIX

**Multiplying**Roman Numerals up to C

**Multiplying**Roman Numerals up to M

**Multiplying**Roman Numerals up to MMMCMXCIX

**Dividing**Roman Numerals up to C

**Dividing**Roman Numerals up to M

**Dividing**Roman Numerals up to MMMCMXCIX

**Mixed Operations**with Roman Numerals up to C

**Mixed Operations**with Roman Numerals up to M

**Mixed Operations**with Roman Numerals up to MMMCMXCIX

## Factors and Factoring Worksheets

Factors and factoring worksheets including listing factors of numbers and finding prime factors of numbers using a tree diagram.

What would factoring be without some factoring trees? They are probably the most elegant and convenient way to find the prime factors of a number, but they take a little practice, which is where we come in. The worksheets below are of two types. The first is finding all of the factors of a number. This is great for students who know their multiplication/division facts. If they don't, they might find this a little frustrating, so go back and work on that first. The second type is finding prime factors which we've chosen to do with tree diagrams. Among other things, this is a great way to find prime numbers and to practice divisibility rules.

**Factors of numbers**

**List of Factors**of Numbers

**2 to 99**(Informational)

**List of Factors**of Numbers

**100 to 999**(Informational)

**List of Factors**of Numbers

**1000 to 9999**(Informational; CAUTION 166 Pages)

**Determining Factors**of Numbers (range 4 to 50)

**Determining Factors**of Numbers (range 50 to 100)

**Determining Factors**of Numbers (range 100 to 200)

**Determining Factors**of Numbers (range 200 to 400)

**Prime factors of numbers**

**List of Prime Factors**of Numbers

**2 to 99**(Informational)

**List of Prime Factors**of Numbers

**100 to 999**(Informational)

**List of Prime Factors**of Numbers

**1000 to 9999**(Informational; CAUTION 137 Pages)

**Determining Prime Factors**Using a

**Tree Diagram**(range 4 to 48)

**Determining Prime Factors**Using a

**Tree Diagram**(range 4 to 96)

**Determining Prime Factors**Using a

**Tree Diagram**(range 4 to 144)

**Determining Prime Factors**Using a

**Tree Diagram**(range 48 to 192)

**Determining Prime Factors**Using a

**Tree Diagram**(range 48 to 240)

**Greatest Common Factors**

**Using Prime Factors**; Range

**4 to 100**(Sets of 2) Calculating Greatest Common Factors

**Using Prime Factors**; Range

**100 to 200**(Sets of 2) Calculating Greatest Common Factors

**Using Prime Factors**; Range

**200 to 400**(Sets of 2) Calculating Greatest Common Factors

**Using Prime Factors**; Range

**4 to 400**(Sets of 2) Determining Greatest Common Factors

**Using All Factors**; Range

**4 to 100**(Sets of 2) Determining Greatest Common Factors

**Using All Factors**; Range

**100 to 200**(Sets of 2) Determining Greatest Common Factors

**Using All Factors**; Range

**200 to 400**(Sets of 2) Determining Greatest Common Factors

**Using All Factors**; Range

**4 to 400**(Sets of 2)

## Multiples and Least Common Multiple Worksheets

Multiples and least common multiple (LCM) worksheets including determining the LCM using multiples and prime factors.

### Determining LCM **using multiples**

**of Numbers to 10 (LCM Not One of the Numbers or the Product)**Determine LCM From Multiples

**of Numbers to 10 (LCM Not One of the Numbers)**Determine LCM From Multiples

**of Numbers to 10**Determine LCM From Multiples

**of Numbers to 15 (LCM Not One of the Numbers or the Product)**Determine LCM From Multiples

**of Numbers to 15 (LCM Not One of the Numbers)**Determine LCM From Multiples

**of Numbers to 15**Determine LCM From Multiples

**of Numbers to 25 (LCM Not One of the Numbers or the Product)**Determine LCM From Multiples

**of Numbers to 25 (LCM Not One of the Numbers)**Determine LCM From Multiples

**of Numbers to 25**

### Determining LCM **using prime factors**

## Roots and Exponents Worksheets

Roots and exponents worksheets including squares and cubes and writing exponents in factor form.

**Squares and square roots**

**Squares**of Numbers from

**0 to 9**

**Squares**of Numbers from

**1 to 12**

**Squares**of Numbers from

**1 to 20**

**Common Squares**(Squares of 1 to 15, 20, 25, and multiples of 10 to 90)

**Squares**of Numbers from

**1 to 32**

**Squares**of Numbers from

**1 to 99**

**Square Roots 0 to 9**

**Square Roots 1 to 12**

**Square Roots 1 to 20**

**Common Square Roots**(1 to 15, 20, 25, and multiples of 10 to 90)

**Square Roots 1 to 32**

**Square Roots 1 to 99**Squares and Square Roots of Numbers 1 to 16 Squares and Square Roots of Common Numbers (1 to 15, 20, 25, and multiples of 10 to 90)

**Cubes and cube roots**

## Binary and Other Base Number Systems

Binary and other base number systems worksheets for learning about number systems with bases other than 10.

**Binary numbers** worksheets

The binary number system has broad applications, but it is most known for its predominance in computer architecture. Learning about the binary system not only encourages higher order thinking, but it also prepares students for further studies in mathematics and computer studies. The chart below may be useful for students who need some help lining things up and learning about place value as it relates to the binary system. We included a base 10 number column, so you can use the chart for converting between decimal and binary systems.

