Converting Base-ten Numbers to Binary Numbers for JSS 1

Conversion of Base-ten numerals to Binary numbers

Performance Objectives

By the end of the lesson, students should be able to:

1. convert base ten numerals to binary numbers.
2. Identify patterns in binary representation.
3. Identify computing binary digits (bits).

Introduction to Number Bases

Numbers can be written using different systems called number bases. The most common system used in daily life is the base-ten system, while computers use the base-two system to process information.

Meaning of Base-Ten and Binary Systems

A base-ten number system, also called the decimal system, is a system of writing numbers using ten digits from 0 to 9.

The ten digits used in this system are 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9. When counting in base-ten, the value of each position increases by powers of ten. For example, in the number 245, the digit 5 represents units, 4 represents tens, and 2 represents hundreds.

A binary number system, also called base-two, is a system of writing numbers using only two digits, 0 and 1.

The binary system is the fundamental language of modern computers. Since computer hardware runs on electricity, it uses these two digits to represent the presence or absence of an electrical signal.

Place Value in Binary Numbers

In the base-ten system, place values are based on powers of ten, such as units ($10^0 = 1$), tens ($10^1 = 10$), hundreds ($10^2 = 100$), and thousands ($10^3 = 1000$). In the binary system, place values are based on powers of two.

The place values in binary, starting from the right (the units column) and moving to the left, are:

1. Units column: $2^0 = 1$
2. Twos column: $2^1 = 2$
3. Fours column: $2^2 = 4$
4. Eights column: $2^3 = 8$
5. Sixteens column: $2^4 = 16$
6. Thirty-twos column: $2^5 = 32$

To understand how a binary number is structured, observe the place value table below for the binary number $1101_2$:

The value of $1101_2$ in base-ten is calculated by multiplying each binary digit by its corresponding place value and adding the results together:

$$1101_2 = (1 \times 8) + (1 \times 4) + (0 \times 2) + (1 \times 1)$$
$$1101_2 = 8 + 4 + 0 + 1 = 13_{10}$$

Converting Base-ten Numbers to Binary

To convert a number from base-ten to binary, the continuous division method is used. This method involves repeatedly dividing the base-ten number by 2 and recording the remainders.

The steps for the continuous division method are:

1. Divide the base-ten number by 2.
2. Write down the quotient under the number and write the remainder (which will always be either 0 or 1) to the side.
3. Divide the new quotient by 2 and record the new remainder.
4. Repeat this division process until the quotient becomes 0.
5. Write down the remainders starting from the very last remainder (at the bottom) up to the first remainder (at the top). This sequence of 0s and 1s forms the binary equivalent.

Worked Examples

The worked examples of converting base-ten numbers to binary are:

Example 1

Convert $8_{10}$ to base 2.

The division steps are shown in the table below:

Reading the remainders from the bottom to the top gives:

$$8_{10} = 1000_2$$

Example 2

Convert $13_{10}$ to base 2.

The division steps are shown in the table below:

Reading the remainders from the bottom to the top gives:

$$13_{10} = 1101_2$$

Example 3

Convert $25_{10}$ to base 2.

The division steps are shown in the table below:

Reading the remainders from the bottom to the top gives:

$$25_{10} = 11001_2$$

Patterns in Binary Representation

By studying binary numbers, several mathematical patterns can be identified. Recognizing these patterns helps in understanding binary representation quickly.

The main patterns in binary numbers include:

1. Even and Odd Numbers: Every even base-ten number ends in 0 when converted to binary (for example, $2_{10} = 10_2$, $4_{10} = 100_2$, $8_{10} = 1000_2$). Every odd base-ten number ends in 1 when converted to binary (for example, $1_{10} = 1_2$, $3_{10} = 11_2$, $13_{10} = 1101_2$).
2. Powers of Two: Numbers that are exact powers of two always consist of a single digit 1 followed only by zeros. For example, $2_{10} = 10_2$, $4_{10} = 100_2$, $8_{10} = 1000_2$, and $16_{10} = 10000_2$.
3. Number of Digits (Bits): The number of digits in a binary number increases by one every time the base-ten value reaches a new power of two. For example, numbers from 1 to 3 require up to 2 bits, numbers from 4 to 7 require up to 3 bits, and numbers from 8 to 15 require up to 4 bits.

Binary Digits (Bits) and Computer Logic Gates

A binary digit, commonly called a bit, is the smallest unit of data in a computer, representing either a 0 or a 1.

Computers are electronic machines made of millions of tiny switches called transistors. These switches can only be in one of two states: ON (electricity is flowing) or OFF (no electricity is flowing). The digit 1 represents the ON state, and the digit 0 represents the OFF state.

A logic gate is an electronic circuit that controls the flow of electricity based on simple rules, using binary inputs to produce a binary output.

Logic gates act like smart switches. They receive binary signals (0s and 1s) from other parts of the computer, process them, and send out a new binary signal. By combining millions of these logic gates, a computer can perform complex calculations, run programs, and store information.

To make computers useful for humans, everything typed, seen, or heard must first be converted into binary. For example, when a key is pressed on a keyboard, the computer does not see the letter “A”. Instead, it converts that letter into a specific pattern of eight bits, such as $01000001_2$. This process of converting all data into binary is how basic programming and computing work.

Evaluation Exercises

Solve the following questions to test understanding of the topic:

1. Convert $8_{10}$ to base 2. Show all the division steps clearly.
2. Convert the following base-ten numbers to binary:
   1. $15_{10}$
   2. $18_{10}$
   3. $30_{10}$
3. Explain the difference between an even binary number and an odd binary number based on their last digits.
4. What is a binary digit (bit), and how do computers use bits to represent ON and OFF states?
5. Describe how logic gates use the binary digits 0 and 1 to control electrical signals in a computer.