The mystery number trick below is actually based on binary numbers. As you may know, each place in the binary system is a power of 2 (1, 2, 4, 8, 16, etc.). Since every decimal (base 10) number can be expressed as a binary number, each decimal number can therefore be expressed as a sum of a unique set of powers of 2. It is this concept that makes this trick work. You might notice that the largest decimal number on the cards is 63 which is also the largest 6-digit binary number (111111). The target position on each version of the mystery number trick contains the powers of 2 associated with the first 6 place values in the binary system (1, 2, 4, 8, 16, 32). Each of the 6 cards represents a specific place value. All 32 numbers on each card contain a 1 in the associated place when written in binary. Basically, when the "friend" identifies the cards that contain the mystery number, they are giving you a binary number that simply needs converting into a decimal number. Just for fun, we mixed up the numbers on the cards and the target position on versions C to J. Version A includes numbers in ascending order and version B includes numbers in descending order. The other versions (B to J) will be available once you click on the A version below.

**Converting** between base number systems worksheets

**Decimal to Binary**Converting from

**Decimal to Octal**Converting from

**Decimal to Hexadecimal**Converting from

**Decimal to Various Other Base Sytems**Converting from

**Binary to Decimal**Converting from

**Binary to Octal**Converting from

**Binary to Hexadecimal**Converting from

**Binary to Various Other Base Sytems**Converting from

**Octal to Decimal**Converting from

**Octal to Binary**Converting from

**Octal to Hexadecimal**Converting from

**Octal to Various Other Base Sytems**Converting from

**Hexadecimal to Decimal**Converting from

**Hexadecimal to Binary**Converting from

**Hexadecimal to Octal**Converting from

**Hexadecimal to Various Other Base Sytems**Converting from

**Various Base Systems to Decimal**Converting from

**Various Base Systems to Binary**Converting from

**Various Base Systems to Octal**Converting from

**Various Base Systems to Hexadecimal**Converting Between

**Various Base Systems**

Help with Converting Between Base Number Systems:

There are shortcuts for converting between some bases. For example, converting from binary to octal takes little effort since 8 is a power of 2. Each group of 3 digits in a binary number represents a single digit in an octal number. For example, 111_{2} (the 2 stands for binary or base 2) is 7_{8} (the 8 stands for octal or base 8). The simple way to convert binary numbers to octal numbers is to group the binary number into groups of three digits. For example, 111010101000111_{2} could be written as 111 010 101 000 111. Converting each group into octal means multiplying the first digit of each group by 4, the second digit by 2 and the third digit by 1 then adding the results together. This will result in digits no larger than 7 (since 4 + 2 + 1 = 7) and the number will be converted to base 8. In octal, therefore, the number is 72507_{8}. If you can express the octal numbers from 0 to 7 in binary, you can easily convert the other way. For example 7223_{8} = 111010010011_{2} since 7 is 111, 2 is 010, and 3 is 011 in binary.

A similar shortcut for converting between binary and base 4 numbers involves looking at binary numbers in groups of 2. Similarly, converting from base 3 to base 9 and base 4 to base 16 involves groups of two. Converting from binary to hexadecimal would involve groups of 4.

For other conversions, a commonly used tactic is to convert to decimal as an intermediate step since this is the base system that is probably ingrained in your brain, so it is much more intuitive. For example, converting from a base 5 number to a base 7 number would involve first converting the base 5 number to base 10. To convert, it is only necessary to know the place values of the system that you are converting from and to. In base 5, the lowest place value (furthest to the right) of whole numbers is 1 followed by 5, 25, 125 and so on. In base 7, the place values are 1, 7, 49, 343 and so on. First multiply the digits in the base 5 number by its place values, then divide the resulting decimal number by the base 7 place values and you will have your conversion. For example 4331_{5} is expanded to (4 × 125) + (3 × 25) + (3 × 5) + (1 × 1) = 500 + 75 + 15 + 1 = 591 (in base 10). To continue into base 7, there are at least two ways, the second method is in the next paragraph. For simplicity's sake, take the largest base 7 place value that will divide into 591 at least once. In this case it is 343 which goes into 591 exactly once (1) with a remainder of 248. Divide the remainder by the next place value down, 49, to get (5) with a remainder of 3. Divide 3 by 7 which is (0) with a remainder of 3. Finally, divide by 1 which should leave no remainder, and it is (3) in this case. Put all those digits together and you should have your number in base 7: 1503_{7}.

A method to convert directly from one base system to another involves knowing how to divide in the base system you want to convert from. It is fairly easy if you are familiar with the base system. Simply divide the number by the base you want to convert to (but express it in the original base system). Repeat until the division results in 0 with or without a remainder. Convert the remainders and put them in reverse order for the number in the new base system. For example, convert 3750_{8} to hexadecimal (base 16). 16 in base 8 is 20_{8}. The first step is to divide 3750_{8} by 20_{8} = 176_{8} R 10_{8}. Next, divide 176_{8} by 20_{8} to get 7_{8} R 16_{8}. Finally, 7_{8} divided by 20_{8} is 0_{8} R 7_{8}. Convert the remainders to base 16 (which you may have to think of in terms of decimal numbers, or you can use your fingers and some toes) and write the digits in reverse order. 7_{8} is 7_{16}, 16_{8} is (14 in decimal) E_{16}, and 10_{8} is 8_{16}. So, the number 3750_{8} is 7A8_{16}.